138191351-t3-y11-Maths-Ext-1-Theorybook-2013
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Y11 MATHEMATICS EXT 1
Copyright © MATRIX EDUCATION 2012
TERM 3 THEORY BOOKLETS
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Y11 MATHEMATICS EXT 1
Copyright © MATRIX EDUCATION 2012
TERM 3 THEORY BOOKLETS
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Y11 MATHEMATICS EXT 1
TERM 3 THEORY BOOKLETS
CONTENTS LESSON 1:
LOCUS & PARABOLA 2.......................................................... 7
1. OVERVIEW OF LOCUS & PARABOLA 2 .................................................................. 8 2. THE PARABOLA AS A LOCUS .................................................................................. 9 3. FEATURES OF THE PARABOLA ............................................................................ 17 4. PARABOLAS WITH VERTEX NOT AT THE ORIGIN ............................................... 23
LESSON 2:
INTRODUCTORY CALCULUS 1............................................ 35
1. OVERVIEW OF INTRODUCTORY CALCULUS 1.................................................... 36 2. LIMITS OF A FUNCTION ......................................................................................... 37 3. CONTINUITY ............................................................................................................ 46 4. GRADIENTS OF SECANT AND TANGENTS .......................................................... 51
LESSON 3:
INTRODUCTORY CALCULUS 2............................................ 63
1. OVERVIEW OF INTRODUCTORY CALCULUS 2.................................................... 64 2. DIFFERENTIATION OF
...................................................................................... 65
3. THEOREMS ON DERIVATIVES .............................................................................. 69 4. COMPOSITE FUNCTION (CHAIN) RULE ................................................................ 74 5. PRODUCT RULE ..................................................................................................... 77 6. QUOTIENT RULE..................................................................................................... 81 7. TANGENTS & NORMALS TO CURVES .................................................................. 85
LESSON 4:
PARAMETRIC REPRESENTATION 1 ................................... 95
1. OVERVIEW OF PARAMETRIC REPRESENTATION .............................................. 96 2. CARTESIAN AND PARAMETRIC EQUATIONS ...................................................... 97 3. PARAMETRIC EQUATIONS OF A PARABOLA..................................................... 100 4. THE EQUATION OF A CHORD OF
....................................................... 102
5. EQUATIONS OF TANGENTS TO THE PARABOLA .............................................. 106 6. EQUATION OF NORMAL TO THE PARABOLA .................................................... 110
LESSON 5:
PARAMETRIC REPRESENTATION 2 ................................. 119
1. OVERVIEW OF PARAMETRIC REPRESENTATION 2 ......................................... 120 2. DERIVATIVES OF PARAMETRIC EQUATIONS .................................................... 121 3. POINT OF INTERSECTION OF TANGENTS & NORMALS ................................... 124 4. FOCAL CHORD PROPERTIES.............................................................................. 130 5. REFLECTION PROPERTIES ................................................................................. 132 6. EQUATION OF THE CHORD OF CONTACT......................................................... 134
LESSON 6:
PARAMETRIC REPRESENTATION 3 ................................. 143
1. OVERVIEW OF PARAMETRIC REPRESENTATION 3 ......................................... 144 2. LOCUS OF A POINT .............................................................................................. 145 3. LOCUS PROBLEMS INVOLVING ONE VARIABLE ............................................... 149 Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
TERM 3 THEORY BOOKLETS
4. LOCUS PROBLEMS INVOLVING TWO VARIABLES ............................................ 155
LESSON 7:
POLYNOMIALS 1................................................................. 165
1. OVERVIEW OF POLYNOMIALS 1 ......................................................................... 166 2. DEFINITION AND NOTATION ............................................................................... 167 3. OPERATIONS WITH POLYNOMIALS ................................................................... 169 4. THE DIVISION TRANSFORMATION ..................................................................... 173 5. THE REMAINDER THEOREM ............................................................................... 174 6. FACTOR THEOREM .............................................................................................. 176 7. DEDUCTIONS FROM THE FACTOR THEOREM .................................................. 179
LESSON 8:
POLYNOMIALS 2................................................................. 189
1. OVERVIEW OF POLYNOMIALS 2 ......................................................................... 190 2. POLYNOMIAL EQUATIONS .................................................................................. 191 3. GRAPHS OF POLYNOMIAL FUNCTIONS............................................................. 194 4. RELATIONSHIPS BETWEEN THE ROOTS & COEFFICIENTS ............................ 201
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Y11 MATHEMATICS EXT 1
Copyright © MATRIX EDUCATION 2012
TERM 3 THEORY BOOKLETS
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Y11 MATHEMATICS EXT 1
1.
LESSON 1: LOCUS & PARABOLA 2
OVERVIEW OF LOCUS & PARABOLA 2
Locus
Quadratic Functions
Locus and the Parabola
Conics
Completing the Square
We begin the term by redefining the standard parabola as a locus. This topic is often referred to as “
Locus in the Complex Plane
” and is an extremely common question in the 2 unit papers.
You will need to be able to generate the equation of a parabola given various pieces of locus information. You will also be asked to identify a parabola’s geometrical features such as vertex, focus and directrix.
This topic also extends naturally into the parametric (
) topic in the Extension 1
course and even to the conics section of Extension 2 mathematics.
A typical example is the following question from the 2011 H.S.C. paper. 2011 H.S.C. Mathematics Q3b A parabola has focus (
(2 marks)
) and directrix
. Find the coordinates of the vertex.
Be alert to the possibility that the parabola may sit upside down or left to right rather than in its standard position.
We will also make extensive use of the technique of completing the square to deal with situations where the vertex of the parabola is not at the origin. If you are a little rusty on completing the square, you should revise the topic before attempting the harder problems.
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Y11 MATHEMATICS EXT 1
2.
LESSON 1: LOCUS & PARABOLA 2
THE PARABOLA AS A LOCUS
A parabola is a locus of points equidistant from a fixed point called the focus and a fixed line called the directrix.
The vertex or turning point of the parabola lies midway between the focus and the directrix.
Equations of Parabolas with Vertex the Origin and Focus (
In the diagram, the focus, , has coordinates ( The axis of symmetry is the
)
) and the directrix has equation
.
axis.
From the definition of a parabola as a locus, all points lying on the parabola must be equidistant from the focus and the directrix. Hence
( are (
) is midway between the focus and the directrix. Hence the coordinates of the vertex ).
Using the graph shown above and the definition of a parabola as a locus, it can be shown that the equation of the parabola is
In the diagram, the point ( (
.
) lies on the parabola, (
) is the focus and the point
) lies on the directrix.
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Y11 MATHEMATICS EXT 1
The locus definition of a parabola is distance of
LESSON 1: LOCUS & PARABOLA 2
the distance of
from the focus must equal the
from the directrix. i.e.
Using the distance formula: )
√(
(
)
Also:
From the definition of a parabola:
Hence (
)
(
)
Hence, the equation of the parabola with the focus at (
), vertex (
) and directrix
is
Note to Students:
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
(
Equations of Parabolas with Vertex the Origin and Focus at
The parabola shown in the diagram has focus (
) and directrix
The vertex is at (
symmetry the
)
.
) and axis of axis.
Using the definition of a parabola as a locus, it can be shown that the equation of the parabola is:
Using the distance formula: √(
)
(
)
√(
)
(
)
From the definition of a parabola:
Hence (
)
(
)
Hence, the equation of the parabola with the focus at (
), vertex (
) and directrix
is
Note to Students:
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
Equations of Parabolas with Vertex the Origin and Focus (
The parabola shown in the diagram has focus (
) and directrix
The vertex is at (
the
)
.
) and axis of symmetry
axis.
Using the definition of a parabola as a locus, it can be shown that the equation of the parabola is
Using the distance formula: √(
)
(
)
√(
)
(
)
From the definition of a parabola:
Hence (
)
(
)
Hence, the equation of the parabola with the focus at (
), vertex (
) and directrix
is
Note to Students:
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
Equations of Parabolas with Vertex the Origin and Focus (
)
The parabola shown in the diagram has focus (
) and directrix
.
The vertex is at ( symmetry the
) and axis of axis.
Using the definition of a parabola as a locus, it can be shown that the equation of the parabola is
Using the distance formula: √(
)
(
)
√(
)
(
)
From the definition of a parabola:
Hence (
)
(
)
Hence, the equation of the parabola with the focus at (
), vertex (
) and directrix
is
Note to Students:
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Y11 MATHEMATICS EXT 1
Concept Check 2.1 (i)
LESSON 1: LOCUS & PARABOLA 2
i
Find the equation of the locus of the point ( point (
) that moves so that its distance from the
) is equal to its distance from the line
Note to Students: ( ) is the focus
is the directrix
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(ii)
Hence show that (4, 1) lies on the parabola …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(iii)
Sketch the parabola.
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
ii
Concept Check 2.2
In the following questions, the coordinates of the focus and the equation of the directrix are given. Find the equation of the parabola: (i)
Focus (
);
Directrix
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………………….……………………………………………………………
(ii)
Focus (
);
Directrix
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(iii)
Focus (
);
Directrix
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………………….……………………………………………………………
(iv)
Focus (
);
Directrix
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………………….…………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 2.3
LESSON 1: LOCUS & PARABOLA 2
iii
Write down the coordinates of the focus and the equation of the directrix for the following parabolas: Note to Students: Always start with a sketch! (i) …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(ii)
…………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(iii) …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(iv) …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
(v) …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………….
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
3.
FEATURES OF THE PARABOLA
Focal Length
The focal length of a parabola is the distance between the focus and the vertex. It is equivalent to the distance between the vertex and the directrix.
Axis of Symmetry
The axis of symmetry of a parabola is called the axis of the parabola. It passes through the vertex and the focus of the parabola and is perpendicular to the directrix.
Chord
An interval joining any two points on a parabola is called a chord.
Focal Chord
A chord that passes through the focus is called a focal chord.
Latus Rectum
A focal chord that is perpendicular to the axis of the parabola is called the latus rectum.
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Y11 MATHEMATICS EXT 1
Concept Check 3.1 (a)
LESSON 1: LOCUS & PARABOLA 2
iv
A parabola has equation
. Find the
Note to Students: Your first job is to find the focal length “ ” You should also pick between , (i)
,
,
coordinates of the vertex.
…………………………………….……………………………………………………........................
(ii)
equation of the axis of symmetry.
…………………………………….……………………………………………………........................
(iii)
coordinates of the focus.
…………………………………….……………………………………………………........................
(iv)
equation of the directrix.
…………………………………….……………………………………………………........................
(v)
focal length.
…………………………………….……………………………………………………........................
Note to Students: The focal length is always positive. (vi)
equation of the latus rectum.
…………………………………….……………………………………………………........................
Note to Students: The latus rectum is a chord passing through the focus, parallel to the directrix Its length is
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Y11 MATHEMATICS EXT 1
(vii)
LESSON 1: LOCUS & PARABOLA 2
length of the latus rectum. …………………………………….…………………………………………………………………… …………………….…………………………………………………………………………………… …….………………………………………………………………………………………….………… ………………………………………………………………….………………………………………
(b)
A parabolic arch has equation (i)
. Find the:
equation of its latus rectum.
…………………………………….…………………………………………………………………… ……………………………………………….…………………………………………………….........
(ii)
focal length of the parabola.
…………………………………….……………………………………………………........................
(iii)
span of the arch when its height is 5 metres.
…………………………………….…………………………………………………………………… …………………………….…………………………………………………………………………… ………………………………………………………….………………………………………………
(c)
For the parabola
, show that the length of the latus rectum is equal to
…………………………………….…………………………………………………………………… …………………….…………………………………………………………………………………… …………………………….…………………………………………………………………………… …………………………….…………………………………………………………………………… …………….……………………………….…………………………………………………………… …………………………………………….……………………………………….…………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 3.2 (a)
LESSON 1: LOCUS & PARABOLA 2
v
A parabola has its vertex at the origin, its axis along the point ( ). (i)
axis and it passes through the
Find the equation of the parabola.
…………………………………….…………………………………………………………………… ………………….………………………………………………………………………………………. ………………….………………………………………………………………………………………. ………………….……………………………………………………………………………………….
(ii)
Write down the focal length of the parabola.
…………………………………….……………………………………………………........................
(iii)
Find the coordinates of the focus and the equation of the directrix.
………………….……………………………………………………………………………………… ……………….……………………………………………………………………………………….… ……………….……………………………………….…………………………………………………
Note to Students: The focus must be presented as a point – For example, say “the focus is ( )” The directrix must be presented as a line – For example, say “the directrix is (b)
”, not “the directrix is
”
A parabola has its vertex at (0, 0), the equation of its axis is and its latus rectum has length 4 units. Find the possible equations of the parabola with these features. …………………………………….…………………………………………………………………… …………………………….…………………………………………………………………………… …………………….…………………………………………………………………………………… ………………….……………………………………………….……………………………………… ……………....................................…………………………………….…………………………… ……………....................................…………………………………….……………………………
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Y11 MATHEMATICS EXT 1
Concept Check 3.3
LESSON 1: LOCUS & PARABOLA 2
vi
Find the equation of the following parabolas.
(i)
Vertex (
) focal length 2 units, axis of symmetry
.
…………………………………….……………………………………………………………………. …………….…………………………………………………………………………………………… …………….………………………………………………………………………………………….…
(ii)
Vertex (
), focus (
).
…………………………………….……………………………………………………………………. …………….…………………………………………………………………………………………… …………….……………………………………….…………………………………………………….
(iii)
Focus (
) directrix
.
…………………………………….……………………………………………………………………. …………….…………………………………………………………………………………………… …………….……………………………………….…………………………………………………….
(iv)
Focus (
), directrix
.
…………………………………….……………………………………………………………………. …………….…………………………………………………………………………………………… …………….……………………………………….…………………………………………………….
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Y11 MATHEMATICS EXT 1
Concept Check 3.4 (a)
LESSON 1: LOCUS & PARABOLA 2
vii
A chord of the parabola chord.
has equation
. Find the length of the
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………… …………………………………………..…………………………………….………………………… ………………………………………….…………………………………….…………………………
(b)
A chord of the parabola the midpoint of the chord.
lies along the line
. Find the coordinates of
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………………………………….…………………………………………………………………… …………………………………………..…………………………………….…………………………
Discussion Question 1: Can you do this question without actually solving the quadratic equation?viii (c)
A focal chord of the parabola
has slope
. Find the equation of the focal chord.
…………………………………….……………………………………………………………………. …………………………………….……………………………………………………………………. …………….…………………………………………………………………………………………… …………….………………………………………………………………………………………….… …………….……………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
4.
PARABOLAS WITH VERTEX NOT AT THE ORIGIN
Parabolas with Vertex (
) and Axis Parallel to the
In the diagram, the parabola has vertex ( ) and its directrix has equation
With reference to the
From the diagram,
Hence with reference to the
) and focal length . Its focus is at (
.
axes, the equation of the parabola is of the form and
. axes, the equation of the parabola becomes (
axis
)
(
)
Hence we can conclude that the equation of a parabola with focus ( directrix
) and
is (
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)
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(
)
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Y11 MATHEMATICS EXT 1
Concept Check 4.1
(i)
LESSON 1: LOCUS & PARABOLA 2
ix
On a number plane, plot the point (
) and draw the line
𝑦
O
(ii)
𝑥
Find the locus of a point which moves in a plane so that its distance from the point ( equal to its distance from the line .
) is
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………………………………….…………………………………………… ……………………….…………………………………….…………………………………… ……………………………….…………………………………….…………………………… ……………………………………….…………….…………………………………………… …………………………………………………………….…………………………………… …………………………………………………….……………………………………….…… ……………………………………………………………….………………………………… ….…………………………………………………………………….………………………… ………….…………………………………………………………………….…………….… ………………………………………………………………………………………………… …….………………………………………………………………………………………….…
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Y11 MATHEMATICS EXT 1
Concept Check 4.2
LESSON 1: LOCUS & PARABOLA 2
x
Find the equation of the following parabolas. Note to Students: Always start with a sketch! (i)
Focus (
) and directrix
.
…………….…………………………………………………………………………………… …………………….……………………………………………………………………………
(ii)
Vertex (
), axis parallel to the
– axis and passing through the point (
).
…………….…………………………………………………………………………………… …………………….……………………………………………………………………………
(iii)
Vertex (
) focal length 2 units, axis of symmetry
…………….…………………………………………………………………………………… …………………….…………………………………………………………………………… …………….……………………………………….……………………………………………
(iv)
Vertex (
) focus (
).
…………….…………………………………………………………………………………… …………………….…………………………………………………………………………… …………….……………………………………….……………………………………………
(v)
Axis of symmetry
, Vertex (
) and
intercept .
…………….…………………………………………………………………………………… …………………….…………………………………………………………………………… …………….……………………………………….…………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 4.3
(a)
xi
A parabola has equation
(i)
LESSON 1: LOCUS & PARABOLA 2
.
Find the coordinates of its vertex and focus.
Note to Students: Complete the square.
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………………………………….…………………………………………… ……………………….…………….…………………………………………………………… …………………………………………….……………………………………………………
(ii)
Find the equations of its directrix and latus rectum.
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(b)
Find the coordinates of the focus and the equation of the directrix of the parabola
Note to Students: Before completing the square, make the coefficient of
equal to .
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………………………………….…………………………………………… ……………………….…………………………………….…………………………………… ……………………………….…………….…………………………………………………… …………………………………………………….…………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 4.4
xii
A parabola has equation (i)
LESSON 1: LOCUS & PARABOLA 2
(
)
Draw a neat sketch of the parabola and clearly indicate on it the equation of its directrix, the coordinates of its focus and the coordinates of all points of intersection of the parabola with the coordinate axes. 𝑦
𝑥
0
(b)
Another parabola with equation the coordinates of
cuts the parabola
(
) at
and
Find
and
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………………………………….…………………………………………… ……………………….…………………………………….…………………………………… ……………………………….………………………………….……………………………… …………………………………….……………….…………………………………………… …………………………………………………………….…………………………………… …………………………………………………….……………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 4.5
(i)
LESSON 1: LOCUS & PARABOLA 2
xiii
Sketch the parabola whose focus is the point ( ) and whose directrix is Indicate on your diagram the coordinates of its vertex.
.
𝑦
𝑥
(ii)
Find the equation of the parabola
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(iii)
The parabola cuts the axis at the point . Find the coordinates of the point . Hence find the equation of the focal chord passing through the point .
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….…………………………………………………………………… …………………….……………………………………….…………………………………… ……………………………….…………….…………………………………………………… …………………………………………………….…………………………………………… …………………………………………………….……………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
Parabolas with Vertex (
A parabola with vertex (
) and Axis Parallel to the
), focal length , focus at (
and axis parallel to the (
A parabola with vertex (
(
Concept Check 4.6
), directrix with equation
axis has equation )
(
)
), focal length , focus at (
and axis parallel to the
–axis
), directrix with equation
axis has equation )
(
)
xiv
) that moves in a plane so that its distance from the Find the equation of the locus of a point ( )is equal to its distance from the line point ( .
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………………………………….…………………………………………… ……………………….…………………………………….…………………………………… ……………………………….…………….…………………………………………………… …………………………………………………….…………………………………………… …………………………………………………….…………………………………………… …………………………………………………….…………………………………………… …………………………………………………….…………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 4.7
LESSON 1: LOCUS & PARABOLA 2
xv
Find the equation of the following parabolas
(i)
Focus (
) and Directrix
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(ii)
vertex (
) axis parallel to the
– axis and passing through the point (
)
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(iii)
Vertex (
), focal length
units, axis of symmetry
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(iv)
Vertex (
) focus (
)
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….……………………………………………………………………
(v)
Axis of symmetry
, Vertex (
) and focal length
…………………………………….…………………………………………………………… ……….…………….…………………………………………………………………………… …………………………….…………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 4.8
xvi
A parabola has equation
(i)
LESSON 1: LOCUS & PARABOLA 2
. Find
the coordinates of its vertex Note to Students: Complete the square.
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………….…………………………………………………………………… …………………………………….……………………………………………………………
(ii)
its focal length
…………….…………………………………………………………………………………… …………………….……………………………………………………………………………
(iii)
the equation of its directrix
…………….…………………………………………………………………………………… …………………….……………………………………………………………………………
(iv)
Hence sketch the parabola
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
Past H.S.C. Questionsxvii
Question 1
(2011 H.S.C. Mathematics Q3b)
A parabola has focus (
) and directrix
. Find the coordinates of the vertex.
2
…………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………….…………………………………………………………………… …………………………………….…………………………………………………………… …………………………………….…………………………………………………………… ……….…………………………………….…………………………………………………… ……………….…………….…………………………………………………………………… …………………………………….……………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 1: LOCUS & PARABOLA 2
SOLUTIONS i
Concept Check 2.1
(i) ii
Concept Check 2.2
(i)
(ii)
(iii)
(iv) iii
Concept Check 2.3
(i)
(
)
(ii)
(
)
(iv)
(
)
(v)
(
)
iv
(iii)
(
(ii)
(iii)
(
(v)
(vi)
(ii)
(iii)
(ii)
(iii)
(ii)
(iii)
)
Concept Check 3.1
(a)
(i)
(
)
(iv)
)
(vii) (b) v
(i)
√
Concept Check 3.2
(a)
(i)
(
)
(b) vi
Concept Check 3.3
(i) (iv) vii
Concept Check 3.4
(a) viii
√
(b)
(
)
(c)
Discussion Question 1
Yes ix
Concept Check 4.1
(ii)
x
22 12 y 1
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Y11 MATHEMATICS EXT 1
x
LESSON 1: LOCUS & PARABOLA 2
Concept Check 4.2 (
(i)
(
(iv)
xi
)
(
)
)
(
(
(ii)
)
(v)
)
(
)
(
)
(
)
(
)
(
)
(iii)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Concept Check 4.3
(a)
(
(b)
xii
)
(
)
(ii)
)
Concept Check 4.4 (
(b) xiii
)(
)
Concept Check 4.5 (
(ii) xiv
)
(
)
(iii)
)
(ii)
Concept Check 4.6
( xv
(
(i)
)
(
)
Concept Check 4.7 (
(i)
(
(iv) (i)
) )
( (
(iii)
(
)
y 12 16x 2
(ii)
y
(iii) S 2, 2 ;
32 4x 2
y 22 S 6, 2 ; (iv) S 0,1 ;
y 12
(v)
16x 2
y 22 16x
2
y 12 4x 2
16x 4
S 8,1 ;
y 12 16x xvi
(i)
xvii
1.
4
Concept Check 4.8
V 16,2
(ii)
a
1 4
(iii)
x 16
1 4
Past H.S.C. Questions (
)
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)
Y11 MATHEMATICS EXT 1
1.
LESSON 2: INTRODUCTORY CALCULUS 1
OVERVIEW OF INTRODUCTORY CALCULUS 1
Area and Volume
Functions
Limits
Motion
Calculus
Algebra
Tangents and Normals
Rates of Change
Maxima and Minima
This lesson introduces the central topic in the 2 unit, 3 unit and 4 unit mathematics courses, the calculus. Indeed these three subjects are often referred to as the calculus based H.S.C. courses and a quick survey of any of their major papers will reveal that at least half of the exam questions will deal with calculus in one way or another. Calculus is the backbone of your future mathematical study.
The theory of calculus falls into two neat halves, differentiation and integration. The development of the theory of rates of change via differentiation will take many lessons to unfold. Once these have been mastered we will move on to integration theory and the calculation of areas.
The process of differentiation involves dozens of formulae and algorithms. These will be carefully and methodically presented to you over the next handful of lessons. It is crucial however that you do not reduce the theory to a mere collection of facts. You must also have a strong understanding of the underlying concepts which drive calculus and the manner in which they used, not only in mathematics but also in engineering and the sciences.
The theory of calculus is one of the most astounding achievements of the human mind. Once uncovered in the late 1600’s by Sir Isaac Newton and Gottfried Leibnitz, it changed forever the face of both mathematics and science.
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Y11 MATHEMATICS EXT 1
2.
LESSON 2: INTRODUCTORY CALCULUS 1
LIMITS OF A FUNCTION
Note to Students: You will need to evaluate two types of limits: ( ) and ( )
They are done completely differently.
of
[
To determine
] we need to see what happens to the value of [
close to but not equal to 2.
By choosing values of
that are slightly less than or slightly greater than 2, we can see what
happens to the value of [
] for values
].
Complete the following table:
1.9
1.99
1.999
2.1
2.01
2.001
( )
From the table, we can say that as
gets closer and closer to 2, then [
] gets closer
and closer to …………………
This is written ( )
................... as
We say the limiting value of [
Hence we write
[
]
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. ] is …………………….
........................................
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
Properties Of Limits
1.
[ ]
2.
[ ]
3.
[
4.
[ ( )
( )]
[ ( )]
[ ( )]
5.
[ ( )
( )]
[ ( )]
[ ( )]
6.
[
7.
[ ( )]
{
8.
[ ( )]
( ) if ( ) is a polynomial function
9.
[
where
is a constant.
[ ( )]
( )]
( ) where
[ ( )]
( ) ] ( )
[ ( )]
( ) ] ( )
( ) ( )
provided
is a constant.
[ ( )]
[ ( )]}
if ( ) and ( ) are both polynomial functions and ( )
Limits of Quotients
Type 1:
[
( ) ] ( )
and ( )
[
Example:
Evaluate
Solution:
In this case, ( ) Hence
using property 9 of limits.
[
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]
and ( ) ]
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Y11 MATHEMATICS EXT 1
Note:
LESSON 2: INTRODUCTORY CALCULUS 1
Both the numerator and denominator can be factorised and common factors cancelled out to simplify the function
[
(
]
(
[we can cancel (
)(
)
)(
)
) if
, we are investigating the behaviour of around
]
as expected.
So it is not necessary to factorise first in this case. [
Type 2:
( ) ] ( )
and ( )
( )
Example:
Consider the limit of the function
as
Solution:
We are interested in the behaviour of ( ) as We are not determining the value of ( ) when
approaches 3. is exactly equal to 3.
Fill in the table shown below to determine the value of ( ) approaches as
2 .9
.99
2 .999
2 .9999
2 .99999
2
2
3
3
.999999
( )
3 .1
.01
3 .001
3 .0001
3 .00001
.000001
( ) approaches 3 from either side, the value of ( )approaches
Therefore, as ……………..
Mathematically this is written: [
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]
[
]
Page 39 of 213
.................
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
In this case, both the numerator and denominator are polynomial functions and ( )
so this limit cannot be evaluated using property 9 of limits as this
will give which is undefined.
[
To evaluate limits of quotients,
( ) ], ( )
when ( )
, try factorising first
then evaluate the limit
[
]
[
Note that (
(
)(
)
]
) in the numerator will cancel out with the (
) in the
denominator if Since we are concerned about the behaviour of ( ) the actual value of ( ) when
[
] [
[
(
)(
as
and not
then we can cancel them out. )
]
] since
as shown in the table of values.
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Y11 MATHEMATICS EXT 1
Concept Check 2.1
LESSON 2: INTRODUCTORY CALCULUS 1
i
Did you know: ( ) is “well-behaved”, If
Problems arise when
( )
( )
( )
Determine the limits of the following: [
(a)
]
…………………………………………………………………………………………………………
[
(b)
]
………………………………………………………………………………………………………… …………………………………………………………………………………………………………
[
(c)
]
………………………………………………………………………………………………………… …………………………………………………………………………………………………………
[
(d)
]
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Discussion Question 1:ii You have shown:
Do we say:
equals , or
is approximately 6, or
approaches 6
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Y11 MATHEMATICS EXT 1
(e)
[
LESSON 2: INTRODUCTORY CALCULUS 1
]
………………………………………………………………………………………………………… …………………………………………………………………………………………………………
(f)
[
]
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Discussion Question 2:iii What is the value of: a) b) c) (g)
[
]
Did you know: When faced with
( ) , ( )
you need factors of (
) in the numerator and denominator.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(h)
[
]
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
The Concept of Infinity
Consider the function ( )
As
where
.
approaches zero from the positive side, the values of ( ) become larger and larger. We say ( ) increases without bounds or become positively infinite. [ ]
We write
As
. This tells us that there is no limit.
approaches zero from the negative side, the values of ( ) are negative and become
negatively larger as we move closer to zero. We say ( ) decreases without bounds or become negatively infinite. [ ]
We write
The Limit
As
.
Again there is no limit.
[ ]
becomes larger, the value of [ ] becomes smaller and closer to zero. We say the limit
is zero. [ ]
Hence
To evaluate
[
( ) ], ( )
where ( ) and ( ) are polynomial functions of
Step 1:
Divide each term in the numerator and the denominator by the highest power of .
Step 2:
Use the
Example:
[ ]
to evaluate the limit.
The highest power of
(
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)
(
Page 43 of 213
is 5 so divide each term by
)
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Y11 MATHEMATICS EXT 1
Concept Check 2.2
LESSON 2: INTRODUCTORY CALCULUS 1
iv
Determine the limits of the following
Note to Students: ( ) ( )
is all about “power”. ( ( ( (
) ) ) )
If ( ) is strong
If ( ) is strong
If ( ) and ( ) have equal strength, we get a finite non-zero answer
(or unbounded)
(a)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………..
(b) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 2.3
LESSON 2: INTRODUCTORY CALCULUS 1
v
Determine the following limits:
(i)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Note to Students: ( ) and ( ) are done completely differently.
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
3.
CONTINUITY
Continuous or Discontinuous Functions
Functions that have smooth unbroken curves or lines are called continuous functions.
Discontinuous functions have gaps in their graphs.
Graphs of continuous and discontinuous functions are shown below.
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
The diagrams below show the different types of discontinuity. Type 1:
–
A function that is continuous everywhere except at a point ( )
as
where
is said to have an infinite discontinuity at
Type 2:
–
A function that is not defined at a point
but the
[ ( )] exists and
its value is not equal to ( ). This type of discontinuity is called point discontinuity at
The discontinuity at the point may be removed by
redefining the function to include ( )
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[ ( )] .
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
Type 3:
–
A function that is not defined at
and
[ ( )]
[ ( )] is
said to display a jump discontinuity at
Test for Continuity at a Point
A function ( ) is continuous at a point
– – –
if:
( ) exists ( ) exists ( )
Copyright © MATRIX EDUCATION 2012
( )
( )
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Y11 MATHEMATICS EXT 1
Concept Check 3.1
LESSON 2: INTRODUCTORY CALCULUS 1
vi
Determine which of the following functions are discontinuous and state the type and point of discontinuity. Note to Students: Roughly speaking, pen off the paper. (i)
( ) is continuous if you can draw its graph without taking your
( ) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( ) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
( ) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iv)
( )
√
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(v)
( ) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(vi)
( )
LESSON 2: INTRODUCTORY CALCULUS 1
{
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Concept Check 3.2
(a)
vii
A function is defined as ( )
for
. What is the value of ( ) if ( ) is a
continuous function? …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(b)
The function ( )
{
Find the values of
and .
is a continuous function.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Discussion Question 3:viii (i) Is the sum of two continuous functions always continuous? (ii) Is the sum of two continuous functions always discontinuous? Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
4.
GRADIENTS OF SECANT AND TANGENTS
Definition of a Secant
A secant is a line that intersects a curve at two points.
The Gradient of the Secant
( )
but
( )
and (
)
( )
(
)
( )
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
Gradient of a Tangent at a Point (
) on the Curve
A tangent is a line that touches a curve only at one point.
The gradient of the tangent at a point on the curve is the gradient of the curve at that point.
To find the gradient of a line, we need two points. The tangent to a curve is a line and the only known point on the tangent is its point of contact with the curve. Hence the gradient of the tangent can only be found by using the gradient of the secant.
The diagram shows secants and the tangent at . The value of
decreases as
moves closer to .
The tangent is the limiting position of the secant
.
Hence the gradient of the tangent is the limiting value of
(
)
Page 52 of 213
)
( )
as
( )
The gradient of tangent is represented by the notation
Copyright © MATRIX EDUCATION 2012
(
or
( )
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
( ), of the curve
( ) is given by
The gradient function,
Therefore, the gradient of the tangent to the curve
( )
(
)
( )
( ) at a point on the curve where
is denoted by:
(
( )
)
( )
A normal to the curve is a line drawn perpendicular to the tangent at the point of contact of the tangent with the curve.
If the gradient of the tangent at the point where normal at the point where
Concept Check 4.1
( ), then the gradient of the
( )
ix
( ) where ( )
Consider the curve
(a)
is
is
.
Use first principles to find the gradient function
( ). Complete the following:
( ) ( ( )
)
…………………………………………………………………………………………… (
)
( )
………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… …………………………………………………………………………………………………… ……………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
(b)
Draw the tangent to the curve
at
(c)
Determine the gradient of tangent to the curve at
by evaluating
( ).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 4.2 (i)
LESSON 2: INTRODUCTORY CALCULUS 1
x
Find, from the first principles, the gradient of the tangent to the curve point ( ) on it.
at the
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Hence determine the equation of the tangent at (
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… (iii)
Write down the gradient of normal at the point (
)
……………………………………………………………………………………………………………
(iv)
Hence find the equation of the normal to the curve at the point (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Note to Students: The alternative definition of ( )
( )
( )
( )
works very well on ( )
√ . Its disadvantage is that the answer is in terms of
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and
rather than
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
Alternate Method for Finding the Gradient of the Tangent at a Point
The gradient of the tangent to a curve
In the diagram, the point
The gradient of the secant
As
, the secant
( ) at the point [
has coordinates [
( )] on the curve.
( )]. ( )
is given by
( )
approaches the tangent at . Hence the limiting position of the
secant is the tangent at .
Therefore the gradient of the tangent at [
( )] is given by:
( )
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( )
( )
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Y11 MATHEMATICS EXT 1
Concept Check 4.3 (i)
LESSON 2: INTRODUCTORY CALCULUS 1
xi
at the point (
Find the gradient of the tangent to the curve
)
Complete the solution
( )
( )
( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
( )
(ii)
Hence the gradient of the tangent at P4,29 is 12.
Hence find the equation of the tangent at (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Find the equation of the normal at (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
xii
Concept Check 4.4
Complete the following statements on the gradient function: 1. The gradient at any point [
2.
(
( )
)
( )
or
( )] on the curve
( )
( )
( ) is given by……………………............. ( )
at the point where
on the curve.
3. If this limit exists, it is called the d……………… or differential coefficient of the function any point [
( ) at
( )].
4. The derivative is also called the ………………… function. 5. Differentiation is the process of finding the ……………… of the curve.
6.
7.
( ) ( )
is the equation of the …………………………. is called the ………………… function of the curve or the ………………… of the
function ( ). 8. The derivative of a function ( ) is a formula for the gradient of the ……………… at any point on the curve.
Note to Students: You must be familiar with both limit definitions of
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
Past H.S.C Questionsxiii Question 1
(2000 H.S.C. Mathematics Extension 1 Q3a)
Use the definition
( )
(
)
( )
to find the derivative of
where
.
2
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
COMMENTS FROM THE MARKING CENTRE Although candidates recognised the given formula as relating to differentiation from first principles, very few had the algebraic skills to complete the substitution and simplification. ) which was frequently mistaken for Confusion reigned over the expansion of ( or . Many candidates did not know how to handle the function notation. Failed )( ); coefficients of instead expansion attempts often involved the use of ( of ; the omission of and the confusion of with . The majority of the candidature were unable to score full marks on this part. Even though most knew that the answer should be or , very few were able to derive it from the formula. In attempting to find the limit many candidates divided by powers of , hoping for relevant cancellations from their incorrect method. It was also difficult to discern whether the term, after factorisation, was being dropped from the expression because candidates knew that powers of were infinitesimally small or because their algebra was careless. Justification for such a step should always be stated in the proof. The quality of responses to this part were unusually dependent on the candidate’s centre, with many centres completing this part well while other centres had almost no attempts for this part.
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Y11 MATHEMATICS EXT 1
Question 2 Let ( )
LESSON 2: INTRODUCTORY CALCULUS 1
(2001 H.S.C. Mathematics Extension 1 Q2a) . Use the definition
(
( ) to find the derivative of ( ) at the point
)
2
( )
.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
MARKING GUIDELINES Criteria
Marks
( )
Shows that
Correctly substitutes (
Copyright © MATRIX EDUCATION 2012
2 ) for
in the expression for
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( )
1
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
SOLUTIONS i
Concept Check 2.1
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
ii
Discussion Question 1
Equals 6 iii
Discussion Question 2
(i) (ii)
or unbounded
(iii)
iv
is called an indeterminate form. It could be anything.
Concept Check 2.2
(i)
v
(ii)
0
(ii)
0
(iii)
Concept Check 2.3
(i)
vi
Concept Check 3.1
(i)
Continuous
(ii)
Continuous
(iii)
Infinite discontinuity at
(iv)
Continuous
(v)
Point discontinuity at
(vi)
Continuous
vii
Concept Check 3.2
(a) viii
(b)
Discussion Question 3
(i)
Yes
(ii)
No
ix
Concept Check 4.1
(a)
(
)
(c)
( ) x
Concept Check 4.2
(i) (ii) Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 2: INTRODUCTORY CALCULUS 1
(iv) xi
Concept Check 4.3
(ii)
(iii)
xii
Concept Check 4.4
1.
( )
5.
Gradient
8.
Tangent
xiii
1.
3.
Derivative
4.
Gradient
6.
Curve
7.
Gradient, derivative
Past H.S.C. Questions 2.
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Y11 MATHEMATICS EXT 1
1.
LESSON 3: INTRODUCTORY CALCULUS 2
OVERVIEW OF INTRODUCTORY CALCULUS 2
Area and Volume
Differentiation from First Principles
Motion
Calculating the Derivative
Index Laws
Lines
Tangents and Normals
Rates of Change
Maxima and Minima
( ) can be perceived in many different ways. It is:
The derivative
( ), or
–
the slope of the curve
–
the gradient of its tangent, or
–
the instantaneous rate of change of the function.
In the previous lesson you learnt how to calculate
from first principles. This approach is
slow and clumsy. In this lesson we develop algorithms which will help us calculate efficiently.
Our 3 major laws are:
–
the product rule
–
the quotient rule
–
the chain rule.
Different schools will approach this theory in varied ways. However you must become skilled in the use of the product, quotient and chain rules. You need to be able to differentiate accurately, quickly and with minimum effort. Many simpler questions in the Advanced (2 unit) and Extension 1 (3 unit) papers will simply demand the calculation of a derivative.
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
2.
DIFFERENTIATION OF
General Rule for Differentiation (
Factorisation of ( – )(
)
–
–
–
–
–
)
Examples (i)
If ( )
, then from first principles at any point
( )
(
( )
( )(
)
If ( )
(
)
then
( )
, then from first principles at any point
( )
on the curve
,
)
)
Hence if ( )
(ii)
,
)
(
(
( )
on the curve
(
( )
( (
)
) )(
( Hence if ( ) Copyright © MATRIX EDUCATION 2012
( )
)
(
)
) then
( )
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Y11 MATHEMATICS EXT 1
(iii)
If ( )
LESSON 3: INTRODUCTORY CALCULUS 2
, then from first principles at any point
( )
( )
(
( )
( )(
)
(
then
( )
, then from first principles at any point ( )
(
( )
( ( – )(
(
,
)
)
Hence if ( )
( )
on the curve
)
(
If ( )
,
)
(
(iv)
on the curve
–
Hence if ( )
) –
–
–
–
–
then
This rule for the differentiation of
Copyright © MATRIX EDUCATION 2012
)
–
–
)
–
(
) –
)
( ) applies to all values of .
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Y11 MATHEMATICS EXT 1 i
Concept Check 2.1 Use
( )
LESSON 3: INTRODUCTORY CALCULUS 2
(
( )
( )
) to differentiate with respect to , from first principles, the following
functions: (i)
( ) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iii)
( )
√
using the fact that
LESSON 3: INTRODUCTORY CALCULUS 2
(√
√ )(√
√ )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
3.
THEOREMS ON DERIVATIVES
Theorem 1 If
then
Note to Students: Using this theorem, you can differentiate any power of .
Concept Check 3.1
ii
Use the rule to find the derivatives of the following.
(i)
( )
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( )
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
( )
√
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Theorem 2 If
( ) then
( )
Note to Students: In all of calculus, any constant sitting in front of a function has no impact.
Concept Check 3.2
iii
Use the rule to find the derivatives of the following.
(i) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………….......
(ii)
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………….......
(iii)
√ …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………..........
Note to Students: Under calculus, we always convert √ to
.
(iv) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………….......... Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Theorem 3
If
Concept Check 3.3
then
iv
Consider the function
(i)
Sketch the line on a number plane.
(ii)
What is the gradient of the line
?
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Use the rule to find the derivative of
.
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iv)
Explain why the answers in (ii) and (iii) are the same. …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Theorem 4
If
, where
is a constant, then
Note to Students: The derivative of a constant function is always . To be a constant means to have no rate of change.
Concept Check 3.4
v
Consider the function
(i)
Sketch the line on a number plane.
(ii)
What is the gradient of the line
?
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Hence find
given
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Theorem 5 ( )
If
Concept Check 3.5
( ) then
( )
( )
vi
Use the rule to find the derivatives of the following
(i)
(
)(
)
by expanding first
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( )
√ (
)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
by dividing first
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
4.
LESSON 3: INTRODUCTORY CALCULUS 2
COMPOSITE FUNCTION (CHAIN) RULE
If
If
Proof of the composite function rule
( ) and
( ) then
[ ( )] then
[ ( )]
[ ( )] then Let
This is the chain rule
( ). This is the composite function rule.
[ ( )]
( )
( ) then
Then
( )
and
Using the chain rule ( ) [ ( )]
( )
Complete the following for the processes used in applying the chain rule for differentiation: Multiply by the ………….. of the bracket, lower the power of the bracket by ………………, then multiply by the ………………………. of the expression inside the bracket.
Example:
Differentiate
(
Solution:
Let ( )
, therefore
Hence
(
)
)
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( ) (
)
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
vii
Concept Check 4.1
Use the chain rule to differentiate the following with respect to :
(
(i)
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(
(ii)
)
h …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
√ …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iv)
LESSON 3: INTRODUCTORY CALCULUS 2
√
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
√
(v)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………..........
( )
(vi)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………….......... ( )
(vii)
√
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………..........
(viii)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ...................................................................................................................................................
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Y11 MATHEMATICS EXT 1
5.
LESSON 3: INTRODUCTORY CALCULUS 2
PRODUCT RULE
If
( )
( ) where
and
are both functions of
( )
( )
( )
Proof {
{
{
{
(
)
(
)
( )
( )
(
)
(
)
( )
(
)
(
)
(
)
( )
(
)
(
)[ (
[ ( ( )
( )
then
)
)]
( )
{ ( )
}
( )] } (
)
{ ( )
( )
(
}
{
( )[ (
( )
( )
)
( )
}
)
(
( )
)
}
( )
( )
}
( )] }
{
(
)
( ) }
( )
Applying the Product Rule:
Leave the second function times differentiate the first function plus leave the first function times differentiate the second function.
Note: Both functions are never differentiated at the same time.
Example:
Given ( )
Solution:
This is a product of two functions of ( )
(
)(
(
)
(
)
(
( )
Copyright © MATRIX EDUCATION 2012
(
)
( )
( )
) find the value of
so we use the product rule. )
(
(
)
(
)
) (
(
)
)
( )
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
viii
Concept Check 5.1
Use the product rule to differentiate the following:
(i)
(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( )
(
)(
√ )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Discussion Question 1:ix Try differentiating by writing it as work?
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and using the product rule. Does it
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
x
Concept Check 5.2
Use the product and chain rule to differentiate the following: (
(i)
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Did You Know: Sometimes you need to use more than one rule.
(ii)
( )
(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
√ …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 5.3
LESSON 3: INTRODUCTORY CALCULUS 2
xi
Use the chain rule and the product rule to differentiate the following: (i)
[
(
)]
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
( )
√(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
6.
LESSON 3: INTRODUCTORY CALCULUS 2
QUOTIENT RULE
( )
–
If
To prove the quotient rule, use the product rule and chain rule on
Complete the following:
and
are both functions of
( ) ( )
then
( ) ( )
( )[ ( )]
................. [ ( )]
( ) ....................................................................
……………………………………………………………………………………………. …………………………………………………………………………………………….
Applying the Quotient Rule:
Example:
Given
find
.
Solution: (
(
) (
)
)
(
)
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Y11 MATHEMATICS EXT 1
Concept Check 6.1
LESSON 3: INTRODUCTORY CALCULUS 2
xii
Use the quotient rule to differentiate the following: (i)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii) Note to Students: Some students re-write
as
(
)(
)
and then use the product rule
instead of the quotient rule. This is not a good idea. If you see a quotient, use the quotient rule.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
xiii
Concept Check 6.2
Use the quotient rule and the chain rule to differentiate the following: (i)
( )
(
)
Did You Know: Sometimes you need to use more than one rule. …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
√
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 6.3
LESSON 3: INTRODUCTORY CALCULUS 2
xiv
Use the chain rule and the quotient rule to differentiate the following:
(i)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
√
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
7.
TANGENTS & NORMALS TO CURVES
Equation of the Tangent to the Curve to find the gradient of the tangent at any point (
)
Step 1:
Find
Step 2:
Substitute the coordinate of the point of contact into to find the numerical value of the gradient of the tangent at the point of contact.
Step 3:
The point (
) is the point of contact of the tangent.
Use the formula point of contact. Example:
(
) to find the equation of the tangent at the
Find the equation of the tangent to the curve point (
at the
) by completing the following solution.
Solution: …………………………………..
The gradient of the tangent at any point is At the point (
), the gradient of the tangent is
Therefore the equation of the tangent at ( Concept Check 7.1 (a)
) is
…………………………….. )
………..(
xv
Consider the parabola (i)
Find the coordinates of the points where the parabola crosses the
axis.
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Tangents are drawn to the parabola at the points where it crosses the the gradients of the tangents.
axis. Find
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(b)
LESSON 3: INTRODUCTORY CALCULUS 2
Find the equation of the tangent to the following curves at the given points: (
(i)
) at (
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
√
at
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Discussion Question 2: Can a tangent meet a curve more than once? Discussion Question 3: Can a tangent cut across a curve?xvi
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Y11 MATHEMATICS EXT 1
Concept Check 7.2
(a)
LESSON 3: INTRODUCTORY CALCULUS 2
xvii
Find the coordinates of the points on the curve
(i)
parallel to the line
where the tangent is:
. Hence find the equations of the tangents.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
perpendicular to the line
. Hence find the equations of the tangents.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Equation of the Normal to the Curve
A normal is a line perpendicular to the tangent at the point of contact of the tangent with the curve. Step 1:
Find This gives the gradient of the curve at any point (
Step 2:
Substitute the
)
coordinate of the point of contact into
.
This gives the numerical value of the gradient of the tangent at the point of ). contact, (
Step 3:
The normal is perpendicular to the tangent at the point of contact of the tangent. Use of the tangent.
Step 4:
Use the formula (
Example:
to find the gradient of the normal at the point of contact (
) to find the equation of the normal at
).
Find the equation of the normal to the curve ) by completing the following solution. point (
at the
Solution: The gradient of the tangent at any point is
At the point (
…………………………………
), the gradient of the tangent is
………………………...
Therefore the gradient of the normal at the point (
) is ……………………
Therefore the equation of the normal at (
) is
………..(
).
In general form the equation of the normal is ………………………………………………………………………………………… …………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 7.3
(a)
xviii
Consider the curve
(i)
LESSON 3: INTRODUCTORY CALCULUS 2
(
)
Determine the gradient of the tangent to the point (
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Hence find the equation of the tangent at (
).
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Write down the gradient of the normal to the point (
)
……………………………………………………………………………………………………………
(iv)
Hence find the equation of the normal at (
) in the general form.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(b)
Find the equation of the normal to the curve
at the point where
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
Past H.S.C. Questionsxix Question 1
(2011 H.S.C. Mathematics Q2c)
Find the equation of the tangent to the curve
(
) at the point where
.
3
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
NOTES FROM THE MARKING CENTRE Nearly all candidates recognised this as a calculus question and showed logical working that demonstrated a good understanding of the required steps. This allowed for part marks for ) or the occasional imperfect responses. There were errors in differentiating, with ( ( ) being the usual incorrect answers, but often the rest of the working followed correctly. Sometimes notation was poor and lack of parentheses resulted in the wrong gradient or point. Candidates are encouraged to show clear substitutions to avoid careless ) in their errors. Several candidates substituted and either used ( ) or ( equation of the line. In better responses, candidates clearly showed the derivative, the gradient m, the point and finally the equation of the line. A small number of candidates provided only the gradient of the tangent rather than the equation of the tangent (stopping at ). Some also correctly evaluated ( ) then used this as the value of the point rather than as the gradient. Candidates need to take care when copying the value into the writing booklet as some used instead of . MARKING GUIDELINES Criteria
Correct solution Finds correct gradient and attempts to find equation of tangent Correctly differentiates or finds correct for
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Y11 MATHEMATICS EXT 1
Question 2
LESSON 3: INTRODUCTORY CALCULUS 2
(2009 H.S.C. Mathematics Q1d)
Find the gradient of the tangent to the curve
at the point (
).
2
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
COMMENTS FROM THE MARKING CENTRE Almost all candidates recognised that this part was a calculus question. Common errors were incorrect derivatives obtaining or . The use of the derivative was also problematic with some responses attempting to calculate stationary points and others solving . MARKING GUIDELINES
Criteria Correct answer Differentiates correctly, or equivalent merit
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 3: INTRODUCTORY CALCULUS 2
SOLUTIONS i
Concept Check 2.1 ( )
(i)
ii
( )
(iii)
(ii)
2 x3
(iii)
( )
√
Concept Check 3.1
1 x2
(i)
iii
5 2 x7
Concept Check 3.2
(i) (iv) iv
(ii)
(
(ii)
(iii)
(iii)
(iii)
√
)
Concept Check 3.3
(ii)
v
3
Gradient =
Concept Check 3.4
(ii) vi
0
(ii)
0
Concept Check 3.5
(i)
vii
2x
√
(ii)
4
√
(iii)
√
Concept Check 4.1 (
(i) (iv) (vii)
viii
(
)√
(
)√
)
(
(ii)
)
(
(v)
√(
√ (
(vi)
)
(
(viii)
(iii)
)
)(
(
) )
)
Concept Check 5.1
(i)
√
(ii)
√
(iii) ix
Discussion Question 1
Yes, but this is not the way to do it. x
Concept Check 5.2
(i)
(
)(
)
(ii)
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(
)( Page 93 of 213
)
(iv)
√
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Y11 MATHEMATICS EXT 1
xi
LESSON 3: INTRODUCTORY CALCULUS 2
Concept Check 5.3 [
(i)
(
)] (
)
36 x 2 128 x 7
(ii)
33 3x 5 2 x 7 2
xii
Concept Check 6.1 dy 2 dx 1 x 2
(i)
xiii
dy 18 x 2 dx x 9
(ii)
(iii)
2
dy 7 dx 3x 22
Concept Check 6.2 f x
(i)
xiv
2 9x 5 3x
dy dx
(ii)
2
23x 7 35 x 9 (iii) 2 5x 2 3
dy dx
x
x
2
x
3
Concept Check 6.3 4 x 7 x 10
(i)
xv
4
(ii)
x 5 2
√
(
)
(
)
Concept Check 6.1
(a)
(i)
(b)
(i)
xvi
(
)(
)
At (
(ii)
)
At (
)
(ii)
Discussion Question 2
Yes, consider xvii
at
Concept Check 7.2
(i)
At (
(ii)
(
xviii
(a)
)
; At (
)
)
Concept Check 6.3 (i)
8
(iv)
x 8 y 33 0
(ii)
y 8x 4
(iii)
1 8
(b) xix
1.
Past H.S.C. Questions 2.
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Y11 MATHEMATICS EXT 1
1.
LESSON 4: PARAMETRIC REPRESENTATION 1
OVERVIEW OF PARAMETRIC REPRESENTATION
Conics
Parametrics
Calculus, Normals and Tangents
Proofs
Coordinate Geometry
The parametric (
Parametric Relations
) section of the Extension 1 syllabus has a deserving reputation as
being one of the hardest topics in Extension 1 mathematics as it synthesises several techniques and concepts from both 2 unit and 3 unit mathematics.. It builds upon the theory in 2 unit mathematics and presents the parabola in parametric rather than Cartesian form. Almost every Extension 1 Trial and H.S.C. exam contains one of these questions, usually quite late in the paper.
This is one of the few sections of the syllabus where you will need to construct for the examiner convincing proofs of mathematical theorems. Watch your teacher carefully and copy their style.
Some issues to keep in mind are:
–
Well prepared candidates will memorise all of the fundamental equations in the chapter.
–
Make sure you know the 2 unit coordinate geometry.
–
Generally speaking, any minor algebraic error in your proof will stop you. You must watch the algebra extremely carefully and eliminate mistakes.
–
These questions often have several parts building to a final result. If you cannot do a particular part, just jump over it! DO NOT abandon the entire question just because you are stuck on a minor subpart.
–
Always draw a large accurate diagram of the parabola and transfer all data to the graph.
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Y11 MATHEMATICS EXT 1
2.
LESSON 4: PARAMETRIC REPRESENTATION 1
CARTESIAN AND PARAMETRIC EQUATIONS
In Mathematical exercises it is sometimes convenient to express two related variables, such as
and , in terms of a third variable say ( ) and
or
( ) and
so that
( ) or
( ).
The third variable,
The use of the third variable simplifies a proof, which could be otherwise complicated.
( ) and
( ) are called parametric equations.
When the parameter is eliminated from the parametric equations, the resulting equation is of the form
or , is called a parameter.
( ). This equation is called a Cartesian equation.
When a curve is expressed in the Cartesian form, the curve are expressed in the form (
( ), the coordinates of the points on
). These points are called Cartesian coordinates.
When the parametric equations of a curve are given, then all points on the curve are expressed in terms of the parameter and in the form [ ( )
( )]. These points are called
parametric coordinates. Example: The parametric equations of a curve are equation of the curve.
and
. Find the Cartesian
Solution: The parametric equations are in terms of the parameter . To find the Cartesian equation, we need to eliminate the parameter, and express the equation of the curve in terms of the and only Express in terms of
Step 1:
and
and
Step 2:
Use the identity ( )
to link
and .
( )
Therefore the Cartesian equation of the curve is
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Y11 MATHEMATICS EXT 1
Concept Check 2.1 (a)
LESSON 4: PARAMETRIC REPRESENTATION 1
i
The parametric equations of a curve are of the curve.
and
. Find the Cartesian equation
Did You Know: To move from parametric to Cartesian form you must eliminate the parameter. ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(b)
Find the Cartesian equation of a curve whose parametric equations are .
and
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(c)
( ) and The parametric equations of a curve are where constant and . Find the Cartesian equation of the curve.
is a
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(d)
Show that points (
LESSON 4: PARAMETRIC REPRESENTATION 1
) and (
) lie on the parabola
.
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(e)
(i)
Show that the point (
) lies on the parabola
.
................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(ii)
Hence find the coordinates of the point P on the parabola
. Where
.
................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(f)
A parabola has parametric equations and the equation of its directrix.
and
. Find the coordinates of its focus
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
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Y11 MATHEMATICS EXT 1
3.
LESSON 4: PARAMETRIC REPRESENTATION 1
PARAMETRIC EQUATIONS OF A PARABOLA
The parametric equations of the parabola
The point (
are
) is a variable point on the parabola
depending on the value
of parameter,
The advantage of the parametric coordinates becomes obvious when we consider 2 variable points. In Cartesian form the points on the parabola where and (
Concept Check 3.1 (a)
and
are (
) and (
whereas in parametric form the points are (
) )
) which are easily distinguishable in a mathematical solution. ii
With reference to the parabola
, find the coordinates of the points with parameter:
Did You Know: Every point on the curve corresponds to a unique value of . (i)
................................................................................................................................................... ...................................................................................................................................................
(ii)
................................................................................................................................................... ...................................................................................................................................................
(iii)
................................................................................................................................................... ...................................................................................................................................................
(iv)
................................................................................................................................................... ...................................................................................................................................................
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Y11 MATHEMATICS EXT 1
(b)
LESSON 4: PARAMETRIC REPRESENTATION 1
Find the parametric equations of the following parabolas: Note to Students: Find the focal length
first.
(i) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iv) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(v) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(vi) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
4.
LESSON 4: PARAMETRIC REPRESENTATION 1
THE EQUATION OF A CHORD OF
Let (
) and (
Gradient of the chord
(
.
:
)( (
)be two points on
) )
Equation of chord ( ( (
) )
)
Therefore the equation of the chord (
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is
)
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Y11 MATHEMATICS EXT 1
Concept Check 4.1
(a)
( (i)
LESSON 4: PARAMETRIC REPRESENTATION 1
iii
) and
(
) are two points on the parabola
. Find:
the equation of the chord
Note to Students: Remember
(
)(
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Show that the chord
is a focal chord
Note to Students: A focal chord passes through the focus (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(b)
LESSON 4: PARAMETRIC REPRESENTATION 1
is a point on the parabola
(i)
Write down the coordinates of
in terms of the parameter, .
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Find the equation of the focal chord passing through the point .
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Hence write down the equation of the focal chord drawn through another point the parabola.
on
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iv)
Find the coordinates of
in terms of
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 4.2
( (i)
) and (
LESSON 4: PARAMETRIC REPRESENTATION 1
iv
) are two points on the parabola
.
Find the equation of the chord
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
The chord and
passes through the focus of the parabola. Find the relationship between
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
5.
LESSON 4: PARAMETRIC REPRESENTATION 1
EQUATIONS OF TANGENTS TO THE PARABOLA
Gradient of the tangent at (
): is the equation of the parabola. Hence At (
),
Note to Students: You can differentiate parametrically or directly. Either way, for a standard situation
Equation of the tangent at (
.
): (
)
Therefore the equation of the tangent at (
), is
Note to Students: Remember this formula! Discussion Question 1:v Which point on has a tangent Discussion Question 2:vi Which point on has a tangent Copyright © MATRIX EDUCATION 2012
?
?
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Y11 MATHEMATICS EXT 1
Concept Check 5.1 (a)
LESSON 4: PARAMETRIC REPRESENTATION 1
vii
is a point on the parabola (i)
Write down the coordinates of
in terms of the parameter, .
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………….......
(ii)
Find the equation of the tangent to the parabola
at the point .
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………….................
(b)
(
) and
the tangents at
(
) are two points on the parabola
. Find the equations of
and .
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 5.2
LESSON 4: PARAMETRIC REPRESENTATION 1
viii
is a point on the parabola parabola at intersects the tangent at .
and is the focus of the parabola. The tangent drawn to the axis at . The perpendicular from to the tangent intersects the
(i)
Draw a diagram of the parabola and show the positions of
(ii)
Write down the parametric coordinates of the point
and .
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………….......
(iii)
Find the equation of the tangent at …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iv)
LESSON 4: PARAMETRIC REPRESENTATION 1
Hence find the equation of the perpendicular from
to the tangent at .
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(v)
Find the coordinates of
and .
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
6.
LESSON 4: PARAMETRIC REPRESENTATION 1
EQUATION OF NORMAL TO THE PARABOLA
(
Gradient of normal at (
)
):
is the equation of the parabola. Hence At (
),
Therefore, gradient of normal is
Equation of normal at (
since
): (
)
Therefore the equation of the normal at (
) is
Note to Students: Remember this formula!
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Y11 MATHEMATICS EXT 1
Concept Check 6.1 (a)
LESSON 4: PARAMETRIC REPRESENTATION 1
ix
( ) and ( the normals at and .
) are two points on the parabola
. Find the equation of
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(b)
LESSON 4: PARAMETRIC REPRESENTATION 1
is a point on the parabola (i)
Write down the coordinates of
in terms of the parameter
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Find the equation of the normal to the parabola at the point
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 6.2
x
is a point on the parabola
(i)
LESSON 4: PARAMETRIC REPRESENTATION 1
.
Write down the coordinates of the point
in terms of the parameter
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
A normal is drawn to the parabola at
Find the equation of the normal.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
The normal at
passes through the focus. Find the coordinates of
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 4: PARAMETRIC REPRESENTATION 1
Past H.S.C. Questionsxi Question 1
(2009 H.S.C. Mathematics Ext 1 Q2c)
The diagram shows points ( The tangents to the parabola at
(i)
) and ( ) which move along the parabola and meet at .
Show that the equation of the tangent at
is
.
. 2
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Write down the equation of the tangent at , and find the coordinates of the point in terms of t. 2
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 4: PARAMETRIC REPRESENTATION 1
NOTES FROM THE MARKING CENTRE The lead-in, part (i), where the candidates were given the result, was instrumental in allowing a significant number of candidates to correctly obtain the equation of the tangent in part (ii). The progression through the three parts saw most obtaining the correct coordinates of the point . Yet again many arithmetic errors were seen. Most candidates knew how to find the locus of . MARKING GUIDELINES (i) Criteria
Marks 2
Correct solution Uses the equation of the parabola to show that the gradient at is (or equivalent progress)
1
(ii) Criteria
Marks 2
Correct solution Attempts to solve simultaneously the equations for tangents at and
Question 2
1
(1999 H.S.C. Mathematics Ext 1 Q4b)
The diagram shows the graph of the parabola The tangent to the parabola at ( the axis at . The normal to the parabola at
(i)
),
cuts the
, cuts
axis at .
Derive the equation of the tangent
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(ii)
Show that the coordinates of
LESSON 4: PARAMETRIC REPRESENTATION 1
are (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Find the coordinates of the midpoint of
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… NOTES FROM THE MARKING CENTRE (i)
Deriving the tangent, , was a simple application of bookwork using the parametric locus . This part was very well done, although some candidates made it more complicated by attempting to reproduce remembered bookwork for the locus , without letting . As only two marks were assigned to this part, examiners did not insist that candidates use calculus to determine that the slope of the tangent at was . Candidates who simply quoted this fact were able to obtain full marks.
(ii)
Candidates needed to find the equation of the normal in terms of the parameter and substitute to obtain . Although this was well done, many responses showed attempts to ‘fudge’ the final step from an incorrect equation. Candidates making genuine attempts in questions where the answer is given should ensure that each step is clearly presented to avoid any doubt about its authenticity. For instance, candidates should explicitly state ‘let ’ and show the corresponding substitution clearly.
(iii)
The majority of the candidature found the midpoint (
), but then found it very difficult to
get the second mark by solving simultaneously using
, to obtain the Cartesian form of
locus,
. Many candidates obtained the equation
and then made errors
in attempts to simplify this expression. Such errors were ignored, and these candidates were awarded both marks. A number of candidates tried to use the distance formula with This approach had no chance of success, as all points on the perpendicular bisector of AB are equidistant from and . Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 4: PARAMETRIC REPRESENTATION 1
SOLUTIONS i
Concept Check 2.1
(a)
(b)
(e) ii
(
(ii)
(f)
(
(ii)
(
)
Concept Check 3.1
(a)
(b)
iii
(i)
(
(iv)
(
)
)
(iii)
(
)
)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Concept Check 4.1 (
(a)
(i)
(b)
(i)
(
(iv)
(
iv
) )
(
)
(ii) (
)
(
)
(iii)
(
)
)
Concept Check 4.2 (
(i)
v
)
(c)
)
(ii)
Discussion Question 1
( vi
)
Discussion Question 2
None. vii
Concept Check 5.1
(a)
(i)
(
)
(ii)
(b)
viii
Concept Check 5.2
(ii) (v) ix
(
) (
(iii) )
(
(iv)
)
Concept Check 6.1
(a) Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(b) x
(i)
(
LESSON 4: PARAMETRIC REPRESENTATION 1
)
(ii)
Concept Check 6.2
(i)
(
)
(ii)
(iii)
(
)
xi
Past H.S.C. Questions 1. (ii) 2.
(i)
(
) (iii)
Copyright © MATRIX EDUCATION 2012
(
)
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Y11 MATHEMATICS EXT 1
1.
LESSON 5: PARAMETRIC REPRESENTATION 2
OVERVIEW OF PARAMETRIC REPRESENTATION 2
Conics
Calculus, Normals and Tangents
Parametrics
Proofs
Coordinate Geometry
Parametric Relations
In this lesson we move deeper into parametric representation (
) of the parabola
. The main issues to consider are:
–
You must have a complete understanding of the 2 unit coordinate geometry. You will be finding tangents, normal, intercepts, points of intersection midpoints and gradients. Almost all of the 2 unit coordinate geometry will appear in one form or another, usually in extremely complicated circumstances.
–
You should memorise the properties and standard parametric equations of tangents, normals and chords. Admittedly the first part of the question usually asks you to prove these results, so they are often on display in the examination. Nevertheless it is sometimes also very handy to be able to quote a particular fact or formula.
–
Algebraic error is your number one enemy. Small errors will stop you from moving forward.
–
These questions often have several parts building to a final result. If stuck on a part always carefully consider the parts above, the key will usually be hidden there.
–
If you cannot do a particular part just jump over it! DO NOT abandon the entire question just because you are stuck on a minor subpart.
–
If you are tasked with the sketching of a graph take extreme care to correctly transfer the data from the question. A misunderstanding in the sketch will immediately block your progress.
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Y11 MATHEMATICS EXT 1
2.
LESSON 5: PARAMETRIC REPRESENTATION 2
DERIVATIVES OF PARAMETRIC EQUATIONS
( ) and
Given
the first derivative
( ) are the parametric equations of a curve
( ) then to find
:
Step 1:
Find
and
Step 2:
By the chain rule
Example:
Given
etc.
and
, find
in terms of .
Complete the following solution: …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………
Concept Check 2.1
i
For each of the following parametric equations, find
in terms of
Note to Students: ⁄ (a)
and
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(b)
and
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(c)
(
LESSON 5: PARAMETRIC REPRESENTATION 2
) and
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
Did You Know:
Concept Check 2.2
Find
)
(
)(
(
)⁄
)
ii
Consider the curve with parametric equations (i)
(
and
.
.
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
Find the equation of the tangent and normal to the curve where
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(iii)
LESSON 5: PARAMETRIC REPRESENTATION 2
Find the coordinates of the points on the curve where the tangent is horizontal.
…………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………
Discussion Question 1:iii For a horizontal tangent, do we set
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,
or
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?
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
3.
POINT OF INTERSECTION OF TANGENTS & NORMALS
The Intersection of Tangents
Tangents drawn to the parabola
Equation of tangent at (
at the points
and
intersect at the point .
) is (
Hence the equation of the tangent at Solve the two equations of tangents at
and
) is simultaneously to determine the coordinates
of , the point of intersection. ………………(1) ………………(2) Substitute (1) into (2) to get Note to Students: Observe the symmetry in and . Swapping and has no effect. This will help you to remember this fact.
Hence (
)
(
)(
(
)
)
Substitute into (1) (
)
The point of intersection of tangents at [ (
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)
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and
are
]
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Y11 MATHEMATICS EXT 1
Concept Check 3.1 (a)
( (i)
LESSON 5: PARAMETRIC REPRESENTATION 2
iv
) and (
) are two points on the parabola
Find the equation of the tangent at
.
in terms of the parameter .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
Hence find the coordinates of the point of intersection of the tangents at
and at .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
(b) ) and ( In the diagram, ( are two points on the parabola Tangents drawn to the parabola at intersect at .
(i)
) . and at
Find the equation of the tangent at .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
Hence write down the equation of the tangent at
………………………………………………………………………………………………… …………………………………………………………………………………………………
(iii)
Find the point of intersection of the tangents at
and .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
The Intersection of Normals
The normals drawn to the parabola
Equation of normal at (
Hence the equation of the normal at
Solve the two equations of the normals simultaneously to determine the coordinates of
at
and
intersect at
.
) is (
) is
………………(1) ………………(2) Equation (1) – Equation (2) (
)
(
)
(
)
( (
)
(
)(
( Substitute
) )
(
(
)
)
)
into equation (1) to determine the value of
(
Note to Students: Observe the symmetry in remember this fact.
)
and . Swapping
and
has no effect. This will help you to
The point of intersection of the normals at
[
(
) (
and
are
)]
Note: Do not memorise this result. However, understand the result and be able to derive it with different parameters as well.
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Y11 MATHEMATICS EXT 1
Concept Check 3.2 (a)
LESSON 5: PARAMETRIC REPRESENTATION 2
v
Normals are drawn to the parabola
at the points (
(i)
in terms of the parameter
Find the equation of the normal at
) and (
).
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
The normals at
and
intersect at the point . Find the coordinates of .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
(b)
The diagram shows a variable point (
) on the parabola
(i)
Find the equation of the normal to the parabola
.
at the point
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
At the point , a normal is drawn perpendicular to the normal at . Find the coordinates of in terms of .
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(iii)
The normals at
and at
intersect at . Find the coordinates of the point .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
4.
LESSON 5: PARAMETRIC REPRESENTATION 2
FOCAL CHORD PROPERTIES
In the diagram, the chord
passes
through the focus of the parabola .
The gradient of
The equation of the chord
Since
(
is
)( (
) )
(
is
passes through the focus, then (
)
(
)
(
)
) must satisfy the equation of the chord.
Hence
Hence the condition for a chord
But the gradient of the tangent at the point
to be a focal chord is is
and the gradient of the tangent at the point
on the parabola is . Since the product of the gradients is
then the tangents drawn to
the end-points of a focal chord will intersect at right angles.
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
Concept Check 4.1 ( ) and ( show that: (i)
) are two points on the parabola
the tangents at the ends of a focal chord
. If
is a focal chord then
intersect at right angles.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
the tangents at the ends focal chord intersect on the directrix of the parabola. Assume the equation of tangent at is
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
5.
LESSON 5: PARAMETRIC REPRESENTATION 2
REFLECTION PROPERTIES
The law of reflection of light states that the angle of incidence of a light ray is equal to the angle of reflection.
The reflection property is: All rays of light emitted from a light source, placed at the focus of a parabolic reflector, are reflected parallel to the axis of the reflector.
Discussion Question 2: How does your car use this fact?
Geometry and the definition of a parabola as a locus are used to show that the reflected ray PM is parallel to the
– axis
In the diagram (Law of reflection: angle of incidence is equal to angle of reflection)
(Locus definition of a parabola,
is equidistant from the focus & directrix)
Note to Students: In any question where the distance from the focus to a point is mentioned, always consider the locus definition of the parabola. Therefore Hence
is an isosceles triangle (2 sides equal) (Base angles of isosceles triangle)
These angles are corresponding angles Therefore, ray PM is parallel to the Copyright © MATRIX EDUCATION 2012
– axis
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Y11 MATHEMATICS EXT 1
Concept Check 5.1 ( (i)
LESSON 5: PARAMETRIC REPRESENTATION 2
vi
) is a point on the parabola Find the equation of the tangent to the parabola at the point .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
S is the focus of the parabola and
is the
intercept of the tangent at . Prove that
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………..............................
(iii)
Hence show that is equal to the acute angle between the tangent and the line passing through parallel to the axis of the parabola.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
6.
LESSON 5: PARAMETRIC REPRESENTATION 2
EQUATION OF THE CHORD OF CONTACT
are drawn from (
Tangents
touch the parabola
at
The chord of contact is
P has coordinates (
.
) and
has coordinates (
The gradient of the chord
The equation of the chord ( (
The gradient of the tangent at (
) is
The equation of the tangent at (
) is
(
) to
)
)
)
)
Point of intersection of the tangents at
and
......................(1) ......................(2)
(
) (
Therefore the tangents at
From the given data,
Equating, gives
(
( ( )
) )
intersect in ( (
has coordinates ( ) and
)
).
).
and substituting into the equation of the chord
( )
The equation of the chord of contact of tangents drawn from (
) to the parabola
is ( Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
Concept Check 6.1
LESSON 5: PARAMETRIC REPRESENTATION 2
vii
(a)
Tangents drawn from the point ( points and .
) to the parabola
(i)
Write down the equation of the chord of contact
touch the parabola at the
………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
lies on the directrix. Show that this chord of contact is a focal chord.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(b)
) to touch the parabola Tangents are drawn from ( Show that the equation of the chord of contact is
at the points and find its length.
and .
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Concept Check 6.2 (a)
(i)
viii
LESSON 5: PARAMETRIC REPRESENTATION 2
Examination-type questions
Find the equation of the line through ( .
) and the vertex 0 of the parabola
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(ii)
This line meets the directrix in . Prove that tangent at .
, where
is the focus, is parallel to the
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(iii)
Write down the equation of the chord in terms of the parameters the condition for to pass through the focus .
and
Deduce
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
(iv)
Hence show that
is parallel to the axis of the parabola.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(b)
(i)
LESSON 5: PARAMETRIC REPRESENTATION 2
Sketch the parabola whose parametric equations are and On your diagram, mark the points and which correspond to respectively.
. and
Did You Know: This is a non-standard parametric representation!
There is no ! You cannot use any standard results on questions of this type. Everything needs to be done from basics.
(ii)
Show that the tangents to the parabola at
and
intersect at
(
).
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(iii)
LESSON 5: PARAMETRIC REPRESENTATION 2
( )is a point on the parabola between and such that the tangent at midpoint of . Show that the tangent at is parallel to .
meets
at the
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
Past H.S.C. Questions Question 1
(2010 H.S.C. Mathematics Ext 1 Q4c) . The point (
The diagram shows the parabola parabola.
The tangent to the parabola at , The point Show that
), where
, is on the 3
, meets the -axis at .
is on the directrix, such that
is perpendicular to the directrix.
is a rhombus.
………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
NOTES FROM THE MARKING CENTRE Although parametric coordinates were involved, this part was a very simple geometry question requiring the candidates to prove that the given shape was a rhombus. Very few candidates showed that they understood the properties of a rhombus. Those who did know the required properties often led themselves astray via weak algebra. Many then fudged their answers. This was particularly noticeable among those trying to prove four sides had equal length. Candidates proved opposite sides equal but did not complete the algebra to show all sides equal. One of the simplest proofs was showing the midpoints of the diagonals equal and that they bisected at right angles. A number of candidates used this method with very few errors. There were other methods attempted with varying results. MARKING GUIDELINES Criteria
Correct proof Makes significant progress Finds one piece of relevant information
Question 2
(2008 H.S.C. Mathematics Ext 1 Q4c)
The points ( parabola at and right angle. (i)
Marks 3 2 1
. The tangents to the ), ( ) lie on the parabola intersect at . The chord produced meets at , and is a
Find the gradient of
, and hence show that
.
2
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Y11 MATHEMATICS EXT 1
(ii)
The chord
LESSON 5: PARAMETRIC REPRESENTATION 2
produced meets
at . Show that
is a right angle.
1
………………………………………………………………………………………… …………………………………………………………………………………………
(iii)
Let be the midpoint of the chord otherwise, show that .
. BY considering the quadrilateral
, or
2
………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………
NOTES FROM THE MARKING CENTRE (i)
Well set out solutions with clear logic were more successful. Mid-range responses found the gradient of but then many multiplied by stating and therefore .
(ii)
It was important to relate the results in (c)(i) to this part. The most successful method simply stated that
(iii)
. Some candidates who could not establish the result in part
(i) nevertheless used the result to successfully complete part (ii). In the better responses, candidates who recognised that was a cyclic quadrilateral were quite efficient and effective at explaining why . Those who tried coordinate geometry formulae found that it was nearly impossible to prove the result and so they spent valuable time completing large amounts of algebra to little benefit. MARKING GUIDELINES (i) Criteria
Correct solution Finds gradient of
or equivalent merit
Marks 2 1
(iii) Criteria
Correct solution Recognises the significance of the fact that
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Marks 2 1
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Y11 MATHEMATICS EXT 1
LESSON 5: PARAMETRIC REPRESENTATION 2
SOLUTIONS i
Concept Check 2.1
(i)
ii
(ii)
(iii)
(
)
Concept Check 2.2
(i)
(ii)
(iii)
iii
(
√
) (
√
,
)
Discussion Question 1 or
iv
Concept Check 3.1
(a)
(i)
(ii)
(b)
(i)
(ii)
v
( (
)
) (iii)
(
(iii)
(
)
Concept Check 3.2
(a)
(i)
(ii)
(
(b)
(i)
(ii)
(
vi
) )
)
Concept Check 5.1
(i) vii
Concept Check 6.1
(a) (b) viii
(i)
(
)
√
Concept Check 6.2
(a)
(i)
(iii)
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(
)
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Y11 MATHEMATICS EXT 1
1.
LESSON 6: PARAMETRIC REPRESENTATION 3
OVERVIEW OF PARAMETRIC REPRESENTATION 3
The Hyperbola as a Locus
Identifying a Locus
Conics
The Ellipse as a Locus
Parametric Curves
As mentioned earlier, the (
) questions in the Extension 1 papers have many parts.
These parts will often lead to the parametric description of a special point (
). The very
last part will then ask you for the Cartesian equation of the locus over which this point travels. This involves eliminating the parameter(s) and establishing a direct relationship between the
and
variables. This can be a tricky algebraic task:
–
If only one parameter is involved try to eliminate it.
–
If two parameters are involved it is sometimes a good idea to square either the
or the
component and look for patterns. The answer is usually another
parabola.
–
If you are asked to verify a particular equation for the locus, simply substitute the
–
and
components into that equation and check that it is satisfied.
Be aware that it is possible that the locus does not fill out the entire curve. For full marks you will then need to specify domains and/or ranges.
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Y11 MATHEMATICS EXT 1
2.
LESSON 6: PARAMETRIC REPRESENTATION 3
LOCUS OF A POINT
Locus is the path traced by a moving point.
A point expressed in parametric form represents a point that changes its position on the coordinate plane as the value of the parameter changes. The path traced by all these points is called the locus of the point.
The Equation of the Locus of Points expressed in terms of One Variable Step 1:
Express the coordinates of the variable point as is the parameter.
Step 2:
Eliminate the parameter from the two equations.
Step 3:
Express the equation of the locus in the form equation of the locus.
( ) and
( ) where
( ), the Cartesian
Example: ( )is a variable point. As the value of the parameter p changes, the position of on the coordinate plane varies. The path that is traced by the varying positions of is called the locus of . Find the equation of the locus of and describe the locus of in geometric terms.
The variable point has coordinates (
) .................(i)
..................(ii) From (i) (
Substitute into (ii) Hence (
)
)
The equation of the locus of
is (
)
(
As varies, it will trace a parabola with vertex ( ). focus at (
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) ) focal length 1 and
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Y11 MATHEMATICS EXT 1
Concept Check 2.1
(a)
i
A variable point ( (i)
LESSON 6: PARAMETRIC REPRESENTATION 3
) and (
) lie on the coordinate plane.
is the midpoint of the interval parameter .
. Find the coordinates of
in terms of the
................................................................................................................................................... ...................................................................................................................................................
(ii)
As varies on the number plane, the position of changes and the point path. This path is called the locus of Find the equation of the locus of
traces a
Note to Students: Eliminate .
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(b)
A point divides the interval joining a variable point ( the ratio 2:1. (i)
) and the origin, (
Show that the coordinates of T, in terms of the parameter t, are (
) in
)
................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(ii)
Find the equation of the locus of
as
varies on the coordinate plane.
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 6: PARAMETRIC REPRESENTATION 3
The Equation of the Locus of Points expressed in terms of Two Variables Step 1:
Express the parametric coordinates as parametric equations.
Step 2:
Find the condition that links the two variables
Step 3:
Use the condition linking the two variables to eliminate the variables from the two equations.
Step 4:
Express
as a function of .
Example: A variable point
has coordinates [ (
As the values of
and
varies,
)
] where
.
traces a path. Find the equation of the locus of
and give a geometric description of this locus. Solution: The parametric equations of the locus of (
are:
) ...................(i)
...................(ii) The condition that links the two variables is Since
(
),then square both sides of the equation and (
)
(
) .................(iii) (
)
(
)
Hence Substitute into (iii)
Hence the equation of the locus of
is
parabola vertex the origin and focal length (
. The point
will trace a
The coordinates of the focus are
) and the equation of the directrix is
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Y11 MATHEMATICS EXT 1
Concept Check 2.2
LESSON 6: PARAMETRIC REPRESENTATION 3
iii
) ( )] where A variable point has coordinates [ ( the locus of . Describe the locus of in geometric terms.
. Find the equation of
............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. ............................................................................................................................................................. .............................................................................................................................................................
Note to Students: If asked to geometrically describe: a parabolic locus, supply the vertex, focus and directrix a circle, give the centre and radius
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Y11 MATHEMATICS EXT 1
3.
LESSON 6: PARAMETRIC REPRESENTATION 3
LOCUS PROBLEMS INVOLVING ONE VARIABLE Note to Students: These are much harder. Example: In the diagram given below, the tangent at ( axis at and the normal to the parabola at
(i)
Find the equation of the tangent
)( cuts the
) to the parabola – axis at .
cuts the
.
;
Equation of tangent at (
(ii)
) is
)
Find the equation of the normal BP. Gradient of the normal at
is
Equation of the normal at (
(iii)
(
) is
Show that B has coordinates ( Coordinates of :
)
).
, (
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(
)
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Y11 MATHEMATICS EXT 1
(iv)
LESSON 6: PARAMETRIC REPRESENTATION 3
is the midpoint of
. Find the coordinates of .
Coordinates of
Hence ( (
(v)
)
(
Coordinates of
)
)
Hence find the Cartesian equation of the locus of C. ; hence
(
(vi)
Describe the locus of The locus of
(a)
A point ( The normal at
(i)
in geometric terms.
is a parabola with vertex at (
directrix has equation
Concept Check 3.1
) is the locus of .
) focal length , focus at (
) and
.
iv
) lies on the parabola cuts the
– axis at point
. .
Find the equation of the normal at .
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
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Y11 MATHEMATICS EXT 1
(ii)
LESSON 6: PARAMETRIC REPRESENTATION 3
Find the coordinates of
.
...................................................................................................................................................
(iii)
is the midpoint of the interval
Find the coordinates of
…………………………………………………………………………………………………………... ...................................................................................................................................................
(iv)
As varies on the parabola, the position of varies and consequently the position of also varies and the points representing traces a path. This path is called the locus of . Find the equation of the locus of .
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(b)
A variable point ( at and the normal at (i)
) lies on the parabola cuts the – axis at .
. The tangent at
cuts the -axis
Find the equation of the tangent at the point
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
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Y11 MATHEMATICS EXT 1
(ii)
LESSON 6: PARAMETRIC REPRESENTATION 3
Find the co-ordinates of
...................................................................................................................................................
(iii)
Find the equation of the normal at .
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
(iv)
Find the coordinates of
.
...................................................................................................................................................
(v)
is the midpoint of interval
. Find the locus of
as
varies on the parabola.
................................................................................................................................................... ................................................................................................................................................... ................................................................................................................................................... ...................................................................................................................................................
Note to Students: Observe how the locus question is almost always the last part.
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Y11 MATHEMATICS EXT 1
(c)
LESSON 6: PARAMETRIC REPRESENTATION 3
The diagram shows a normal drawn to the parabola is the foot of the perpendicular drawn from the focus
(i)
at the point ( to the normal at .
) and
Find the equation of the normal at .
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………..
(ii)
Find the equation of
.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iii)
LESSON 6: PARAMETRIC REPRESENTATION 3
Find the coordinates of
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….… …………………………………………………………………………………………………………… …….…………………………………………………………………………………………………… ……………….…………………………………………………………………………………………
(iv)
Hence find the locus of
as
varies on the parabola.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ………………………………………………….………………………………………………………
(v)
Describe the locus of
in geometric terms.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ………………………………………………….……………………………………………………… …………………………………………………………….…………………………………………… ……………………………………………………………………….………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
4.
LESSON 6: PARAMETRIC REPRESENTATION 3
LOCUS PROBLEMS INVOLVING TWO VARIABLES Example:
The diagram shows the normals drawn to the at the points (
parabola and (
)
), the extremities of the focal
chord
(i)
Find the equation of the chord
(
Equation of (
(ii)
The chord
)
passes through the focus . Show that
Coordinates of the focus:
(iii)
)
(
)
Find the equation of the normal at the point
Gradient of the tangent at (
):
Did You Know: You could also differentiate parametrically. ⁄
Gradient of the normal at
is
the equation of the normal at
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(
)
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Y11 MATHEMATICS EXT 1
(iv)
LESSON 6: PARAMETRIC REPRESENTATION 3
The normal at P and Q intersect at R. Find the coordinates of R.
..................(1) ..................(2) (1) – (2)
(
)
(
)
(
)
(
) after dividing by (
)
Substitute into (1) (
) ( Coordinates of
(v)
Hence find the locus of
as
and
)
(
(
)
(
))
vary on the parabola. (
)
(
) and
(
) ...............(i)
(
)......(ii)
()
(
)
(
)
From (ii) (
(
(vi)
)
)
Describe the locus of R in geometric terms.
Locus of
is a parabola with vertex at (
The focus is at (
). The focal length of the parabola is
.
) and the directrix has equation
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Y11 MATHEMATICS EXT 1
Concept Check 4.1
(a)
v
The points ( angle at the vertex (i)
LESSON 6: PARAMETRIC REPRESENTATION 3
) and ( ) lie on the parabola of the parabola.
Find the gradient of
.
subtends a right
.
…………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….…
(ii)
Hence prove
.
…………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….…
(iii)
is the midpoint of the chord
. Find the coordinates of
…………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….…
(iv)
Find the equation of the locus of as geometric description of the locus of
and
varies on the parabola. Give a
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………………………… …………………………………….……………………………………………………………………
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Y11 MATHEMATICS EXT 1
(b)
LESSON 6: PARAMETRIC REPRESENTATION 3
The diagram shows tangents drawn to the parabola ( ). The tangents intersect at
(i)
at the points (
) and
Find the equation of the tangent at .
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………
(ii)
Find the coordinates of
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………
(iii)
The tangents at and
and
intersect at right angles at
Find a relationship between
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………….………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
(iv)
LESSON 6: PARAMETRIC REPRESENTATION 3
Hence find the equation of the locus of
as
and
vary on the parabola.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………………………………………………
(v)
Describe the locus of
in geometric terms.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ………………………………………………………………………………….………………………
(c)
(
) and (
the parabola at the points
(i)
) are two points on the parabola and
. Tangents drawn to
intersect at
Find the equation of the tangent at
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….…
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Y11 MATHEMATICS EXT 1
(ii)
LESSON 6: PARAMETRIC REPRESENTATION 3
Find the coordinates of the point
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ……………………………………………………………………………………………………….…
(iii)
The point
lies on the line
. Show that
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… ………………………………………………………………………………….………………………
(iv)
is the midpoint of
. Hence find the locus of
.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………
(v)
Describe the locus of M in geometric terms.
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 6: PARAMETRIC REPRESENTATION 3
Past H.S.C. Questions Question 1
(1999 HS.C. Mathematics Extension 1 Q4b) 6
The diagram shows the graph of the parabola . The tangent to the parabola at ( ), , cuts the axis at . The normal to the parabola at cuts the axis at . (i)
Derive the equation of the tangent
.vi
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………
(ii)
Show that
has coordinates (
).
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….……………
(iii)
Let
be the midpoint of
. Find the Cartesian equation of the locus of .vii
…………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… …………………………………………………………………………………………….…………… Discussion Question 2:viii Why is ? Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 6: PARAMETRIC REPRESENTATION 3
NOTES FROM THE MARKING CENTRE (i)
Deriving the tangent, , was a simple application of bookwork using the parametric locus . This part was very well done, although some candidates made it more complicated by attempting to reproduce remembered bookwork for the locus , without letting . As only two marks were assigned to this part, examiners did not insist that candidates use calculus to determine that the slope of the tangent at was . Candidates who simply quoted this fact were able to obtain full marks.
(ii)
Candidates needed to find the equation of the normal in terms of the parameter nad substitute to obtain . Although this was well done, many responses showed attempts to ‘fudge’ the final step from an incorrect equation. Candidates making genuine attempts in questions where the answer is given should ensure that each step is clearly presented to avoid any doubt about its authenticity. For instance, candidates should explicitly state ‘let ’ and show the corresponding substitution clearly.
(iii)
The majority of the candidature found the midpoint (
), but then found it very difficult to
get the second mark by solving simultaneously using
, to obtain the Cartesian form of
locus
. Many candidates obtained the equation
and then made errors
in attempts to simplify this expression. Such errors were ignored, and these candidates were awarded both marks. A number of candidates tried to use the distance formula with no chance of success, as all points on the perpendicular bisector of and .
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. This approach had are equidistant from
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Y11 MATHEMATICS EXT 1
LESSON 6: PARAMETRIC REPRESENTATION 3
SOLUTIONS i
Concept Check 2.1
(a)
(i)
(b)
(ii)
ii
to introduce
and
.
); concave up parabola with vertex (0,18), focus (
Concept Check 3.1
(a)
(b)
(c)
v
(ii)
Concept Check 2.2 (
iv
)
Discussion Question 1
We square iii
(
(i)
x py 2ap ap 3
(iv)
x 2 a y a
(i)
y px 3p 2
(iv)
N 0,6 3 p 2
(i)
x ty 6t 3t 3
(iv)
x 2 3 y 3
(v)
Parabola; Vertex
0,3 ;
(ii)
N 0,2a ap 2
(ii)
T 3p,0
(v)
x2
(ii)
y tx 3
Focus 0,3
), directrix
(iii)
M ap, a ap 2
(iii)
x py 6 p 3 p 3
(iii)
3t,3t
3 y 3 2
2
3
1 3 ; Directrix y 2 4 4
Concept Check 4.1
(a)
(b)
(c)
a p2 q2 a p q , 2
(i)
p 2
(iii)
(iv)
x 2 2a y 4a ;
Parabola ; Vertex
(i)
y px 4 p 2
(ii)
R4 p q ,4 pq
(iv)
y 4
(v)
A line parallel to the x-axis and intersecting the y-axis at
(i)
y px ap 2
(ii)
a p q, apq
(iv)
x 2 2a y 2a
(v)
Parabola; Vertex
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0,4a ;
Page 163 of 213
Focus 0, (iii)
(iii)
7a 9a ; Directrix y 2 2 pq 1
4
pq 2
0,2a ; Focus 0, 5a ; Directrix y 3a
2
2
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Y11 MATHEMATICS EXT 1
LESSON 6: PARAMETRIC REPRESENTATION 3
Past HS.C. Questions vi
(i)
vii
(iii)
viii
Discussion Question 2
You were given that
and
(
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).
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Y11 MATHEMATICS EXT 1
1.
LESSON 7: POLYNOMIALS 1
OVERVIEW OF POLYNOMIALS 1
Complex Numbers
Quadratic Polynomials
Long Division
Conics
Polynomials 1
Roots and Factors
Curve Sketching
Inequalities
Roots of Cubics and Quartics
Polynomials are the simplest and most well behaved of all of the functions. They are trivial to integrate and differentiate and have specific properties which can be exploited to our advantage. In a very real sense they are the building blocks from which all other functions may be constructed.
The general theory of polynomials extends from simple 2 unit quadratic functions all the way to complex numbers in the Extension 2 syllabus. In the Extension 1 papers we see straightforward remainder theorem questions very early in the paper (often in the MCQ section) and deeper division algorithm problems in the middle sections of the exams.
In this lesson we will explore:
–
Polynomial Division
–
The Remainder Theorem
–
Properties of Polynomial Roots and Factors
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Y11 MATHEMATICS EXT 1
2.
LESSON 7: POLYNOMIALS 1
DEFINITION AND NOTATION ( )
is called a polynomial
expression of degree
…,
,
where
is an integer greater than or equal to zero.
are called coefficients and
is called the leading coefficient and
is
called the constant term.
is called the leading term.
When
If
, the polynomial is called a monic polynomial. then the polynomial ( ) is called a zero polynomial.
… ( )
is called a polynomial equation of degree that satisfy the equation ( )
Real values of
are called the real roots or real zeros of
the polynomial.
( ) is a constant polynomial if
Concept Check 2.1 (a)
…
then ( )
.
i
If ( )
then the degree of the polynomial is ………………….
The leading term is………………………… The leading coefficient is ………………….. The constant term is…………………… The coefficient of
is …………………….. and the polynomial has ………… terms.
Note to Students: Always write your polynomials with the biggest power at the front down to the smallest power at the back.
(b)
Write down a monic polynomial of degree 3 that has 4 terms and a constant term of 5.
Note to Students: Monic means that the leading co-efficient is .
……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(c)
Explain why
LESSON 7: POLYNOMIALS 1
is not a polynomial.
……………………………………………………………………………………………………………
(d)
State the degree, leading coefficient and constant term of (
)(
)
Note to Students: Do not expand. Just imagine what would happen if you did expand.
……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
3.
LESSON 7: POLYNOMIALS 1
OPERATIONS WITH POLYNOMIALS Discussion Question 1:ii Let ( ) and ( )
. Are
and
the same polynomial?
Equality
Two polynomials
and are equal if and only if
,
,
, .... ,
,
,
Addition and Subtraction
The sum or difference of two polynomials is found by collecting “like terms”. (
)
given by (
)
(
(
)
) is (
)
(
)
(
)
Multiplication
Use the distributive law to find the product of two polynomials (
)( (
)
)
The degree of ( ) ( ) is the sum of the degrees of ( ) and ( ) where ( ) and ( ) are both polynomials.
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
Division
The long division process may be applied to polynomials in the following way:
Therefore (
and hence where ( (
) is called the quotient, (
)(
)
) is called the divisor and
) is called the remainder
When ( ) is divided by ( ), the degree of ( ) must be less than or equal to the degree of ( ) and the degree of the remainder must be less than the degree of
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( ).
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Y11 MATHEMATICS EXT 1
Concept Check 3.1 (a)
LESSON 7: POLYNOMIALS 1
iii
Expand and simplify (
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ………………………..................................................................................................................
(b)
–
Find the quotient and the remainder when
is divided
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ………………………………………………………........................................................................
Complete the following: –
(
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)(
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)
(
)
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Y11 MATHEMATICS EXT 1
(c)
When values of
LESSON 7: POLYNOMIALS 1
is divided by
the remainder is
. Find the
and
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(d)
Find the values of
given that
is divisible by
.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
4.
LESSON 7: POLYNOMIALS 1
THE DIVISION TRANSFORMATION
When ( ) is divided by ( ) the degree of ( ) must be less than or equal to the degree of ( ).
( ) can be expressed in the form ( )
( ) ( )
( ) where ( ) is called the
quotient, ( ) is called the divisor and ( ) is called the remainder.
The expression, ( )
( ) ( )
( ), is called the division transformation. The
degree of ( ) must be less than the degree of ( ). Concept Check 4.1 (i)
Divide
iv
–
by
and express the result in the division transformation.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Hence find the values of
and
if
is divisible by
.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
Did You Know: The division algorithm should be used as a last resort. There are often much quicker alternatives.
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Y11 MATHEMATICS EXT 1
5.
LESSON 7: POLYNOMIALS 1
THE REMAINDER THEOREM
When ( ) is divided by (
), the degree of the divisor is 1, so the degree of the
remainder must be 0. Hence the remainder is a constant.
The division transformation becomes ( )
Substituting
Hence the remainder theorem states:
(
) ( )
where
is a constant
into the division transformation gives the remainder
When ( ) is divided by (
( )
) the remainder is ( )
Did You Know: The remainder theorem applies exclusively to division by lines.
Concept Check 5.1 (a)
v
Use the remainder theorem to find the remainder when .
–
is divided by
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(b)
When – – is divided by theorem to find the value of .
–
the remainder is 5. Use the remainder
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(c)
LESSON 7: POLYNOMIALS 1
–
Use the remainder theorem to find the remainder when by –
is divided
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(d)
Given that ( – ) ( ) and without actual division.
(
) factorise
–
and hence find
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(e)
Find the remainder when
–
is divided by
–
–
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
6.
LESSON 7: POLYNOMIALS 1
FACTOR THEOREM
When ( ) is divided by (
) and ( )
then ( – ) is a factor of ( ) and a is called
a root of ( ) or a zero of ( )
Proof: By the remainder theorem, ( )
( – ) ( )
( ) Since ( – ) is a factor, then Hence ( ) and ( )
( – ) ( )
Therefore ( –
) is a factor of ( )
Factorisation of Polynomials
Using the remainder theorem, ( )
can be factorised using a as a
possible zero.
( – )(
Hence
)
Therefore ac = 6 and hence a must be a factor of 6 as both a and c are integers. By trial and error the possible zeros of ( ) are
and
.
Testing these values: ( ) Therefore (
( (
)(
and therefore ( )
–
) is a factor. ( )
By inspection,
) )
and hence (
)(
(
)(
since
) )(
)
When factorizing polynomials, the method is basically searching for a factor using trial and error. In this search we try only the factors of the constant term,
to find the value that
makes ( )
If the degree of ( ) is 3, then we need to find only one value as the quotient will be a quadratic and we use inspection to find the quadratic and further factorise if possible.
For polynomials of degree 4, we need to find two zeroes by inspection.
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Y11 MATHEMATICS EXT 1
Concept Check 6.1
(a)
(i)
LESSON 7: POLYNOMIALS 1
vi
Find the value of k if
(
)
(
) is divisible by (
).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
For this value of , factorise
(
)
(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(b)
(i)
Find the values of
LESSON 7: POLYNOMIALS 1
and
given that
is divisible by
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
For these values of a and
factorise
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
7.
DEDUCTIONS FROM THE FACTOR THEOREM
Result 1 ( )
If the polynomial zeroes factor of ( )
) then ( –
(where
)( –
)(
has ) ( –
distinct ) is a
Proof is a zero of ( ) then ( ) = 0 and hence ( ) can be expressed in the form ( )
(
)
is a zero of
( ) by the division transformation ( ) then
(
)
(
)
(
)
. But
since the zeros are
distinct. (
) = 0 and hence ( – ( )
( –
) is a factor of )(
is a zero of ( ) then ( But Hence ( )
)
)
(
( ) and hence
( )
(
)( –
)
)
( )
( ) –
)(
)
(
)
since the zeros are distinct. ( ( –
)
and hence ( – )
) is a factor of
( )
( )
and therefore ( )
( –
)
)(
( )
( –
)(
( )
By repetition of the above argument it can be shown that ( )
( –
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)(
)( –
)
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( –
)
( )
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Y11 MATHEMATICS EXT 1
Concept Check 7.1
(a)
(i)
LESSON 7: POLYNOMIALS 1
vii
Show that 1 and 2 are zeroes of ( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Hence factorise
( ).
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(b)
Show that ( )( ) is a factor of ( ) m and n are positive integers.
(
)
(
)
(
) where
………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… …………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
Result 2 If the polynomial ( )
has n distinct then ( )
zeroes
( –
)(
)( –
( –
)
) where the
is necessary to give the correct leading term of the polynomial i.e.
Proof are distinct zeroes of ( ) then the product
Since ( –
)( –
)(
From result 1, ( )
( –
( –
) is a factor of ( ) )( –
)(
( ) is .
Since the leading term in ( ) is
.
( –
)( –
)(
)
) ( ) where ( ) must have
( –
)
degree zero since the degree of
( )
)
( –
)
Result 3 ( )
A polynomial have more than
of degree
cannot
real zeroes.
Proof Suppose that ( ) has Hence ( )
But ( )
Therefore( –
( –
( –
)(
)(
) ( )
as zeros. )( –
)( –
)
)
( –
( –
for all values of
Hence ( ) cannot have more than
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)( –
) ( )
)
This is impossible since
is a constant.
real zeros if it is a polynomial of degree .
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
Result 4
( )
If polynomial
has more than n and hence ( )
distinct zeroes, then
for all values of .
Proof Assume there are (
) zeroes
and . are zeroes of ( ) then
From result 2, since ( ) But ( )
)(
( –
, and therefore
Hence either But
( –
or
)( –
)
)( –
)(
( – )
)
( –
)
or ………..or
or
is distinct from
therefore
Hence ( )
and since are distinct zeroes of ( ) then ( )
But ( )
, therefore
( –
)(
)( –
( –
)(
)( –
) )
( – ( –
) )
and therefore
since
Proceeding in this manner, we can show that ( )
and therefore
for all values of
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
Result 5
If ( )
( ) are polynomials each of degree , and are equal for all values of x then where
and
are
the coefficients of ( ) and ( ).
Proof Let ( )
and
( )
be two polynomials which are equal
for more than
Then
( )
distinct values of . ( ) = ………………………………………………………………………………….
Therefore { ( )
( ) is a polynomial of degree
with more than
distinct zeroes.
From the remainder and factor theorem, if is a zero of ( ) and of ( ) then () and () ( )
and hence ()
()
. Therefore, is a zero of the polynomial
( )
Using result 4: (
)
, ………………… ……………………… …… (
)
Therefore, , …………………………………………………………
Hence ( )
( ) have equal coefficients for corresponding powers of
and thus ( )
( ) are equal for all values of .
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Y11 MATHEMATICS EXT 1
Concept Check 7.2 (a)
LESSON 7: POLYNOMIALS 1
viii
if the polynomials ( )
Find the values of the constants ( )
(
)(
)
(
)
and
are equal for all values of
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(b)
( )
(i)
Write down (
) and
( )
…………………………………………………………………………………………………………… ………………………………………………………………………………………………………..… ……………………………………………………………………………………………………………
(ii)
If ( ) is an even polynomial, show that
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iii)
LESSON 7: POLYNOMIALS 1
If ( ) is an odd polynomial, show that
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iv)
Hence state general results for the coefficients of odd and even powers of polynomial is even or odd.
when a
Did You Know: Odd polynomials are made up of odd powers. Even polynomials are made up of even powers.
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… Discussion Question 2:ix Let ( ) . Is odd or even? Concept Check 7.3 (i)
Prove that 2,
3 are zeroes of ( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Hence find the linear factors of ( ) x …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
Concept Check 7.4 When the polynomial ( ) is divided by ( ( )
(i)
)(
), the quotient is ( ) and the remainder is
Why is the most general form of ( ) given by ( )
?
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Given that ( )
, show that ( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Further, when ( ) is divided by (
), the remainder is 5. Find ( ).xi
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………..............................…………
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
SOLUTIONS i
Concept Check 2.1
(a) (b) (c)
The degree of
must be an integer equal to or greater than
(d) ii
Discussion Question 1
No iii
Concept Check 3.1
(a) (b)
3x 8
x 7 8x 6 x 5
( )
22 x 4 49 x 3
x2
46 x 45
( )
(c) (d) iv
Concept Check 4.1
(x2
(i)
v
4)( x 3) 8x 11
(ii)
Concept Check 5.1
(a)
(b)
(d)
(e)
vi
(c)
Concept Check 6.1
(a)
(i)
(ii)
(c)
(i)
(ii)
vii
( (
)( ) (
)(
)
)
Concept Check 7.1
(a)
(ii)
( x 1)( x 2)( x 2 5)
viii
Concept Check 7.2
(a) (b) ix
(i)
(
)
( )
Discussion Question
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Y11 MATHEMATICS EXT 1
LESSON 7: POLYNOMIALS 1
x
Concept Check 7.3
(ii)
( x 2)( x 3)( x 1)( x 4)
xi
Concept Check 7.4
(iii)
( )
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Y11 MATHEMATICS EXT 1
1.
LESSON 8: POLYNOMIALS 2
OVERVIEW OF POLYNOMIALS 2
Complex Numbers
Quadratic Polynomials
Long Division
Conics
Polynomials 1
Curve Sketching
Sums and Products of Roots
Inequalities
Roots of Cubics and Quartics
Even though polynomials are relatively simple objects, solving polynomial equations is not a trivial task. Indeed most cubic equations cannot be solved unless we get a lit bit lucky in guessing an initial root. We can then use long division to find the other roots. This process of guessing followed by long division is the only technique we have once we hit the cubic level. Similarly factorising a random polynomial is also an impossible task.
You saw in term 2 however that if a polynomial is presented in factored form, the sketch of its graph is immediately accessible. This theory will be revisited in this lesson.
We will close off this term’s work with the theory of roots and coefficients. Even though most polynomial equations cannot be solved we can still extract surprisingly useful information regarding the roots without actually finding the roots. This is a natural extension of the theory of sums and product of roots from the 2 unit syllabus. It is examined directly in the early stages of the Extension 1 paper and is an essential tool in many of the harder Extension 2 questions where polynomial equations spring naturally from the theory of conics and inequalities.
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Y11 MATHEMATICS EXT 1
2.
LESSON 8: POLYNOMIALS 2
POLYNOMIAL EQUATIONS
The equation ( )
where ( )
is a
polynomial equation of degree .
Any value of Since ( )
that satisfies the equation is called a root or zero of the polynomial. , then we can use the factor theorem to solve the equation.
Example: Solve the equation Note to Students: Our only option is to guess! Try integers which divide into the constant term. Solution: Step 1:
Factorise
Possible factors of 3 are
and
Using the factor theorem, test the values ( (
)
hence (
) or (
) to find a zero.
is a factor
)(
) by inspection
To find the value of , equate the coefficients of
Step 2:
(
)(
(
)(
Use the fact that (
)(
)(
) )(
then
)
or
or
)
or
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Y11 MATHEMATICS EXT 1
Concept Check 2.1
LESSON 8: POLYNOMIALS 2
i
Solve the following polynomial equations Note to Students: Learn to use your calculator efficiently for these repeated calculations. (i)
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(ii)
……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… Copyright © MATRIX EDUCATION 2012
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
(iii) Note to Students: A fourth degree polynomial means that 2 guesses will probably be needed.
……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ……………………………………………………………………………………………………… ………………………………………………………………………………………………………
(iv) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
3.
GRAPHS OF POLYNOMIAL FUNCTIONS
Graphs of the Form
( – )(
)(
)
Note to Students: We have explored this in Term 2, Lesson 2 where we needed to graph equations to solve inequalities. Example: Sketch the curve
(
)(
)(
).
Solution Step 1:
Find the
– intercepts of the curve by letting or 2
Step 2:
Find the
– intercept of the curve by letting (
Step 3:
)(
)( )
Plot these points and sketch the curve (
Curves in the form These curves cut the
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( – )( – axis at
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)(
)(
)(
).
) have distinct zeros.
, and
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Concept Check 3.1 Sketch the following curves (i)
(
)(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
(
)(
)(
)
Note to Students: The does almost nothing. There is certainly no root at
!
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Note to Students: Recall: Even powers bounce off the -axis Odd powers cut the -axis Example: ) (
(
Sketch the curve
)
Solution
Step 1:
Find the
and
intercepts of the curve.
intercepts:
hence
intercept:
Step 2:
or 1
( ) (
,
)
Identify the single and double roots. There is a double root at
. The curve will touch the
There will be a turning point at
There is a single root at
Step 3:
axis at
.
, the curve will cross the
axis at
Sketch the curve Note: Test (
The curves in the form The curve cuts the
( – )(
– axis at
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) (
)
) have a single root at and a double root at
and touches the – axis at Page 196 of 213
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Concept Check 3.2 Sketch the following: (i)
(
) (
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
(
)(
)(
)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
Example:
LESSON 8: POLYNOMIALS 2
(
Sketch the curve
) (
)
Solution:
Step 1:
Find the
and
– intercepts of the curve.
intercepts: intercept:
Step 2:
( )(
) ( )
Identify single and double roots and triple roots There is a double root at be a turning point at
There is a single root at
There is a triple root at
. The curve will touch the
axis at
There will
.
, the curve will cross the
axis at
. The curve has a horizontal point of inflexion on the
axis at
Step 3:
Plot the points and sketch the curve Note: Test (
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) (
)
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Concept Check 3.3 Sketch the following polynomial curves clearly showing any intercepts with the coordinate axes. (i)
(
)(
)
(ii)
(
)(
) (
)
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Y11 MATHEMATICS EXT 1
(iii)
(
)(
LESSON 8: POLYNOMIALS 2
)
Note to Students: Testing a single point beyond the roots will clarify the orientation of the sketch.
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
4.
RELATIONSHIPS BETWEEN THE ROOTS & COEFFICIENTS
Cubic Equations
The roots of the cubic equation
are
The roots of the equation (
)(
Hence
)(
)
(
)(
[
(
)
(
)
are also
)(
and . and .
) (
) (
] )
Equating coefficients of: (
gives
)
(
gives
)
the constant term gives
Hence if
and
are the roots of the cubic equation
The sum of the roots
The sum of the roots taken in pairs
The product of the roots
then:
Did You Know: will help you remember the signs.
(
Important Result to remember:
)
(
)
Proof: (
)
(
(
)(
)
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) (
)
(
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Example: and
are the roots of
. Find the values of:
(i) (ii) (iii)
Did You Know: This is a very common exam question.
(iv) (v) (vi)
(
)(
)(
)
Solution: (i)
(ii)
(iii)
( (
(
)
)
)
(iv) (
)
(v)
(vi)
(
)(
)(
) ( (
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)
(
)
(
)
)
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Y11 MATHEMATICS EXT 1
Concept Check 4.1 (a)
If
LESSON 8: POLYNOMIALS 2
ii
are the roots of
(i)
find:
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii) …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
………………………………………………………………………………………………………… ………………………………………………………………………………………………………… ………………………………………………………………………………………………………… …………………………………………………………………………………………………………
(b)
If
(i)
are the roots of
find:
……………………………………………………………………………………………………………
(ii)
.
……………………………………………………………………………………………………………
(iii)
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
(iv)
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………….…………………………………………………………… ……………………………………………………….…………………………………………
(v)
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………….…………………………………………………………… ……………………………………………………….…………………………………………
(vi)
(
)(
)(
)
………………………………………………………………………………………………… ………………………………………………………………………………………………… …………………………………….…………………………………………………………… ……………………………………………………….………………………………………… ………………………………………………………………………….……………………… …………………………………………………………………………………………….……
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Y11 MATHEMATICS EXT 1
Concept Check 4.2 (a)
LESSON 8: POLYNOMIALS 2
iii
The polynomial equation to 4. Find the roots of the equation.
has the product of two of its roots equal
............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................... ............................................................................................................................................................
(b)
The polynomial ( ) (i)
has real roots √
√ , and .
Explain why
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Show that
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
(iii)
LESSON 8: POLYNOMIALS 2
Show that
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Quartic Equations
If
are the roots of the quadratic equation,
and The equation (
)(
)(
)( (
Hence [ )
(
)
also has roots
) )(
)(
)(
(
.
) )
(
]
Equating coefficients of: (
gives
gives
gives
)
(
)
(
)
The constant term gives
Did You Know: will help you to remember the signs.
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Y11 MATHEMATICS EXT 1
Concept Check 4.3 (a)
If
and
LESSON 8: POLYNOMIALS 2
iv
are the roots of
find:
(i) ……………………………………………………………………………………………………………
(ii) ……………………………………………………………………………………………………………
(iii) ……………………………………………………………………………………………………………
(iv) ……………………………………………………………………………………………………………
(v) …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………….…………………………………………………………………………………………
(vi)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………….…………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
(vii)
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………….………………………………………………………………………………………… ………………………….………………………………………………………………………………
(b)
The roots of the equation 2 x 4
(i)
4 x 2 5x 3 0 are
and
. Find the values of:
……………………………………………………………………………………………………………
(ii)
…………………………………………………………………………………………………………… ……….………………………………………………………………………………………………… ………………….……………………………………………………………………………………… …………………………….……………………………………………………………………………
(iii)
...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
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Y11 MATHEMATICS EXT 1
(iv)
(
LESSON 8: POLYNOMIALS 2
)(
)(
)(
)
...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ...................................................................................................................................................... ......................................................................................................................................................
(v)
The roots of the equation terms of and , the value of (
)(
are , , and . Find, in )( )( )
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
Past H.S.C. Questions Question 1
(2011 H.S.C. Mathematics Q2a)v
The quadratic equation (i)
Find
has roots
and .
.
1
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Find
.
1
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(iii)
Find
.
1
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………….………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
NOTES FROM THE MARKING CENTRE This part was generally done well. The main error was to incorrectly quote the rules for the sum and product of the quadratic roots. When this occurred, the mark for part (iii) could still be obtained for evaluating
from the previous answers. In some responses the irrational
roots were calculated using the quadratic formula and full marks were possible at the expense of time, working and the likelihood of errors. There were a number of non-attempts for part (iii) and quite a few incorrect attempts to add the fractions
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Y11 MATHEMATICS EXT 1
Question 2
LESSON 8: POLYNOMIALS 2
(2006 H.S.C. Mathematics Ext 1 Q4a)vi
The cubic polynomial ( ) real zeroes and – . (i)
, where
Find the value of .
and are real numbers, has three
1
…………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
(ii)
Find the value of
.
2
…………………………………………………………………………………………………………… …………………………………………………………………………………………………………… …………………………………………………………………………………………………………… ……………………………………………………………………………………………………………
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Y11 MATHEMATICS EXT 1
LESSON 8: POLYNOMIALS 2
SOLUTIONS i
Concept Check 2.1 (i) √
(ii)
(iii)
(ii) (ii)
(iii) (iii)
(v)
(vi)
(ii) (v)
(iii) (vi)
(i)
(ii)
(iii)
(iv)
(v)
(iv) ii
Concept Check 4.1 (a) (i) (b) (i) (iv) iii
Concept Check 4.2 (a) iv
Concept Check 4.3 (a) (i) (iv) (vii) (b)
Past H.S.C. Questions v 1. (i) vi 2. (i)
(ii) (ii)
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(iii)
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