Math 360 Notes

Additional Maths Notes (20 Oct 2014) Visit sleightofmath.com for latest notes, solutions and math tuition. Authors: Daniel and Samuel from Sleight of Math Ex 3.1

Table of Contents Ex 1.1

Simultaneous Equations ................................................. 5

Solve a Pair of Linear & Non-linear Eqns .............................. 5 Form Relation ................................................................................... 5 Ex 1.2

Sum and Product of Roots.............................................. 5

Sum & Product of Roots................................................................ 5 Form a Quadratic Equation from its Roots ........................... 5 Useful Formulae............................................................................... 5 Prove Identities involving Roots............................................... 5 Ex 1.3

Discriminant ........................................................................ 5

Complete the Square ...................................................................... 5 Sketch Quadratic graphs .............................................................. 6 Discriminant & Nature of Roots/Number of xintercepts/Number of Intersections ....................................... 6 Conditions for ax 2 + bx + c to be always positive or negative ............................................................................................... 6

Polynomials and Identities.............................................8

Definition of Polynomials ............................................................ 8 Multiply Polynomials .................................................................... 8 Find Unknown(s) in an Identity ............................................... 8 Ex 3.2

Division of Polynomials ...................................................8

Long Division .................................................................................... 8 Division Algorithm ......................................................................... 9 Ex 3.3

Remainder Theorem.........................................................9

Remainder Theorem ..................................................................... 9 Ex 3.4

Factor Theorem ..................................................................9

Factor Theorem ............................................................................... 9 Sum/Difference of Cubes ............................................................. 9 Ex 3.5

Cubic Polynomials and Equations ...............................9

Factorize Cubic Expressions ...................................................... 9 Form Cubic Polynomial ................................................................ 9 Ex 3.6

Partial Fractions .................................................................9

Quadratic Inequalities ..................................................... 6

Break into Partial Fractions ....................................................... 9

Solve Quadratic Inequality .......................................................... 6

Cover-up Rule ................................................................................. 10

Solve Simultaneous Inequalities ............................................... 6

Compare Coefficients .................................................................. 10

Form Quadratic Inequality from Solution............................. 6

Juggling ............................................................................................. 10

Ex 1.4

Ex 2.1

Surds ...................................................................................... 6

Surds Properties .............................................................................. 6

Proper & Improper fraction ..................................................... 10 Ex 4.1

Modulus Functions and their Graphs ...................... 10

Simplify Surds................................................................................... 7

Modulus Definition ...................................................................... 10

Rationalize Denominator ............................................................. 7

Modulus Properties ..................................................................... 10

Solve Surds Equation ..................................................................... 7

Solve Modulus Equations .......................................................... 10

Method of Difference ..................................................................... 7

Sketch y = f(|x|) ............................................................................ 10

Ex 2.2

Indices ................................................................................... 7

Law of Indices ................................................................................... 7 Ex 2.3

Index equations.................................................................. 7

Ex 4.2

Power Graphs ................................................................... 11

Sketch Power Graphs .................................................................. 11 Ex 5.1

Binomial Expansion of (𝟏 + 𝐛)𝐧 ............................... 11

Equality of Indices .......................................................................... 7

Factorial ............................................................................................ 11

Different Types of Manipulation ............................................... 8

Combination ................................................................................... 11

Exponential Functions..................................................... 8

Use Pascal’s Triangle ................................................................... 11

Sketch Exponential Functions.................................................... 8

Expand (1 + b)n ............................................................................ 11

Ex 2.4

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Binomial Theorem Cross-applications ................................ 12 Ex 5.2

𝐧

Binomial Expansion of (𝐚 + 𝐛) ................................12

Equate Coordinates ...................................................................... 17 Ex 8.2

Linear Law ......................................................................... 17

Expand (a + b)n ............................................................................ 12

Linearization................................................................................... 17

Use Tr+1 ............................................................................................ 12

Gradient & Y-intercept ............................................................... 17

Ex 6.1

Mid-point of a Line Segment .......................................12

Scale.................................................................................................... 17

Distance Formula ......................................................................... 12

Graphical Reading ........................................................................ 17

Gradient ........................................................................................... 12

Intersection ..................................................................................... 18

Find line ........................................................................................... 12 Use point on line/curve ............................................................. 13 Ratio of Diagonal Segments ..................................................... 13

Ex 9.1

Graphs of Parabolas of the Form 𝐲 𝟐 = 𝐤𝐱............. 18

Sketch y 2 = kx ............................................................................... 18 Ex 9.2

Coordinate Geometry of Circles ................................ 18

Use Vectors ..................................................................................... 13

Circle Equation .............................................................................. 18

Find Intersection .......................................................................... 14

Circle Equation Cross-applications ....................................... 19

Mid-point Formula ...................................................................... 14

Ex 10.1 Triangle Theorems ............................................................ 20

Ex 6.2

Parallel Lines.....................................................................14

Use Line Addition and Subtraction........................................ 20

Angle of Inclination ..................................................................... 14

Angle Properties of Line(s) ...................................................... 20

Parallel Lines.................................................................................. 14

Angle Properties of Triangles .................................................. 20

Collinearity ..................................................................................... 15

Congruency Tests ......................................................................... 20

Find Parallel Line ......................................................................... 15

Similarity Tests .............................................................................. 20

Ex 6.3

Perpendicular Lines .......................................................15

Perpendicular Lines .................................................................... 15

Mid-point Theorem ...................................................................... 21 Ex 10.2

Quadrilaterals Theorems ........................................ 21

Find Perpendicular Line ............................................................ 15

Definition & Properties of Quadrilaterals........................... 21

Find Perpendicular Bisector .................................................... 15

Prove Quadrilaterals ................................................................... 22

Ex 6.4

Areas of Triangles and Quadrilaterals.....................15

Shoelace Formula ......................................................................... 15 Ex 7.1

Ex 10.3

Circles Theorems........................................................ 22

Angle Properties of Circle.......................................................... 22

Introduction to Logarithms .........................................15

Chord Properties of Circle ......................................................... 22

Logarithm Definition .................................................................. 15

Tangent Properties of Circle .................................................... 22

Special Log Values........................................................................ 15 Convert between Log & Index Form .................................... 16 Ex 7.2

Ex 11.1

Trigo Ratios of Acute Angles .................................. 22

Special Angles................................................................................. 22

Laws of Logarithms ........................................................16

Convert between Degrees and Radians ............................... 23

Laws of Logarithm ....................................................................... 16

Complementary ∡s....................................................................... 23

Ex 7.3

Logarithmic Equations ..................................................16

Supplementary ∡s ........................................................................ 23

Equality of Logarithms............................................................... 16

Identify Quadrant ......................................................................... 23

Solve Log Equations .................................................................... 16

Find Basic Angle α ........................................................................ 23

Ex 7.4

𝐱

Log and Eqns of the form 𝐚 = 𝐛 ..............................16

Find General Angle θ ................................................................... 23

x

Use ⊿ .................................................................................................. 23

Solve a = b .................................................................................... 16 Solve Index Equations ................................................................ 16 Ex 7.5

Ex 11.2

Trigo Ratios of any Angles ...................................... 23

Logarithmic Graphs ........................................................17

Trigo Function Definition .......................................................... 23

Draw Logarithmic Graphs ........................................................ 17

Use ⊿ in Quadrant(s) .................................................................. 23

Ex 8.1

Reducing Equations to Linear Form ........................17

Reciprocal Identities ................................................................... 24

Linearize .......................................................................................... 17

Negative Angles ............................................................................. 24

Form Non-linear Equation ....................................................... 17

ASTC Rule ......................................................................................... 24

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Solve Trigo Eqn f(x) = k ........................................................... 24 Ex 11.3

Trigo Graphs .................................................................25

Solve Trigo Eqn f(x) = k by Graph........................................ 25

Ex 15.2

Increasing and Decreasing Functions ................ 31

Increasing/Decreasing function ............................................. 31 Ex 15.3

Rates of Change........................................................... 31

Range of Sine & Cosine............................................................... 25

Rate of Change ............................................................................... 31

Find Unknowns of Trigo Function af(bx) + c................... 25

Quantity & Constant Rate .......................................................... 31

Sketch Trigonometric Functions ........................................... 25 Use Symmetrical/Cyclical Nature of Trigo Graphs ........ 26 Inverse Trigo Function .............................................................. 27 Ex 12.1

Ex 15.4

Connected Rates of Change .................................... 32

Connected Rates of Change ...................................................... 32 Ex 16.1

Nature of Stationary Points .................................... 32

Simple Identities .........................................................27

Stationary Point/Value ............................................................... 32

Questions involving Identities ................................................ 27

1st Derivative Test ....................................................................... 32

Ratio Identities .............................................................................. 27

2nd Derivative Test...................................................................... 32

Pythagorean Identities............................................................... 28 Square Root of Trigo Function f(x) ....................................... 28 Ex 12.2

Further Trigo Eqns.....................................................28

Simplify to Tangent Eqn ............................................................ 28 Factorize Trigo Eqn ..................................................................... 28 Solve Trigo Eqn f(ax + b) = k................................................. 28 Ex 13.1

The Addition Formulae ............................................28

Addition Formulae ....................................................................... 28 Ex 13.2

The Double Angle Formulae ...................................29

Double ∡ Formulae...................................................................... 29 Half ∡ Formulae ............................................................................ 29 Ex 13.3

Ex 16.2

Maxima and Minima.................................................. 32

Maxima/Minima ............................................................................ 32 Ex 17.1

Derivatives of Trigo Functions.............................. 32

Derivatives of Trigonometric Functions ............................. 32 Ex 17.2

Derivatives of Exponential Functions ................ 32

Derivatives of Exponential Functions .................................. 32 Ex 17.3

Derivatives of Log Functions ................................. 33

Derivatives of Log functions..................................................... 33 Use Logarithmic Differentiation ............................................. 33 Ex 18.1

Indefinite Integrals .................................................... 34

Integral Rules ................................................................................. 34

The R-Formulae ..........................................................29

Find Integral from Derivative .................................................. 34

R-Formulae ..................................................................................... 29

Find Curve from Derivative ...................................................... 34

Ex 14.1

The Derivative and its Basic Rules .......................29

Derivative as Gradient ............................................................... 29

Integrals of Power Functions ................................................... 34 Ex 18.2

Definite Integrals........................................................ 35

Power Rule ...................................................................................... 30

Definite Integrals .......................................................................... 35

Constant Multiple Rule .............................................................. 30

Definite Integrals Rules .............................................................. 35

Sum/Difference Rule .................................................................. 30

Integrals of Modulus Functions .............................................. 35

Differentiation from First Principles.................................... 30 Ex 14.2

The Chain Rule.............................................................30

Chain Rule ....................................................................................... 30 Ex 14.3

Ex 18.3

Integrals of Trigo Functions................................... 35

Integrals of Trigonometric Functions .................................. 35 Ex 18.4

Integrals of Exponential Fns & 1/x ..................... 35

The Product Rule ........................................................30

Integrals of Exponential Functions ....................................... 35

Product Rule ................................................................................... 30

Integrals of 1/x & 1/(ax + b)................................................... 35

Ex 14.4

The Quotient Rule ......................................................31

Quotient Rule ................................................................................. 31 Ex 15.1

Tangents and Normals..............................................31

Find Tangent .................................................................................. 31 Find Normal.................................................................................... 31

Ex 19.1

Area by Integration ................................................... 36

Area by integration ...................................................................... 36 Ex 19.2

Area bounded by Curves ......................................... 36

Strategies to find area bounded by curves ......................... 36 Ex 20.1

Kinematics .................................................................... 36

Tangent Properties ...................................................................... 31

Kinematics Relation ..................................................................... 36

Normal Properties ....................................................................... 31

Implications of Kinematics Statements ............................... 37

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Distance ............................................................................................ 37 Appendix 1

Geometric Formulae .............................................38

2D Shapes ........................................................................................ 38 3D Shapes ........................................................................................ 38 Appendix 2

Trigonometric Identities .....................................39

Appendix 3

Calculus Formulae .................................................40

Differentiation ............................................................................... 40 Integration ...................................................................................... 40

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Additional Math Notes (20 Oct 2014) Useful Formulae

Ex 1.1 Simultaneous Equations

α2 + β2 = (α + β)2 − 2αβ Solve a Pair of Linear & Non-linear Eqns

α−β

Step 1: Subject variable in linear eqn

= ±√(α − β)2

(α − β)2 = (α + β)2 − 4αβ

Step 2: Substitute it into non-linear eqn

α4 + β4 = (α2 + β2 )2 − 2(αβ)2 Form Relation Prove Identities involving Roots

Step 1: Assign variables

To form useful equations (to be substituted),

Step 2: Form relation between variables Ex 1.2 Sum and Product of Roots Sum & Product of Roots Step 1: Simplify to ax 2 + bx + c = 0

Step 3: Find SOR/POR

Product of roots = αβ =

b a

α

a

(iii)

Apply power of n to (1)

(ii) (1)2 :

α4 = (α − 3)2 α4 = α2 − 6α + 9 α4 = α2 − 6(α2 + 3) + 9 [use (1) to make α the subject] α4 = α2 − 6α2 − 18 + 9 α4 = −5α2 + 9 α4 + 5α − 9 = 0 (shown) ✓

α+β 2

Find roots and unknowns e.g. the equation x 2 − 4x + c = 0 has roots which differ by 2. Find the value of each root and c.



To prove existence of positive & negative root, use αβ < 0 Convert to quadratic equation in y by substitution

−(1)

α3 = α2 − 3α −(2) sub (1) into (2): α3 = (α − 3) − 3α α3 = −2α − 3 α3 + 2α + 3 = 0 (shown) ✓

β



2 3

Multiply αn to (1)

(1) × α:

Given context e.g. the heights of two men satisfy 40x 2 − 138x + 119 = 0. Without solving the equation, find the average height of these two men. Average height =



(ii)

Solution (i) ∵ α is root, α2 = α − 3

c

Applications  Evaluate expressions involving its roots 2 2 e.g. find + 

∵ α is a root of ax 2 + bx + c = 0, aα2 + bα + c = 0 −(1)

Question Given that α is a root of the equation x 2 = x − 3, show that (i) α3 + 2α + 3 = 0 (ii) α4 + 5α2 + 9 = 0

Step 2: State roots

Sum of roots = α + β = −

(i)

Ex 1.3 Discriminant

1 3

e.g. x − 2x + 3 = 0 has roots α & β

Complete the Square

1

sub y = x 3 : 1

1

y 2 − 2y + 3 = 0 has roots α3 & β3

k 2

k 2

2

2

x 2 + kx = (x + ) − ( )

Form a Quadratic Equation from its Roots Step 1: State roots Step 2: Find SOR/POR Step 3: Form equation x 2 − (SOR)x + (POR) = 0

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Additional Math Notes (20 Oct 2014) Sketch Quadratic graphs Step 1: Step 2: Step 3: Step 4: Step 5:

Ex 1.4 Quadratic Inequalities

Express as y = a(x − h)2 + k Obtain turning point (h, k) Determine ∪ or ∩ −shape from a Sub x = 0 to find y-intercept Sub y = 0 to find x-intercept

Solve Quadratic Inequality Step 1: Simplify to ax 2 + bx + c vs 0, a > 0 Step 2: Factorize Step 3: Draw sign diagram (Arrange roots & alternate signs with + at left) + − + 𝑥1 𝑥2

Note: x = h is the line of symmetry e.g.

𝑦 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐

Solve Simultaneous Inequalities

𝑥

𝑥1 𝑂2 (5, −3) To find x1 ,

Step 4: Find range of x

2+x1 2

f(x) < g(x) < h(x) ⇒ f(x) < g(x) and = 5 ⇒ x1 = 8

Step 1: Split into 2 inequalities using ‘and’ Step 2: Solve each inequality

Discriminant & Nature of Roots/Number of xintercepts/Number of Intersections

Step 3: Take intersection of both solutions

Step 1: Simplify to ax 2 + bx + c = 0 (by substituting line into curve)

Form Quadratic Inequality from Solution

Step 2: Use relation between b2 − 4ac & nature of roots/x-intercepts/intersections Discriminant 2

b − 4ac > 0 b2 − 4ac = 0 b2 − 4ac ≥ 0 b2 − 4ac < 0

Nature of roots 2 distinct ℝ 2 equal ℝ 2ℝ 0ℝ

g(x) < h(x)

No. of x-intercepts/ intersections 2 1 (tangent) 1 or 2 (meet) 0

Conditions for ax 2 + bx + c to be always positive or negative ax 2 + bx + c > 0 for all x ↔ a > 0,

b2 − 4ac < 0

ax 2 + bx + c < 0 for all x ↔ a < 0,

b2 − 4ac < 0

Step 1: Simplify inequality to ax 2 + bx + c vs 0 Step 2: Use 2 conditions (i) a > 0 or a < 0 (ii) b2 − 4ac < 0

⇒ k(x − x1 )(x − x2 ) < 0

x1 < x < x2

x < x1 or x > x2 ⇒ k(x − x1 )(x − x2 ) > 0 Ex 2.1 Surds Surds Properties For a > 0 and b > 0, 

√a × √b = √ab



√a √b



√a × √a = a

=√

a

b

Notation n For √x, n ≡ index x ≡ radicand √ n

√x

≡ radical sign or radix or root symbol n ≡ surd (if √x is irrational) n

Note: For √x and x < 0, Even n results in non-real number Odd n results in real number 3 e.g. √−4 does not exist but √−8 exists

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Additional Math Notes (20 Oct 2014) Simplify Surds

Ex 2.2 Indices

Factor out largest square number Law of Indices

e.g. √45 = √9 × 5 = 3√5 Prime factorize (for more challenging numbers) e.g. √540 = √22 × 33 × 5 = 2 × 31.5 × √5 = 2 × 3√3 × √5 = 6√15 Rationalize Denominator 1



a0 = 1



a−n =



a n = ( √a) = √am



(am )n = amn



am × an = am+n

m

√a √a = a √a √a 1 1 1 × = 2 a√h + b√k a√h − b√k a h − b 2 k

n

= am−n

an



m

n

am



×

1 an

Same Base

an × bn = (ab)n a n

an



=( )

bn

Same Power

b

When you multiply/divide terms, identify common base/power

Solve Surds Equation

1

1

Square both sides √a = b ⇒ a = b2

e.g.

Equate rational & irrational terms a + b√k = c + d√k ⇒ a = c, b = d

e.g. ( √√a3 + b 2 + b) ( √√a3 + b 2 − b)

33 ×30 ×93 2

(common base is 3)

273 3

3

1

Note: Check the answer mentally by substituting it into the original equation. e.g. √6 − 5x = −x 6 − 5x = x2 2 x + 5x − 6 = 0 (x + 6)(x − 1) = 0 x = −6 or x = 1 (rej) When x = 1, LHS = √6 − 5 = 1 RHS = −1 LHS ≠ RHS

(common power is ) 3

When you add/subtract terms, identify highest common factor e.g. 8x+2 −34(23x ) 3 x+2 = (2 ) −2 × 17(23x ) = 23x+6 −17(23x+1 ) (HCF is 23x+1 ) = 23x+1 (25 − 17) = 23x+1 (15) Note: Equations involving even power functions may have multiple solutions e.g. x 4 = 16 ⇒ x = 2 or x = −2

If you cannot simplify to √a = b or a + b√k = c + d√k, consider solving surds equation by substitution e.g. 2x + 3√x + 1 = 0 sub u = √x: 2u2 + 3u + 1 = 0

Ex 2.3 Index equations Equality of Indices ax = an , for a > 0, a ≠ 1 ⇒x =n

Method of Difference Step 1: Break each term into partial sums Step 2: Arrange partial sums vertically Step 3: Cancel diagonally

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Additional Math Notes (20 Oct 2014) Different Types of Manipulation Manipulate Simplify Express Evaluate Show Solve Given

Find Unknown(s) in an Identity

Key Words Complex to simple In terms of … Find numerical value Work towards distinct characteristic Equation Consider rearranging given equation.

Ex 2.4 Exponential Functions Sketch Exponential Functions y = ax , a > 1 (slopes up) x

y=a , 02 −2a < −4 3 − 2a < −1 1 (slopes up)

Question

𝑂 1

𝑦 y = log a x, 0 b > c > 1, 𝑦 𝑦 = log 𝑎 𝑥 𝑦 = log 𝑏 𝑥 𝑂 𝑥 𝑦 = log 𝑐 𝑥 1

Ex 8.2 Linear Law Linearization Step 1: Simplify to Y = mX + c Step 2: Complete table Gradient & Y-intercept

Ex 8.1 Reducing Equations to Linear Form

Step 1: State 2 points: (i) On y-axis (ii) Halfway-down

Linearize Contains x&y X&Y ✓ m&c ✘

Contains constants ✘ ✓

Step 2: Equate gradient & Y-intercept Scale a

1

b

b

e.g. if ax 2 + by 3 = 1, then y 3 = − x 2 + a

1

b

b

i.e. Y = y 3 , X = x 2 , m = − , c = b+x a

e.g. if y = e

1

b

a

a

, then ln y = x + 1

b

a

a

i.e. Y = ln y , X = x, m = , c =

Step 1: Estimate Y-intercept Y1 = mX1 + c c = Y1 − mX1 Step 2: State domain & range Step 3: Find X & Y interval X −X X-Interval = last 1st 10 Ylast −Y1st

Y-Interval = 12 (Round down to 1, 2, 25 or 5)

To find unknowns, Step 1: Linearize to axes variables Step 2: Equate gradient & Y-intercept or use points on line (whichever is given)

Step 4: State X & Y scale Graphical Reading

Step 1: Find point/gradient/ Y-intercept

Step 1: Simplify to (X or Y) Step 2: Identify point Step 3: Equate (Y or X) & solve for desired variable

Step 2: Form linear equation Y = mX + c (If Y-intercept is given) Y − Y1 = m(X − X1 ) (If Y-intercept is not given)

Note: Graphical reading is reliable only within the data range (interpolation) & not reliable outside the data range (extrapolation)

Form Non-linear Equation

Step 3: Form non-linear eqn by replacing X & Y with axes variables

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Additional Math Notes (20 Oct 2014) Intersection

Ex 9.2 Coordinate Geometry of Circles

Step 1: Work towards 2 curves on each side Step 2: Plot 2nd curve & use intersection

Circle Equation 

Standard form (x − a)2 + (y − b)2 = r 2

Ex 9.1 Graphs of Parabolas of the Form y = kx



General form

x 2 + y 2 + 2gx + 2fy + c = 0

Sketch y 2 = kx



Centre

(a, b) = (−g, −f)



Radius

r = √g 2 + f 2 − c

2

2

𝑦

y = kx, k>0 𝑂

Note: It appears to be a counter-intuitive convention that g comes before f in the formula

𝑥

Note: y = 0 is the line of symmetry

Trigger/Setup

Action

Question Given the graph y 2 = 2x, draw a suitable line to solve x 2 − 8x + 9 = 0.

2 points

Find ⊥ bisector of chord where centre lies on

Centre & point

Use distance formula to find radius.

Solution x 2 − 8x + 9 x 2 − 6x + 9 (x − 3)2 ⇒y=x−3

Diameter

Use midpoint formula

=0 = 2x = 2x or y = −(x − 3) ✓

Touches Sketch graph. Deduce horizontal/vertical coordinates, centre, radius or line point on circle. (see example) Right angle triangle drawn

Use Pythagoras’ Theorem

0,1 or 2 intersections

Use discriminant.

Touches another circle

Connect centres with a line.

Line is tangent to circle

Identify right angle (tan ⊥ rad) Find normal.

If the centre cannot be found from the approaches above (or only 1 coordinate can be deduced),  use given information about centre (if any) e.g. centre C(h, k) lies on line y = f(x) ⇒ C is (h, f(h)) e.g. centre C(h, k) is 6 units away from point A(1,2) ⇒ √(h − 1)2 + (k − 2)2 = 6 

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insert parameters into (x − h)2 + (y − k)2 = r 2 and solve for unknowns by elimination.

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Additional Math Notes (20 Oct 2014) Examples of sketching graph to deduce information Touch axis Given: centre (3, −2), touches x-axis Deduce: Cut axis Given: Deduce:

𝑂

radius = 2

𝑥 2 (3, −2) 𝑦

Cuts y-axis at −2 and −5 y − coordinate of centre −1+(−5) = = −3 2

Touch line(s) Given: Touches x = 2 & x = 8 Deduce:

𝑦

radius =

8−2 2

𝑂 𝑥 −1 𝐶(𝑐1 , −3) −5

2

3 𝐶(5, 𝑐2 ) 𝑂𝑥=2 𝑥=8𝑥

Pythagoras Theorem  Find length of PT, given radius is √13. C(2, −1)

Find ⊥ line  Find tangent/normal at point of contact e.g. Find AB

T

P(3, −10) 2√10

Idea: Find PC by distance formula PT = √PC 2 − CT 2 (Pythagoras’ thm) 

Use Distance Formula  Find radius  To check if point A lies within circle, compare distance between A and centre with the radius Use Midpoint Formula  Given that A(2,3) and B(4,5) are points on the circle, the find the centre.  Given A is (2,3), the centre C is (4,5) and AB is the diameter of the circle, find the point B

Circle Equation Cross-applications

P(9,2)

Find Intersection Point  Find point on circle  Find point of contact between tangent & normal  Find centre where line through centre meets perpendicular bisector of chord

𝑦

=3

x − coordinate of centre 2+8 = =5

Use Discriminant  Find number of intersections between line & circle (you can also compare the perpendicular distance with the radius to determine the number of points of intersection)  Find unknown c in line eqn given line is tangent to circle

AB

Find AC 𝑦

C(1, −4)

Find ⊥ bisector  Whenever two points on circle are given, consider finding the perpendicular bisector. The perpendicular bisector of the chord passes through the centre of the circle

6B r = 5 A C 2 𝑥 𝑂 Idea: AC = √r 2 − AB 2 (Pythagoras’ thm)

Use Properties of Circle (refer to Ex 10.3) Solve System of Equations  To find circle equation given 3 points on the circle, insert the points into general form of circle

𝑂 A B ⊥ bisector of chord

tan ⊥ rad

Complete the Square  Convert general form to standard form

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Additional Math Notes (20 Oct 2014) Similarity Tests

Ex 10.1 Triangle Theorems Use Line Addition and Subtraction A

B

C

SSS 3 ∝ sides

AB + BC = AC AC − AB = BC

Question (prove product of sides) Given △ ABC ~ △ DEF, prove that AB × DF = AC × DE

Question Given AB = CD, prove AC = BD A B

Solution Whenever you encounter product of multiple line Given △ ABC ~ △ DEF, segments, consider using the prove that property of similar triangles: AB × DF = AC × DE ratios of corresponding sides are equal.

C D

Solution AB = CD AB + (BC) = CD + (BC) AC = BD ✓ Angle Properties of Line(s) a b

SAS AA 2 ∝ sides, 2 eq. ∡ 1 included ∡

b

a

a

b a

b

AB AC

…

Identify which line segments in the above product correspond to the triangle ABC. AB and AC. AB Take ratio at the left. Note the AC sequence. AB is 12 and AC is 13. 12 over 13.

=

Use same sequence on the other triangle DEF at the right. 12 is DE DE and 13 is DF. Take ratio at DF the right.

ab

∡s in line opp. ∡ int. ∡ corr. ∡ alt. ∡  

Prove straight lines by ∡s in line = 180° Prove parallel lines by int.∡, corr. ∡ & alt. ∡

AB AC

DE DF

Angle Properties of Triangles b a c ∡s in △ = 180° 

a

bc

ext. ∡ = sum of int. opp. ∡s

ab ⊿

ab

c ab

AB × DF = AC × DE [proven] ✓

iso.△ eq.△

Prove equal sides/angles using iso.△ & eq.△ Congruency Tests

Question (Prove relation/ratio of line segments) Given △ ABC ~ △ DEF & DE: EF = 1: 2, prove that 1 AB = BC (or AB: BC = 1: 2) 2

Solution AB BC

= =

SSS 3 eq. sides

SAS AAS RHS 2 eq. sides, 2 eq. ∡s, 1 rt ∡, 1 included ∡ 1 corr. 1 eq. hyp, sides 1 eq. side Note: Order of Points matter e.g. △ ABC ≅ △ XYZ is not the same as △ ACB ≅ △ XYZ 

Cross multiply.

DE EF 1 2 1

⇒ AB = BC ✓ 2

⇒ AB: BC = 1: 2 ✓

Prove equal sides/angles using congruent △s

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Additional Math Notes (20 Oct 2014) Question (Use ratio of area of similar triangles) Given △ ABC ~ △ EDC, B 1 E BC: CD = 1: 2 & C area of △ ABC = x, A 2 D find the area of △ DEC Solution 2 2

Area of △ DEC = ( ) x = 4x ✓ 1

[use

Mid-point Theorem D = MAB , E = MAC 1 ⇒ DE ∥ BC, DE = BC 2

A2

l

2

= ( 1) ] l2

Definition & Properties of Quadrilaterals Kite Quad. with two pairs of equal adjacent sides  ∡s between unequal sides are equal (angle)  One diagonal bisects the other (diagonal)  Longer diagonal bisects ∡s (diagonal)  Diagonals are ⊥ (diagonal) Note: Concave kite have interior ∡s > 180°

A D B

A1

Ex 10.2 Quadrilaterals Theorems

E C

Trapezium Quad. with exactly one pair of parallel sides  supplementary interior ∡s Parallelogram Quad. with two pairs of parallel sides  Opp. sides are equal  Opp. ∡s are equal  interior ∡s are supplementary  Diagonals bisect each other

(side) (angle) (angle) (diagonal)

Rectangle Quad. with four right angles  Opp. sides are parallel  Opp. sides are equal  Diagonals bisect each other  Diagonals are equal

(side) (side) (diagonal) (diagonal)

Rhombus Quad. with four equal sides  Opp. sides are parallel  Supplementary interior ∡s  Diagonals bisect ∡s Diagonals are ⊥ bisector of each other

(side) (angle) (diagonal) (diagonal)

Square Quad. with four equal sides & four right angles  Diagonals bisect angles (diagonal)  Diagonals are equal (diagonal)  Diagonals are ⊥ bisector of each other (diagonal)

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Additional Math Notes (20 Oct 2014) Tangent Properties of Circle

Prove Quadrilaterals Parallelogram  2 pairs of ∥ sides  2 pairs of equal & opp. sides  1 pair of equal & ∥ sides  2 pairs of equal opp. ∡s  Diagonals bisect each other

(definition) (side) (side) (angle) (diagonal)

Rectangle  4 right ∡s  Parallelogram + 1 right ∡

(definition) (angle)

Q b

P

a

R tangents from ext. point

alt. segment tan ⊥ rad thm

Ex 11.1 Trigo Ratios of Acute Angles Special Angles

Rhombus  4 equal sides (definition)  Parallelogram + eq. adj. sides (side)  Parallelogram + bisecting diagonals (diagonal)  Parallelogram + ⊥ diagonals (diagonal)

Table

Square  4 equal sides & 4 right ∡s  Rectangle + eq. adj sides  Rhombus + 1 right ∡

tan θ 0

0° 30° 45° 60° 90° 0 sin θ 0 cos θ 1

(definition) (side) (angle)

π

π

π

π

6 1

4

3

2

√2 2

√3 2 1

1

2 √3 2 1 √3

√2 2

2

0

1 √3 ∞

Triangle

√2

Trapezium  Parallel opposite sides

O

1

(definition)

Kite  2 pairs of equal adjacent sides (definition)

45° 1 Unit circle

30°

2

√3

60°

60°

1

Ex 10.3 Circles Theorems Angle Properties of Circle a

b

a b

a ∡ in semicircle

b a

∡ at centre ∡s in same ∡s in opp. = 2 ∡ at segment segment circumference

Chord Properties of Circle

A

O B

C

⊥ bisector of chord passes through centre 

A C

X

B

O Y D

Equal chords are equidistant from centre

Equal arcs results in equal chords

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Additional Math Notes (20 Oct 2014) Find Basic Angle α

Convert between Degrees and Radians π rad = 180° 

To convert from degrees to radians, multiply



To convert from radians to degrees, multiply

π 180° 180°

Tip: Track the unit conversion to avoid the mistake of multiplying the wrong fraction e.g. 60° = 60°

×

π

1

180°

= π 3

[rad]

[deg] = [deg] × [deg] = [rad]

60° = 60°

×

[deg] = [deg] ×

180° π [deg] [rad]

= =

10800 π [deg]2 [rad]

π

Step 1: Add or subtract 360° until 0° ≤ θ ≤ 360° Step 2: Use table Quadrant α 1 θ 2 180° − θ 3 θ − 180° 4 360° − θ Find General Angle θ Quadrant 1 2 3 4

✓ ✓

✘ ✘

θ α 180° − α 180° + α 360° − α

Use ⊿ Step 1: Draw ⊿ Step 2: Find all 3 sides (by Pythagoras’ Thm)

Complementary ∡s

Ex 11.2 Trigo Ratios of any Angles



sin(90° − θ) = cos θ



cos(90° − θ) = sin θ



tan(90° − θ) =

1 tan θ

Supplementary ∡s

Trigo Function Definition y



sin θ =



cos θ =



tan θ =

r = √x 2 + y 2

r x r y

r θ

y x



sin(180° − θ) = sin θ



cos(180° − θ) = − cos θ





tan(180° − θ) = − tan θ

Angles measured anti-clockwise from the positive x-axis are positive. On the contrary, angles measured clockwise from the positive x-axis are negative.

Identify Quadrant Step 1: Add or subtract 360° until 0 ≤ θ ≤ 360° Step 2: Use table Angle 0° < θ < 90° 90° < θ < 180° 180° < θ < 270° 270° < θ < 360°

Quadrant 1 2 3 4

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x

Use ⊿ in Quadrant(s) Step 1: Identify quadrant Step 2: Draw ⊿ in quadrant Step 3: Find coordinates

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23

Additional Math Notes (20 Oct 2014) Question

Reciprocal Identities

Given that tan A = −

5 12

and that tan A and cos A have

opposite signs, find the value of each of the following. (i) sin(−A) (ii) cos(−A) π

(iii) tan ( − A)



sec θ =



csc θ =



cot θ =

1 cos θ 1 sin θ 1 tan θ

2

Negative Angles

Solution Thought Process Step 1: Identify quadrants Observe that ratio for tan is negative. 5 tan A = − < 0 Tan is only positive in 1st or 3rd quad. 12 Therefore, it is in 2nd or 4th quad. ⇒ 2nd or 4th quad. tan A & cos A have In 3rd quad., only tan is positive opp. signs In 4th quad., only cos is positive ⇒ 3rd or 4th quad. Therefore, it is in 3rd or 4th quad. Take overlap of above deductions. Therefore it is in 4th quadrant.

∴ 4th quadrant Step 2: Draw ⊿ in quadrant

cos(−θ) = cos(θ)



sin(−θ) = − sin(θ)



tan(−θ) = − tan(θ) ASTC Rule

sin is + S A all are + tan is +T C cos is + All trigo functions can be converted to trigo function of basic angle with positive or negative sign depending on ASTC rule. e.g. sin(210°) = − sin(30°) Solve Trigo Eqn f(x) = k Quadrants method Step 1: Find α = f −1 (|k|) & identify quadrants Step 2: State interval Step 3: Find x using quadrants

Draw ⊿ in 4th quadrant.

12 r



−5

Step 3: Find coordinates 5 y tan A = − = 12

180° − α 2 1 α 180° + α 3 4 360° − α

y

tan A = by definition. x

π−α2 1α π + α 3 4 2π − α

x

y = −5,

Equate numerator, 𝑦 = −5. y-coordinate is negative in 4th quad. Equate denominator, 𝑥 = 12. x-coordinate is positive in 4th quad.

x = 12,

r = √122 + (−5)2 Find hypotenuse r by Pythagoras’ Theorem = 13 y

5

r x

13 12

r

13

sin A = = − cos A = =

,

Find other trigo ratios to serve as useful inputs. The rest of the question makes use of the 3 basic trigo ratios: sin A , cos A & tan A.

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Additional Math Notes (20 Oct 2014) Sketch Trigonometric Functions

Ex 11.3 Trigo Graphs

Step 1: Simplify to y = af(bx) + c Solve Trigo Eqn f(x) = k by Graph

Step 2: Find amplitude & period Sin/Cos Tan |a| Amplitude Nil

Graphical method When α = 0° or 90°, i.e.

Period

sin f(x) = 0, ±1

tan f(x) = 0 Step 1: State interval Step 2: Find x using graph y = cos x y = tan x

𝑦

𝑦

𝑦

1

1

1

180° 90°

360°

180°

𝑥

270°

−1

90°

270°

180°

𝑥

b

b

y = sin x

360°

−1

π

Step 3: Complete table and sketch graph Domain x1 ≤ x ≤ x2 Axis with y = c ± |a| Amplitude Shape ±sin/cos/tan x2 −x1 Cycle T

cos f(x) = 0, ±1

y = sin x

2π

270°

90°

𝑥 1

−1

−1

Range of Sine & Cosine

1

𝑦

1

1

𝑥

−1

90° 180° 270° 360°

𝑥

90° 180° 270° 360°

𝑥

−1

90° 180° 270° 360°

𝑥

−1

sin x = −1 at x = 270° sin x = 1 at x = 90°

Max

𝑥

sin x = 0 at x = cos x = 0 at x = tan x = 0 at x = 0°, 180°, 360° 90°, 270° 0°, 180°, 360° sin x = −1 cos x = −1 Min Nil at x = 270° at x = 180° sin x = 1 cos x = 1 Max Nil at x = 90° at x = 0°, 360°

y = cos x

𝑦

Min

𝑦

0

−1 ≤ sin x ≤ 1 −1 ≤ cos x ≤ 1 y = sin x

90° 180° 270° 360°

y = tan x

𝑦

90° 180° 270° 360°

 

y = cos x

𝑦

360°

cos x = −1 at x = 180° cos x = 1 at x = 0°, 360°

Find Unknowns of Trigo Function af(bx) + c Sine/Cosine

Tangent

max A = |a|

c

c

A = |a| min T=

360°

T=

b

Amplitude A = |a| =

max−min

360°

Period

T=

Axis

c= 2 = min + A = max − A

2

180° b

Period T =

180° b

b max+min

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Additional Math Notes (20 Oct 2014) Question

Use Symmetrical/Cyclical Nature of Trigo Graphs

Sketch y = 3(1 − 2 cos 4x) for 0° ≤ x ≤ 270°

Symmetrical Question Given α & β are roots of 3 cos x + 2 = 2 where 3 < k < 4. Find β in terms of α, given that α < β

Solution Step 1: Simplify to 𝐲 = 𝐚𝐟(𝐛𝐱) + 𝐜 y = 3(1 − 2 cos 4x) = 3 − 6 cos 4x = −6 cos 4x + 3

Solution y

Step 2: Find amplitude & period A = |−6| = 6 T=

360° 4

𝑦 = 3 cos 𝑥 + 2

5

= 90°

2

Step 3: Complete table and sketch graph Domain 0° ≤ x ≤ 270° Follow the Axis with y=3±6 sequence from Amplitude top down to Shape −cos sketch the graph. 270−0 =3 Cycle 90

𝑦

𝑂

𝜋

2𝜋

x

-1 ∵ x = π is line of symmetry, α+β =π 2

β = 2π − α ✓ Cyclical

𝑂

270°

Mark the endpoint of domain, 270°. 𝑥

𝑦 9 3 𝑂

270°

−3

𝑥

Mark the axis 3. Add and subtract 6 to get max 9 and min −3.

Question Given that α is the smallest positive root of the equation √2 cos 4x = −3.1 tan 2x, where 0° ≤ x ≤ 360°, state the other roots in terms of α. Solution 𝑦 𝑦1 = −3.1 tan 2𝑥 √2

𝑦 9

Draw 1 cycle of negative cosine.

3 𝑂

270°

−3

𝑂 −√2

𝑥

90°

𝑥 = 45°

𝑦2 = √2 cos 4𝑥 𝑥 180°

𝑥 = 135°

∵ Period = 90°, ⇒ x = α, α + 90°, α + 180°, α + 270° ✓

𝑦 9 3 𝑂

−3

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270°

There are 3 cycles in total. Draw 2 more. 𝑥

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26

Additional Math Notes (20 Oct 2014) Note: If basic angles cannot be found, use ⊿ in quadrant

Inverse Trigo Function

Question (Evaluate compound inverse trigo functions)

Principal values π

π

2

2



− ≤ sin−1 x ≤



0 ≤ cos −1 x ≤ π π



− < tan

−1

2

x<

1

Find the exact value of sin [cos −1 (− )] 5

Solution π

Step 1: Identify quadrant

2

1

Let A = cos −1 (− ) 5 ⇒ 2nd quadrant

Step 1: Identify quadrant Step 2: Find basic angle Step 3: Find general angle

Since 0 ≤ cos −1 x ≤ π, it is in the 1st or 2nd quad. Because of the negative sign 1 of − , it is in 2nd quad. 5

Step 2: Draw ⊿ in quad.

Question 1

Evaluate cos −1 (− ) without using the calculator. 2

𝑦 5 −1

Solution

✘ ✘✘

1

cos −1 (− ) 2

π

= (π − ) 3

=

2π 3

✓

𝜋

Step 3: Find coordinates

3

cos A = − =

Thought process Step 1: Identify quadrant 0 ≤ cos −1 x ≤ π Strike out 3rd and 4th quadrants

✘✘ 1

Input − is negative. 2 1st quadrant always corresponds to positive ratios. Strike out 1st quadrant

Step 2: Find basic angle Mentally use table of special angles and ignore the negative 1

sign of − , 0 cos θ 1

2 π

π

π

π

6

4

2

√3 2

√2 2

3 1

⇒ basic angle =

✘ ✘✘

x

5

r

x

cos A = by trigo definition r

x = −1,

Equate numerator: x = −1. x-coordinate is negative in 2nd quad.

r=5

Equate denominator: r = 5. r is always positive.

y = √52 − (−1)2 = √24 = √4 × 6 = 2√6

Find y by Pythagoras’ Thm

1

sin [cos −1 (− )] y

5 2√6

r

5

= sin A = =

y

sin A = by trigo definition ✓

r

Ex 12.1 Simple Identities

✘ ✘✘ 𝜋 3

0

2

1

Questions involving Identities 

Simplify using identities



Evaluate using identities



Prove identities

π

Ratio Identities

3

Step 3: Find general angle General angle is the angle wrt the positive x-axis. ACW is positive. CW is negative. π General angle = π − 3

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✘ ✘✘ 𝜋



tan θ =



cot θ =

3

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sin θ cos θ cos θ sin θ

27

Additional Math Notes (20 Oct 2014) Pythagorean Identities

Factorize Trigo Eqn



sin2 θ + cos 2 θ = 1



tan2 θ + 1

= sec 2 θ



cot 2 θ + 1

= csc 2 θ

Question Given that sin x + sin y = a and cos x + cos y = a, where a ≠ 0, express sin x + cos x in terms of a.



Take out common factor 2 sin x cos x − sin x = 0 sin x (2 cos x − sin x) = 0



Express in factor form cos 2 x − cos x − 2 =0 (cos x − 2)(cos x + 1)= 0



Factorize by grouping 3 sin x tan x − 12 sin x −2 tan x + 8 3 sin x (tan x − 4) −2(tan x − 4) (tan x − 4)(3 sin x − 2)

Solution sin x + sin y = a ⇒ sin y = a − sin x cos x + cos y = a ⇒ cos y = a − cos x

−(1) −(2) 

(1)2 + (2)2 : sin2 y + cos 2 y= (a − sin x)2 + (a − cos x)2 2

= (a − sin x) + (a − cos x)

1

tan x =

sin x + cos x

−(−2)±√(−2)2 −4(1)(−2)

= 1 ± √3

2(1)

Solve Trigo Eqn f(ax + b) = k

= 2a2 − 2a(sin x + cos x) + sin2 x + cos 2 x

0

If unable to factorize, use quadratic formula tan2 x − 2 tan x − 2 = 0

2

= (a2 − 2a sin x + sin2 x) +(a2 − 2a cos x + cos 2 x)

Quadrants Method Step 1: Find α & identify quadrants

= 2a2 − 2a(sin x + cos x) + 1

Step 2: Adjust interval

= 2a2 − 2a(sin x + cos x)

Step 3: Find ax + b & x using quadrants

= a − (sin x + cos x)

180° − α 2 1 α 180° + α 3 4 360° − α

=a

Square Root of Trigo Function f(x) 

f(x) ≥ 0: √[f(x)]2 = f(x)



f(x) < 0: √[f(x)]2 = −f(x)

The output of square root is positive by definition e.g. √sin2 x = sin x

for 0 < x < 90°

√sin2 x = − sin x for 180 < x < 270°

Graphical Method When α = 0° or 90° i.e. sin f(x) = 0, ±1 cos f(x) = 0, ±1 tan f(x) = 0 Step 1: Adjust interval Step 2: Find ax + b & x using graphs y = sin x

Ex 12.2 Further Trigo Eqns

y = cos x y = tan x

𝑦

𝑦

𝑦

1

1

1

180° 90°

Simplify to Tangent Eqn

=0 =0 =0

360°

270°

−1

180°

𝑥

90°

360°

270°

−1

180°

𝑥

90°

360°

270°

𝑥

−1

Step 1: Separate sin & cos to opp. sides of eqn Step 2: Divide by cos x

Ex 13.1 The Addition Formulae

e.g. a sin θ + b cos θ = 0 a sin θ = −b cos θ

Addition Formulae

tan θ

=−

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b a



sin(A ± B) = sin A cos B ± cos A sin B



cos(A ± B) = cos A cos B ∓ sin A sin B



tan(A ± B) =

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tan A±tan B 1∓tan A tan B

28

Additional Math Notes (20 Oct 2014) Ex 13.2 The Double Angle Formulae

Ex 13.3 The R-Formulae

Double ∡ Formulae

R-Formulae



a sin θ ± b cos θ = R sin(θ ± α)

sin 2A = 2 sin A cos A 1 ⇒ sin A cos A = sin 2A

a cos θ ± b sin θ = R cos(θ ∓ α)

2





cos 2A = cos 2 A − sin2 A = 2 cos 2 A − 1 = 1 − 2 sin2 A ⇒ cos 2 A =

1+cos 2A

⇒ sin2 A =

1−cos 2A

tan 2A =

Tip:



R = √a2 + b 2



α = tan−1 ( )



min = −R, max = R

b a

2

Ex 14.1 The Derivative and its Basic Rules

2

2 tan A

Derivative as Gradient

1−tan2 A

The gradient of the curve y = f(x) at (x1 , y1 ) is

As cos 2A has 3 possible outputs, the output that eliminates 1 is often chosen

Note: Most of the time,

Question

dy dx

dy

|

dx x=x1

is a function of x.

To find the gradient we need the x-input.

Show 1 + cos 2A = 2 cos 2 A

Question Calculate the gradient(s) of the curve at the point(s) where y is given. y = 2x 2 + 3x, y = 2.

Solution LHS = 1 + cos 2A Do not use: 2 2 [(i) cos 2A = cos A − sin A] 2 (ii) cos 2A = 1 − 2 sin A

= 1 + (2 cos 2 A − 1)

Solution y = 2x 2 + 3x dy dx

= 2 cos 2 A = RHS

= 4x + 3

At y = 2, 2x 2 + 3x =2 2 2x + 3x − 2 = 0 (2x − 1)(x + 2) = 0

Half ∡ Formulae 1

1

2

2

To find sin A, use cos A = 1 − 2 sin2 ( A) 1

1−cos A

2

2

⇒ sin A = ±√ 1

To find cos A, use cos A = 2

1 2 cos 2 ( A) 2

1

1+cos A

2

2

⇒ cos A = ±√

1

1 2 1 1−tan2 ( A) 2 −b±√b2 −4ac

2

2a

1

To find tan A, use tan A = 2

⇒ tan A =

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x= dy

1 2

or x = −2

|

dx x=1

=5

2

dy

−1

|

dx x=−2

= −5

2 tan( A)

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Additional Math Notes (20 Oct 2014) Power Rule

Differentiation from First Principles

d n (x ) = nx n−1 dx

f ′ (x) = lim

Useful shortcuts

Question

d



1

(√x) = dx 2√x d



1

1

dx x

x2

( )=−

d dx d

Solution

dx d dx



d

[x(x + 1)] = ( (

f(x) = √x f(x + δx) = √x + δx

2x2 +4x x x2 +2x x−1

d

)=

)=

dx

dx d

dx

(x 2 + x) = 2x + 1

f ′ (x)

(2x + 4) = 2

(x + 3 +

3

= lim

e.g.

(√x) =

dx d 1

( )

=

dx x



d dx d dx

1 2

) (long division)

x−1

=

(x −1 )

=−

d

[

2x+1

]=

dx x(x+1)

d

1

( +

dx x

1

Constant Multiple Rule d dx

[kf(x)] = k

d dx

[f(x)]

Sum/Difference Rule d dx

[f(x) ± g(x)] = f ′ (x) ± g′(x)

√x+δx−√x δx

⋅

√x+δx+√x √x+δx+√x

x+δx−x

δx→0 δx(√x+δx+√x)

2√x 1

δx

= lim

δx→0 δx(√x+δx+√x)

x2

)=−

x+1

δx

= lim

= lim

Breaking into partial fractions e.g.

= lim

δx→0

1

(x )

f(x+δx)−f(x)

δx→0

Use law of indices d

δx

Find the derivative of f(x) = √x from first principles

Consider simplifying before differentiating  Multiply or divide e.g.

f(x+δx)−f(x)

δx→0

1

δx→0 (√x+δx+√x)

1 x2

1

− (x+1)2

= =

1

∵(

√x+√x 1 2√x

as δx → 0, ) √x + δx → √x

✓

Ex 14.2 The Chain Rule Chain Rule d dx

[fg(x)] = f′g(x)

× g′(x)

= Diff Outer × Diff Inner Ex 14.3 The Product Rule Product Rule d dx

[f(x)g(x)] = f(x) g ′ (x) +f ′ (x) g(x) Keep Diff

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30

Additional Math Notes (20 Oct 2014) Tangent Properties

Ex 14.4 The Quotient Rule  

Quotient Rule Diff Bottom Top d

[

f(x)

dx g(x)

]=

g(x)

Diff Top Bottom

f′ (x) −f(x) [g(x)]2

Normal Properties

g′ (x)

Square Bottom

For the case of

k f(x)

d dx

1

[(2x−3)2 ] =

d dx

[(2x − 3)−2 ]

2

= − (2x−3)3 =

d dx

(2x − 3)

×2

4 − (2x−3)3

Consider converting to proper fraction first if it is an improper fraction d dx

(

Normal intersects curve



mnorm =

, it is preferable to use chain rule instead

= −2(2x − 3)−3 ×

e.g.



3x2 +x+3

d

x2 +1

dx

)= = = =

−1 f′ (x1 )

Ex 15.2 Increasing and Decreasing Functions

of quotient rule. e.g.

Tangent intersects curve mtan = f ′ (x1 )

(3 +

(x2 +1)⋅

x

For increasing function,



For decreasing function,

dy dx dy dx

>0 0 ⇒ min

Ex 17.2 Derivatives of Exponential Functions

= 0 ⇒ inflexion

Derivatives of Exponential Functions  

d dx d dx

(ex ) = ex (eax+b ) = aeax+b

Consider simplifying using indices properties before differentiating d d (e2x ⋅ e1−3x ) = (e1−x ) = −e1−x e.g. dx

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32

Additional Math Notes (20 Oct 2014) Use Logarithmic Differentiation

Ex 17.3 Derivatives of Log Functions

Step 1: Take natural log both sides Step 2: Simplify using laws of log Step 3: Differentiate

Derivatives of Log functions  

d dx d dx

(ln x) =

1 x

[ln(ax + b)]=

a ax+b

Consider simplifying by using laws of logarithm before differentiating

=

dx

= =

d dx d dx 1

(ln x + ln ex ) (ln x + 1)

x

Quotient law d dx

[ln (

x

Question Differentiate y = 2x with respect to x

Product law d (ln xex ) e.g.

e.g.

It is useful when differentiating  functions of the form y = [f(x)]g(x) , f(x) ≠ e  complicated products or quotients

d

)] =

x2 +1

dx 1

[ln x + ln(x 2 + 1)] 2x

= +

y = 2x ln y = ln 2x ln y = x ln 2 Diff wrt x: 1 dy ⋅ = ln 2 y dx dy

x2 +1

x

Solution

dx d

dx

dx

[2 ln(4x − 3)]

= 2(

= = =

d

4

Question

)

4x−3

8

=

dx

= (ln 2)y = (ln 2)2x ✓ Replace y with 2x

Power law d [ln(4x − 3)2 ] = e.g.

Change-of-base law d (log a x) = e.g.

Take ln both sides Simplify using power law Differentiate both sides wrt x

Find

dy dx

if y = (2 + x 2 )(1 − x 3 )4

4x−3

(

ln x

)

dx ln a 1 d

⋅

ln a dx 1 1

( )

ln a x 1 xln a

(ln x)

Solution y = (2 + x 2 )3 (1 − x 3 )4 −(1) ln y = ln[(2 + x 2 )3 (1 − x 3 )4 ] = ln(2 + x 2 )3 + ln(1 − x 3 )4 = 3 ln(2 + x 2 ) +4 ln(1 − x 3 ) Diff wrt x: 1 dy y

⋅

dx

=3⋅ =3⋅ =

dy dx

1 2+x2 1 2+x2

⋅

d dx

(2 + x 2 )

⋅ 2x

+4 ⋅

6x

−

2+x2 6x

=(

2+x2

= 6x ( = 6x [

−

12x2

)

1−x3 1 2x

−

)

2+x2 1−x3 3 (1−x )−2x(2+x2 ) (2+x2 )(1−x3 ) 1−x3 −4x−2x3 1−4x−3x3 (2+x2 )(1−x3 )

1 1−x3 1

1−x3 12x2

⋅

d dx

(1 − x 3 )

⋅ (−3x 2 )

1−x3

⋅y

= 6x [ (2+x2 )(1−x3 ) ] = 6x [

+4 ⋅

]

⋅y ]⋅y ⋅y ⋅y

−(2)

sub (1) into (2): dy dx

1−4x−3x3

= 6x [(2+x2 )(1−x3 )] (2 + x 2 )3 (1 − x 3 )4 = 6x(1 − 4x − 3x 3 )(2 + x 2 )2 (1 − x 3 )3 ✓

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Additional Math Notes (20 Oct 2014) Find Curve from Derivative

Ex 18.1 Indefinite Integrals



∫ f(x) ± g(x) dx = ∫ f(x) dx ± ∫ g(x) dx

To form equations and solve unknowns,  use given equations (unknowns are already present) dy e.g. = x 2 (x − k)



∫ af(x) dx = a ∫ f(x) dx



Integral Rules

dx

Find Integral from Derivative

dx

Integration is the reverse of differentiation



introduce arbitrary constants from integration dy e.g. = 2x + 1 dx ⇒ y = x2 + x + c



use point on curve e.g. (1, −2) lies on y = f(x) ⇒ −2 = f(1)



use gradient dy e.g. at turning point, = 0

Question Given find ∫

d dx

√6x + 5 = 1

√6x+5

3

,

√6x+5

dx.

Solution 1

1

use proportionality e.g. Gradient is proportional to f(x) dy ⇒ = kf(x)

3

∫ √6x+5 dx = 3 ∫ √6x+5 dx

dx

1

= √6x + 5 + c ✓ 3

Integrals of Power Functions

Consider rearranging equation involving derivative. Question

xn+1



∫ x n dx =



∫ ax n dx =



∫(ax + b)n dx =

d

Given (x ln x) = 1 + ln x, dx find ∫ ln x dx.

d

(x ln x) = 1 + ln x

ln x

+c

n+1

(ax+b)n+1 a(n+1)

+c

Note: The rules for above hold for all real values of n except for n = −1

Solution dx

+c

n+1 axn+1

=

d dx

e.g. ∫ x −1 dx ≠

(x ln x) − 1 ✓

x0 0 1

+c

but ∫ x −1 dx = ∫ dx = ln|x| + c x

d

∫ ln x dx = ∫ [dx (x lnx) − 1] dx = x ln x − x + c ✓

Consider simplifying before integrating  Multiply or divide e.g. ∫[x(x + 1)] dx = ∫(x 2 + x) dx =

Question Given

d dx

(x cos x) = cos x − x sin x,

∫(

Find ∫ x sin x dx

∫(

Solution d dx



(x cos x) = cos x − x sin x

x sin x

= cos x −

d dx

x2 +2x x−1

) dx = ∫ (x + 3 +

x2 2

+c

3

) dx (long division)

x−1

3

e.g. ∫ √x dx = ∫ x dx = 1

∫ x2 dx = ∫ x −2 dx = 

+

) dx = ∫(2x + 4) dx = x 2 + 4x + c

1 2

d

= sin x − x cos x + c ✓

x

3

Use law of indices

x cos x

∫ x sin x dx = ∫ (cos x − dx x cos x) dx

2x2 +4x

x3

x2

2

= x√x 3

3 2 x−1

−1

1

+c=− +c x

Breaking into partial fractions x

1

1

e.g. ∫ (x−1)2 dx = ∫ + (x−1)2 dx x−1 = ln|x − 1| −

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+c

34

Additional Math Notes (20 Oct 2014) Ex 18.2 Definite Integrals

Ex 18.3 Integrals of Trigo Functions

Definite Integrals

Integrals of Trigonometric Functions

b

∫a f(x) dx = F(b) − F(a) Definite Integrals Rules 

b ∫a f(x) dx

=

a − ∫b f(x) dx



b ∫a f(x) dx

=

c ∫a f(x)



∫a f(x) dx = 0

dx +

b ∫c f(x) dx



∫ cos x dx = sin x + c



∫ sec 2 x dx = tan x + c



∫ sin(ax + b) dx

 𝑦

Definite integrals can be equal because they have equal area

𝑦 = 𝑥2

under curve 0

e.g. ∫0 x 2 dx = ∫−1 x 2 dx

−1

𝑂 1 Equal areas

Integrals of Modulus Functions ∫ f(x) dx ∫|f(x)| dx = { ∫ −f(x) dx

∫ sin x dx = − cos x + c



a

1



1

= − cos(ax + b) + c a

1

∫ cos(ax + b) dx

= sin(ax + b) + c a

1

∫ sec 2 (ax + b) dx

= tan(ax + b) + c a

Consider simplifying using trigonometric identities before integrating e.g. ∫ tan2 x dx = ∫ sec 2 x − 1 dx = tan x − x + c

𝑥 Use special angles for definite integrals of trigonometric function π 2 π 3

π

e.g. ∫ cos x dx = [sin x]π2 3

π

= sin − sin

if f(x) ≥ 0 if f(x) < 0

=1

2 √3 − 2

π 3

Ex 18.4 Integrals of Exponential Fns & 1/x Integrals of Exponential Functions 

∫ ex dx = ex + c



∫ eax+b dx = a eax+b + c

1

Consider simplifying using indices properties before integrating e.g. ∫ e2x ⋅ e1−3x dx = ∫ e1−x dx =

e1−x −1 1−x

= −e 1

1

x

ax+b

Integrals of &

+c +c

1



∫ x dx = ln|x| + c



∫ ax+b dx = a ln|ax + b| + c

1

1

Consider breaking into partial fractions before integrating 2x−1 5 3 ) dx e.g. ∫ dx = ∫ ( − (x+1)(x+2)

x+2

x+1

= 5 ln|x + 2| − 3 ln|x + 1| + c

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Additional Math Notes (20 Oct 2014) Ex 19.1 Area by Integration

Ex 19.2 Area bounded by Curves

Area by integration

Strategies to find area bounded by curves

Top

𝑦

𝑦 = 𝑓(𝑥) x2

Area = ∫ (Top − Bottom) dx 𝑂 𝑥1

x1

y1

𝑦1 Left 𝑦2

𝑦 = 𝑥2 (1,1) 𝑂

Right Solution Method 1 (integrate wrt y-axis)

𝑥

𝑂

There should be a pair of lines parallel to x or y-axis enclosing the region. If parallel to y-axis, integrate wrt x and vice versa.

2

𝑥 𝑦 =2−𝑥

𝑦 𝑥 = √𝑦

Area of region F 1

y

= ∫y 2 (Right − Left) dy

F

1

Consider finding geometric area without integration

=

1 ∫0 [(2

𝑥 𝑥 =2−𝑦

𝑂

− y) − √y] dy

1



Triangle area



Trapezium area = (sum of bases)(height)

= (base)(height) 2

𝑦

1

Given the the diagram at the right. Find area of (i) Region A (ii) Region B

Area of region G +Area of region H

𝑦 (3,6)

= (base)(height)

1

2

= ∫0 x 2 dx

+ ∫1 (2 − x) dx

𝑂

1

G H 𝑂 1 2

𝑥 𝑦 =2−𝑥

𝑦 𝐵

𝐴 1

𝑦 = 2𝑥

(1,2)

Solution Area of Region A (⊿)

𝑦 = 𝑥2

Method 2 (break)

2

Question

(i)

Break into smaller shapes

Question Find the area bounded by y = x 2 , y = 2 − x and the x − axis.

𝑦 𝑥 = 𝑓(𝑦) y2

Break

𝑦

Bottom

Area = ∫ (Right − Left) dy

Integrate wrt x or y-axis

Complement Subtract area

𝑥

𝑥2

Axis

𝑦 = 𝑥2

Method 3 (complement) 3

𝑥

Area of ⊿ −Area of region I 1

= (2)(2) 2

2

1 − ∫0 (2

2 I

− x) dx

1

𝑂

= (1)(2) 2

1

2

𝑥 𝑦 =2−𝑥

= 1 unit 2 (ii)

Area of Region B (trapezium)

Ex 20.1 Kinematics Kinematics Relation

1

= (sum of bases)(height) 2 1

v=

= (2 + 6)(3 − 1) 2

= 8 unit 2 ✓

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dv dt

a

v = ∫ a dt

36

Additional Math Notes (20 Oct 2014) Implications of Kinematics Statements 

t is time after passing O ⇒ s|t=0 = 0



Rest

⇒v=0



Time to turn around

⇒v=0



Max/min quantity

⇒ 1st derivative = 0



Max/min dist. from O

⇒v=0



Max/Min v

⇒a=0



Total Distance = ∫|v| dt



Average Speed =

total distance total time

Distance t2

Total Distance = ∫ |v| dt t1

(Total distance travelled in between t1 and t 2 ) Method 1 (using s-t graph) Step 1: Let v = 0 to find t Step 2: Find s for each t found Step 3: Find s for start & end Step 4: Draw s-t graph Method 2 (using v-t graph) Step 1: Draw v-t graph Step 2: Use distance = ∫|v| dt Step 3: Split at v = 0 Step 4: Remove modulus |v| = { v for v ≥ 0 −v for v < 0

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Additional Math Notes (20 Oct 2014) 3D Shapes

Appendix 1 Geometric Formulae

Prism 2D Shapes Triangle (𝟑 𝐯𝐞𝐫𝐭𝐢𝐜𝐞𝐬) 

1

= (base)(height)

Triangle area

2 1

= ab sin C 2

 

Isosceles triangle area = sin a

Sine rule:

A

=

sin b B



Cosine rule:



Pythagoras’ theorem :

2

Trigonometric identities:

cos θ = tan θ =

o

h o



Cube volume = x 3



Cube surface area = 6x 2



Cylinder volume = πr 2 h



Cylinder surface area = 2πr 2 + 2πrh = 2πr(r + h)

sin c

Pyramid

C

2

a2 + b2 = c 2 a



h a

=

a = b + c − 2bc cos A

Similar triangles:

Prism volume = (base area)(height)

4

2



sin θ =

s2 √3



A

=

b B

=

c C

⇒ o = h sin θ



1

Pyramid volume = (base area)(height) 3

1



Cone Volume = πr 2 h



Cone area (exclude base) = πrl where l = √r 2 + h2

3

Sphere

⇒ a = h cos θ

a

4



Sphere volume = πr 3



Sphere area = 4πr 2

3

Quadrilateral (𝟒 𝐯𝐞𝐫𝐭𝐢𝐜𝐞𝐬) 

Square area = x 2



Rectangle area = (base)(height)



Parallelogram area = (base)(height)



Rhombus area = (product of diagonals)



Trapezium area = (sum of bases)(height)



1 2

1 2

1

Kite area = (product of diagonals) 2

Circle (∞ 𝐯𝐞𝐫𝐭𝐢𝐜𝐞𝐬) 

Circle area = πr 2



Circumference = 2πr



Arc length = rθ = s



Area of sector = r 2 θ

1 2 1

= rs 2

1

1

2

2



Area of segment = r 2 θ − r 2 sin θ



Circle properties (refer to Ex 10.3)

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Additional Math Notes (20 Oct 2014) Appendix 2 Trigonometric Identities Special Angles 0° 30° 45° 60° 90° π

0

6 1

sin θ 0 cos θ 1 tan θ 0

π

π

4

3

2

√2 2

√3 2 1

1

π

2 √3 2 1

√2 2

0

2

1 √3 ∞

√3

Complementary Angles  sin(90° − θ) = cos θ  

cos(90° − θ) = sin θ tan(90° − θ) =

Pythagorean Identities  sin2 θ + cos 2 θ = 1 

tan2 θ + 1

= sec 2 θ



cot 2 θ + 1

= csc 2 θ

Addition Formulae  sin(A ± B) = sin A cos B ± cos A sin B 

cos(A ± B) = cos A cos B ∓ sin A sin B



tan(A ± B) =

Double Angle Formulae  sin 2A = 2 sin A cos A 1 ⇒ sin A cos A = sin 2A 2

1 tan θ



Trigonometric Function Definition   



sin θ = cos θ = tan θ =

r=

y r x

r θ

r y

y x

x

√x 2

+

tan A±tan B 1∓tan A tan B



y2

cos 2A = cos 2 A − sin2 A = 2 cos 2 A − 1 = 1 − 2 sin2 A ⇒ cos 2 A =

1+cos 2A

⇒ sin2 A =

1−cos 2A

tan 2A =

2 2

2 tan A 1−tan2 A

Half Angle Formulae Reciprocal Identities 

sec θ =



csc θ =



cot θ =

1

1

2

2

To find sin A, use cos A = 1 − 2 sin2 ( A)

1 cos θ

1

1−cos A

2

2

⇒ sin A = ±√

1 sin θ 1

1

1

2

2

To find cos A, use cos A = 2 cos 2 ( A) − 1

tan θ



sin(−θ) = − sin(θ)



tan(−θ) = − tan(θ)

1

1+cos A

2

2

⇒ cos A = ±√

Negative Angles  cos(−θ) = cos(θ)

1

1 2 1 1−tan2 ( A) 2 −b±√b2 −4ac

2

2a

1

To find tan A, use tan A = 2

⇒ tan A =

2 tan( A)

Principal Values π π  − ≤ sin−1 x ≤

R-Formulae a sin θ ± b cos θ = R sin(θ ± α)



0 ≤ cos

a cos θ ± b sin θ = R cos(θ ∓ α)



− < tan−1 x <

2

2

−1

x≤π

π

π

2

2

Ratio Identities 

tan θ =



cot θ =

sin θ



R = √a2 + b 2



α = tan−1 ( )



min = −R, max = R

b a

cos θ cos θ sin θ

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Additional Math Notes (20 Oct 2014) Integration

Appendix 3 Calculus Formulae Differentiation Rules/Functions f(x)

f′(x)

Constant Multiple Rule

kf′(x)

kf(x)

f ′ (x) + g′(x)

Chain Rule

fg(x)

f ′ g(x) × g′(x)

Product Rule

f(x)g(x)

f(x)g ′ (x) + f ′ (x)g(x)

f(x)

g(x)f′ (x) −f(x)g′ (x) [g(x)]2

Power Functions

Trigonometric Functions

g(x)

x

n

nx

n−1

sin x

cos x

cos x

− sin x

tan x

sec 2 x

Exponential Functions

ex

ex

e(ax+b )

aeax+b

ln x Log Functions ln(ax + b)

a ∫ f(x) dx

af(x)

Power Functions (power ≠ −1)

∫ f(x) dx ± ∫ g(x) dx xn+1

∫ x n dx

n+1

∫(ax + b)n dx

+ b)

Exponential Functions Power Functions (power = −1)

+c

(ax+b)n+1 a(n+1)

+c

sin x

− cos x + c

cos x

sin x + c

sec 2 x

tan x + c

sin(ax + b)

− cos(ax + b) + c

1 a

1

cos(ax + b)

cos(ax + b) −a sin(ax + b) tan(ax + b) a sec

Constant Multiple Rule

Trigonometric Functions

sin(ax + b) a cos(ax + b) 2 (ax

∫ f(x) dx

Sum/Difference f(x) ± g(x) Rule

Sum/Difference f(x) ± g(x) Rule

Quotient Rule

Rules/Functions f(x)

a

sin(ax + b) + c

sec 2 (ax + b)

1

ex

ex + c

eax+b

1 ax+b e a

1

a

tan(ax + b) + c

+c

ln|x| + c

x 1

ln|ax + b| + c

ax+b

1 x a ax+b

Definite Integrals b ∫a f(x) dx = F(b) − F(a) Definite Integrals Rules

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b

a



∫a f(x) dx = − ∫b f(x) dx



∫a f(x) dx = ∫a f(x) dx + ∫c f(x) dx



∫a f(x) dx = 0

b

c

b

a

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