Limits
Short Description
Exercises in limits...
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LIMITS In order to understand differentiability and continuity, we must be familiar with the limit notation.
Limit Notation means as
approaches ,
approaches or goes as
close as we like to For simple limits, we can simply substitute the appropriate value of
and evaluate
Question 1 Evaluate the following limits a)
b)
c)
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BASIC OPERATIONS AND LIMITS These theorems need to be known but not proven. They are intuitively obvious.
Limit Theorems
Question 2 Evaluate the following limits a)
b)
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c)
d)
Talent Tip: When doing limit questions, you do not have to explicitly split each limit into smaller limits and then evaluate, as we have done here. Substituting the values directly is enough. However, you must always explicitly show when you substitute your value into the limit, and not evaluate further. E.g. in c), do not directly evaluate
as , but write in
the substitution step. This is so examiners know you understand the limit, and are not doing it on a calculator.
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FINDING LIMITS OF POLYNOMIAL FUNCTIONS While some limits are simple substitutions, other limits will require manipulation before they can be evaluated. Mostly, these involve polynomial functions, and in particular, fractions
Limit as
of
Note that this applies to all powers of as well
Finding Limits of Polynomial Functions When the denominator Factorise the numerator and denominator, cancel any common factors, and then evaluate When Divide the top and bottom by the highest power of
in the numerator,
and use the limit of above
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Question 3 Evaluate the following limits a)
b)
c)
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d)
e)
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GEOMETRIC DEFINITION OF ‘DERIVATIVE’ The derivative of a function is denoted
:
Definition:
SLOPE OF A CURVE The gradient is the slope of the curve (or the slope of the tangent to the curve)
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HORIZONTAL TANGENT Question 4 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively flatter
a) Calculate the gradient of each of the curves. Rise of the curve
units
Run of the curve
units
Gradient
(First curve)
Rise of curve
units
Gradient
(Second Curve)
Gradient
(Third curve)
b) Hence, explain what happens to the value of the gradient as the curve becomes flatter? As the curve becomes flatter, the value of the rise decreases to , while the run stays the same. Hence the gradient will get closer and closer to c) What would the value of the gradient be when the curve is completely horizontal? [HINT:
]
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VERTICAL TANGENT Question 5 – Horizontal Tangent (Conceptual) Consider the two functions below, in which the curves are becoming progressively steeper.
a) Calculate the gradient of each of the curves. Rise of the curve
units
Run of the curve
units
Gradient
(First curve)
Run of curve
units
Gradient
(Second Curve)
Gradient
(Third curve)
b) Hence, explain what happens to the value of the gradient as the curve becomes steeper? As the curve becomes flatter, the value of the run decreases to , while the run stays the same. Hence the gradient will get closer and closer to infinity c) What would the value of the gradient be when the curve becomes vertical? [HINT: ] When the curve is vertical, the tangent is vertical The rise will be
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DIFFERENTIATION FROM FIRST PRINCIPLES The gradient of the function is equal to the slope:
In the following diagram, P and Q are two points on the curve coordinates
and hence have
and
Rise of PQ
…………………………………………..
Run of PQ
………………………………………….. …………………………………………..
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Gradient of Secant
Gradient of Tangent We can find the gradient of the tangent, by considering what happens when Q moves closer and closer to P, or when
Differentiation from First Principles
Talent Tip: When differentiating from first principles, we seek to eliminate the h’s from the denominator. The questions will illustrate.
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Question 6 Find the derivative of the following functions using first principles a) Let
b) Let
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[HINT: rationalise the numerator] Let
d) Let
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THE NOTATION
The graph above shows a point
, and another point Q that is a small distance away
from P. We denote a small change by the sign . Hence, Q has co-ordinates )
For a very small
(i.e.
, the gradient of the secant becomes the gradient of the
tangent. We define:
Different notations of the derivative For a function
, the derivative can be written as
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DIFFERENTIATING POLYNOMIAL FUNCTIONS DIFFERENTIATING Differentiating Powers of
Talent Tip: An easy way to consider this principle is that you “bring down” the power, and then minus one from it
Question 7 Differentiate a)
b)
c)
d)
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Question 8 Differentiate a)
b)
c)
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Question 9 Differentiate the following – a)
b)
c)
d)
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BASIC OPERATIONS Just like there are limit theorems, there are also ones for derivatives. This is not surprising as the definition of the derivative comes from limits. However, note that only the first two laws apply, and not multiplication or division.
Derivative Theorems
Talent Tip: Note that the derivative of a constant is
Question 10 Differentiate the following a)
b)
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c)
d)
e)
f)
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Question 11 Differentiate the following functions a)
b)
c)
d)
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e)
f)
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FINDING THE EQUATION OF TANGENTS AND NORMALS Finding the Tangent/Normal STEP 1: Find the
co-ordinate of the point
STEP 2: Find the derivative
, and then find the gradient of the tangent
or normal (Remember STEP 3: Find the equation of the line using point-gradient form Question 1 Find the equation of the tangent to curve STEP 1: Find the
at
co-ordinate of the point
So the co-ordinates of the point are STEP 2: Find the derivative
, and then find the gradient of the tangent or normal
STEP 3: Now use point-gradient form to find the equation of the tangent
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Question 2 Find the equation of the tangent to the curve STEP 1: Find the
at the point
co-ordinate of the point
The point is
STEP 2: Find the derivative
, and then find the gradient of the tangent or normal
STEP 3: Now use point-gradient form to find the equation of the tangent
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Question 3 Find the equation of the tangent to the curve STEP 1: Find the
at
co-ordinate of the point
So the co-ordinates of the point are
STEP 2: Find the derivative
, and then find the gradient of the tangent or normal
STEP 3: Now use point-gradient form to find the equation of the tangent
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Question 4 Consider the graph of
. The tangent to the curve at the point
intersects the
and
and -axis at the points
respectively
a) Find the equation of the tangent
When
,
Gradient of tangent Equation of tangent:
b) Find the area of the triangle
, where
is the origin
When
When
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THE CHAIN RULE To differentiate functions such as
, we need to recognize that the function is
composed of a chain of two functions
and
that we can differentiate
separately: Square and add one
Cube the new function
We use the chain rule to see how to differentiate the combined function
The Chain Rule
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Question 5 a) Differentiate
with respect to
b) Hence, using the chain rule, differentiate ..................... ............................ .........................
.........................
Question 6 a) Differentiate
with respect to
b) Hence, using the chain rule, differentiate ..................... ............................ .........................
.........................
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Question 7 Differentiate the following functions using the chain rule a) ..................... ............................ .........................
.........................
b) ..................... ............................ .........................
.........................
c) ..................... ............................ .........................
.........................
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d) ..................... ............................ .........................
.........................
e) ..................... ............................ .........................
.........................
f) ..................... ............................ .........................
.........................
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g) ..................... ............................ .........................
.........................
h) ..................... ............................ .........................
.........................
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LINEAR FUNCTIONS Differentiating Linear Functions
Talent Tip: “Take the derivative of the OUTSIDE of brackets times by the derivative of inside the brackets”
Question 8 Find the derivative of the following a)
b)
c)
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d)
e)
f)
g)
h)
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THE PRODUCT RULE The Product Rule Suppose we have a function
, where
and
are simpler functions
Talent Tip: You will often be required to use both the chain rule and the product rule Talent Tip: Unless the question specifies otherwise, you do not have to factorise the derivative.
Question 9 a) Using the chain rule, differentiate
b) Using the product rule, differentiate ................................. ....................................... ................................. ......................................
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Question 10 a) Use the chain rule to differentiate
b) Hence, find the derivative of ................................. ....................................... ..............................… ......................................
Question 11 a) Find
and
where
and
b) Hence differentiate ................................. ....................................... ................................. ......................................
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Question 12 Use the product rule to differentiate the following with respect to a) ................................. ....................................... ................................. ...................................... b) ................................. ....................................... ................................. ......................................
c) ................................. ....................................... ................................
d) ................................. ....................................... ................................
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Question 13 Differentiate and then factorise the following a)
b)
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Question 14 Consider the function a) Find
b) At what -values is the gradient of the tangent perpendicular to the gradient at
?
When
The tangent is perpendicular to the tangent at
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at
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THE QUOTIENT RULE The Quotient Rule Suppose we have a function
, where
and
are simpler functions
The order of differentiation is important Question 15 Differentiate a) Find
and
using the quotient rule when
b) Hence, differentiate ................................. ....................................... ................................. ......................................
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using the product rule, simplifying your answer ................................. ....................................... ................................. ......................................
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Question 16 Use the quotient rule to find the derivatives of the following functions a) ................................. ....................................... ................................. ......................................
b) ................................. ....................................... ................................. ......................................
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c) ................................. ....................................... ................................. ......................................
d) ................................. ....................................... ................................. ......................................
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Question 17 Find the derivatives of the following functions, and find the value(s) of
for which the
gradient of the tangent is equal to a)
Gradient of the tangent is
when
Hence, there are no values of
for which the gradient of the tangent is
b)
Gradient of the tangent is
when
Hence, there are no real values of
for which the gradient is
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Question 18 Consider the function a) If
, find
b) For what values of
is the gradient positive? Negative? Zero?
Gradient positive when i.e.
since
for all
excluding Gradient zero when
Gradient negative when
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Question 19 Consider the curve a) Show that the point When
lies on the curve
and
Hence, the point
lies on the curve
b) Using the quotient rule, find
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c) Hence, find the equation of the normal at the point When
Equation of Normal:
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