Limits

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L

IMITS

In order to understand differentiability and continuity, we must be familiar with the limit notation.

For simple limits, we can simply substitute the appropriate value of and evaluate

Question 1

Evaluate the following limits a) b) c)

Limit Notation

means as approaches , approaches or goes as

close as we like to

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BASIC OPERATIONS AND LIMITS

These theorems need to be known but not proven. They are intuitively obvious.

Question 2

Evaluate the following limits a) b)

Limit Theorems

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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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G

EOMETRIC

D

EFINITION OF

‘D

ERIVATIVE

’

The derivative of a function is denoted :

SLOPE OF A CURVE

The gradient is the slope of the curve (or the slope of the tangent to the curve)

Definition:

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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: ]

When the curve is horizontal, the tangent will be horizontal The rise will be

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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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D

IFFERENTIATION FROM

F

IRST

P

RINCIPLES

The gradient of the function is equal to the slope:

In the following diagram, P and Q are two points on the curve and hence have coordinates 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

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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c) [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

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D

IFFERENTIATING

P

OLYNOMIAL

F

UNCTIONS

DIFFERENTIATING

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)

Differentiating Powers of

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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.

Talent Tip: Note that the derivative of a constant is

Question 10

Differentiate the following a) b)

Derivative Theorems

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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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F

INDING THE

E

QUATION OF

T

ANGENTS AND

N

ORMALS

Question 1

Find the equation of the tangent to curve at STEP 1: Find the 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

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

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Question 2

Find the equation of the tangent to the curve at the point STEP 1: Find the co-ordinate of the point

The point is

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Question 3

Find the equation of the tangent to the curve at STEP 1: Find the co-ordinate of the point

So the co-ordinates of the point are

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Question 4

Consider the graph of . The tangent to the curve at the point intersects the and -axis at the points and 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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Square and add one Cube the new function

T

HE

C

HAIN

R

ULE

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:

We use the chain rule to see how to differentiate the combined function

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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

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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T

HE

P

RODUCT

R

ULE

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

... ... ... ...

The Product Rule

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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 at

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T

HE

Q

UOTIENT

R

ULE

Question 15

Differentiate using the quotient rule a) Find and when

b) Hence, differentiate ... ... ... ...

The Quotient Rule

Suppose we have a function

, where and are simpler functions

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c) Differentiate 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

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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 lies on the curve

When 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: