MODUL AM015 2021 INTAKE

AM015/ 8: Differentiation

TUTORIAL 5 of 7

1. Given that ln x + y = exy , find dy .

dx

( )2. Find dy for e4x x2y + 1 = 1
dx

3. Find dy when:
dx

(a) x2 y3 = 8 (b) y2 = ln(x2 + y 2 ) (c) y 2 x + e2y x2 + 2 y = 0

(d) x2 + e xy + y2 = 1 (e) x2 ln y + y 2 ln x = 2

4. By using the laws of logarithm, find dy ex+ y = (2x −1)2
if x +1 ( x − 3)5 .
dx

5. If y − x2 y = 4, show that  1 − x2  dy = 2xy + y .
x  x  dx x2

6. If y2 = 1 , prove that y3 + dy = 0.
2x −1 dx

( ) ( )7. Given 
ln x + 1 , show that dy = 1  1 ln x + 1 
y= −
dx x  x + 1 2x 
x

Answers: ( )2.
dy −4 x2y +1 − 2xy
2 y(x + y)exy −1 dx = x2
1. 1− 2x(x + y)exy

3. (a) − 2 y x −( y2 + 2xe2 y ) − 2x − yexy
3x (b) (c) 2xy + 2x2e2 y + 2 (d)
xexy + 2 y
(x2 + y 2 −1) y

(−2x2 ln y − y 2 ) y
(e)

(x2 + 2 y 2 ln x)x

4. 4 − 1 − 5 −1 (5, 6, 7 ) Shown
2x −1 2(x +1) x − 3

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AM015/ 8: Differentiation

TUTORIAL 6 of 7

1. Find dy for the following functions. Give your answers in the most simplified for

dx

(a) y = (2x2 − 3)e−4x (b) y = 3x ln  x2 + 1  (c) y = x2
 5x + 2  3x + 5

2. Find dy for 2x2 y + y2 = 2ey by using implicit differentiation.
dx

3. Find the slope of the curve y = 5x2 − 3 at the point = 2 by using the first principle of

differentiation.

4. If y = Ax2 + Bx , show that x2 d 2 y − x dy + y = 0
2 dx2 dx

5. Differentiate the following with respect to x. Give your answer in the simplest form.

a) f (x) = xe3x (b) f (x) = (3x + 2)(x + 4)3
x+3

6. Evaluate dy for the curve 3y2 + xy − 3x2 = 1 at point (1,1)
dx

7. Find f '(x) for f (x) = 4x2 − 7x + 5 by using the first principle.

8. Given y = Ax + Bxex with A and B are constants. Determine the values of A and B
if dy = 0 and d 2 y = 2 at x = 0 .

dx dx2

Answers:

1. (a) dy = −4e−4x (2x − 3)(x + 1)
dx

(b) dy = 6x2 − 15x + 3ln  x2 +1  (c) dy = x(3x +10)
dx x2 +1 5x + 2  5x +2  dx (3x + 5)2

2. dy = − 2xy 3. 20
dx x2 + y − ey

4. shown 5. (a) 3e3x[x2 + 3x +1]
(x + 3)2

(b) 3 (x + 4)(2x + 3)
3x + 2

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AM015/ 8: Differentiation

6. dy = 11 7. f '(x) = 8x − 7
dx 13
8. A = -1, B = 1

TUTORIAL 7 of 7

1. Find f '(2) for f (x) = 2x2 − x by using the first principle.

2. Differentiate the following with respect to x. Give your answer in the simplest form.

a) f (x) = ( x − 9)(x + 2)4 b) f (x) = (2x + 4)3 ( )c) f (x) = ln (2x + 3)3 e3x
x−3

3. a) Find dy for y3 = 2x2 y + xe4y − 2 by using implicit differentiation. Hence, evaluate dy at
dx dx

y =0.

b) Prove that if y = ( Ax + B) e−2x then d2y + 4 dy + 4y = 0 .
dx2 dx

y = (A + Bx)e3x , find dy d2y d2y − 6 dy + 9 y = 0.
dx dx2 . Hence, show that dx2 dx
4. Given and

5. Find f '(x) for f (x) = −2x2 + x +1by using the first principle. Hence, find f '(2).

6. Find dy for y2 + 3x2 y2 − 7xy +1 = 0 by using implicit differentiation.
dx

Hence, determine all values of dy when y = 1.
dx

7. Differentiate the following with respect to x. Give your answer in the simple form.

a) y = (x2 − 3) e5x

b) y = ln  x −1 3
 5x2 + 2 

7 of 11

Answers: AM015/ 8: Differentiation
1 f '(2) = 7
3(2x + 5)
2 ( x + 2)3 (9x − 70) 2(2x + 4)2 (2x −11)
b) ( x − 3)2 c)
a) 1
2x +3
2(x −9)2
b) proved
3 dy = 4xy + e4y dy = − 1 at y = 0
dx 3y2 − 2x2 − 4xe4 y dx 16
a) ,

4 dy = 3e3x ( A + Bx) + Be3x d2y = 6Be3x + 9e3x ( A +
dx , dx2
Bx)

, shown.

5 f '(x) = −4x +1, f '(2) = −7.

6 dy = 7 y − 6xy2
dx 2 y + 6x2 y − 7x

at  1 ,1 , dy = 15,
 3 dx

at (2,1), dy = −5 .

dx 12

( )7
a) dy = e5x (5x2 + x −15)
dx x2 − 3
1
2

b) dy = (3 10x − 5x2 + 2)
dx ( x −1)(5x2 + 2)

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AM015/ 8: Differentiation

EXTRA EXERCISE

1. Use suitable rules of differentiation to find the derivative of the following functions.
Give your answer in simplest form.

(a) f (x) = x3e7x (b) h(x) = x ln x − eln x

2. (a) Apply implicit differentiation to find dy for the equation (x − y)2 = xy − 2 . Hence,
dx

solve for x when dy = 0 . Give your answer in exact value.
dx

(b) Determine the values of A, B and C for y = Ax3 + B(x −1)2 + Cx , if dy = 2 and
dx

d2y =1 at point ( 2,1) .
dx2

3. Find f ' (x) for f (x) = 3x2 − 2 by using the first principle . hence or otherwise, find
the value of f ' (x) at x=1.

4. Given y = Ae−x ln x , where A is a constant.
(a) Show that x2 d 2 y − yx2 + Ae−x (2x +1) = 0
dx2
(b) Find the value of A if dy = 1 at x=1. Give your answer in exact value.
dx

5. If 4y − 3x2 − 5xy2 = 8 , find dy in terms of x and y. Hence, evaluate dy when x=0.
dx dx

[5M]

6. (a) Given y = (x x3 and dy = Ax2(x + B) . Find the values of A and B [3M]
+ 1)2 dx (x +1)3 [4M]

(b) Given 1 , find dy
in the simplest form.
y = 4x(3x3 − 2)2 dx

7. (a) Given y = (4 − 5x)e−3x , find d2y d2y
dx2
in the simplest form. Hence, evaluate

dx2

when x=0.

[6M]

(b) Find dy in terms of x and y, given that ln(2 y) = y ln x , where y>0 and x>0. Give
dx

dy [6M]
your answer in the simplest form. Hence, evaluate when x=1.

dx

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AM015/ 8: Differentiation

8. (a) Use first principle to find the derivative of f (x) = 5x2 − 2x + 7 [4M]

(b) Differentiate 2x − 3 with respect to x. Give your answer in the simplest form.
4x2 −9

[5M]

9. (a) Find the second derivative of y = 1 − 1 [3M]
x x2 [6M]

(b) Given y = xe−2x , calculate the value of x when d2y =0
dx2

10. (a) Using implicit differentiation , find dy for x ln( y +1) = e2y−3x [6M]
dx

(b) Given y = (mx + n)e2x , where m and n are constants. Find the values of m and m

if y = −3 and dy = 8 when x=0. [6M]
dx

Answers: (b) ln x

1 (a) x2e7x (7x + 3)

2 3y − 2x ; x =  3 10 (b) A = − 3 , B = 5,C = 1
4
(a)
2y − 3x 5

3 f '(x) = 6x; f '(1) = 6

4 (a) shown (b) e

5 dy = 6x + 5y2
dx 4 −10xy , 5

6 2(15x 3−4) y2 1
(a) A=1, B=3 (b)
3x3 − 2 (b) ,

7 x(1− y ln x) 4

(a) 3e−3x (22 −15x) , 66

8 (a) 10x-2

(b) 6(2x − 3) @ 6@6
(2x + 3) (4x2 − 9) (2x + 3)3 (2x − 3)
1

(4x2 − 9)2

9 (a) 3 −5 − 6 x −4

4 x2

(b) x=1

10 −3( y +1)e2y−3x − ( y +1) ln( y +1)
(a) x − 2( y +1)e2 y−3x

(b) n = -3, m=14

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AM015/ 8: Differentiation
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AM015/ 9. Application of Differentiation

CHAPTER 9: APPLICATIONS OF DIFFERENTIATION

SUBTOPIC
9.1 Tangent and Normal
9.2 Extremum Problems
9.3 Application of Differentiation in Economics and Business

LECTURE 1 OF 4

Learning Outcome:
9.1 a) Find the gradient of the tangent and normal at a point on a curve.
9.1 b) Find the equations of the tangent and normal to a curve.

*Exclude: Parametric, logarithmic and exponential functions

9.1 Tangent and Normal Equations

The figure below shows the tangent and normal to the curve y = f (x) at the point P.
y y = f (x)

P x
Normal
Tangent

The line that touches the curve only at point P (without crossing the other side of the
curve) is called the tangent at point P.

While the line perpendicular to the tangent at point P is called the normal at point P.

The gradient/slope of the tangent to the curve at point P is defined as

mT = f '(x) = dy
dx

If two lines (tangent and normal) with gradients mT and mN respectively are

perpendicular, then mT mN = −1

Therefore, gradient/slope of the normal at point P is defined as mN = − 1
mT

mN = − 1 , mT mN = −1
mT

Page 1 of 20

AM015/ 9. Application of Differentiation

The equation of the tangent at point ( x , y ) can be obtained by
11

y − y1 = mT ( x − x1 )
The equation of the normal at point ( x , y ) can be obtained by

11

y − y1 = mN ( x − x1 ) y − y1 = − 1 (x − x1 )
mT

Example 1
Find the equation of the tangent and normal to the curve y = x2 + 2x + 2 at the point
(1, 5).

Page 2 of 20

AM015/ 9. Application of Differentiation
Example 2
Find the coordinates of the points on the curve y = 2x3 – 3x2 – 8x + 7 where the
gradient is 4.

Example 3
The gradient of the tangent to the curve y = ax2 + b at the point (2, 3) is 8. Find the
values of a and b.

Page 3 of 20

AM015/ 9. Application of Differentiation
Example 4
Find the equation of the tangent to the curve 2 y − x2 + 4x = 0 which is parallel to the
line y = 4x – 3 .

Exercise:
1. Find the equation of tangent and normal line for the curve 2 y2 + x – 2 = 0 at the

point (0, 1).
2. The tangent to the curve y = px2 +1 at the point (1, q) is parallel to the line

y – 6x = 2 . Find the values of p and q.
Page 4 of 20

AM015/ 9. Application of Differentiation

LECTURE 2 OF 4

Learning Outcome:
9.2 a) Find the stationary points. *Introduce stationary points of a curve
9.2 b) Classify the type of turning points using the first derivative test.

*Using change of from positive to negative or vice versa

9.2 Extremum Problems

Stationary Points / Critical Points

The stationary points to a given function y = f ( x) are defined when

dy = 0 or f ’( x) = 0

dx

Stationary points are the point when
i) the tangent to the curve is horizontal
ii) the gradient of the curve is zero

Local Maximum and Local Minimum

y A (a, k) y = f(x)
k (iii) (i)
(iv) (iii)

j B (b, j) x
a
(ii)
b

From the graph:

(i) The gradient changes from positive to zero at A and then becomes negative.
The turning point at A (a, k) is a local maximum.

(ii) The gradient changes from negative to zero at B and becomes positive. The
turning point at B (b, j) is a local minimum.

(iii) f ( x) is increasing on the set {x : x  a  x  b }

(iv) f ( x) is decreasing on the set { x : a  x  b }.

Page 5 of 20

AM015/ 9. Application of Differentiation

Intervals where the Function is Increasing or Decreasing

A function is increasing, if as x increases (reading from left to right), y also
increases.

By definition, a function is increasing on an interval, if when then
As the x’s get larger, the y’s get larger.

A function is decreasing, if as x increases (reading from left to right), y decreases

By definition, a function is decreasing on an interval, if when then
As the x’s get larger, the y’s get smaller.

Theorem
a) If f '(x)  0 on an interval, then f is increasing on that interval.

b) If f '(x)  0 on an interval, then f is decreasing on that interval.

c) If f '(x) = 0 on a point, then f is constant on the point.

Methods to Determine the Nature of the Stationary Point / Critical Point

1. First derivative test
2. Second derivative test

Method 1 (First Derivative Test)

Find the stationary points and evaluate f ’( x) on both sides of the stationary point.
i) If f ’( x) is negative on the left of the point, and positive on the right, then it is

a local minimum at the point.

ii) If f ’( x) is positive on the left and negative on the right then it is a local

maximum at the point.

f (x)

f ’(x)  0 f ’(x)  0 f ’(x)  0 f ’(x)  0

cx c x

a. Local maximum b. Local minimum

f (x) f (x)

f ’(x)  0 f ’(x)  0

f ’(x)  0 f ’(x)  0

c x x
c. No extremum c
d. No extremum

Page 6 of 20

AM015/ 9. Application of Differentiation

Example 1

Given f ( x) = –x2 – 6x + 8 .

a) Find the stationary point.
b) Identify the nature of the point.
c) Determine the interval where the function is increasing or decreasing.

Example 2

Find the intervals which f is increasing or decreasing of the function

f (x) = 1 3 − 4x + 2.

3 x

Page 7 of 20

AM015/ 9. Application of Differentiation
Example 3

Given y = x3 + 3, find if exist the maximum and minimum point using first derivative

test. Hence, determine the intervals which function is increasing or decreasing.

Exercise

Given f (x) = 3x4 − 4x3 −12x2 + 5 , determine the interval where the function is

increasing or decreasing.
Page 8 of 20

AM015/ 9. Application of Differentiation

LECTURE 3 OF 4

Learning Outcome:
9.2 c) Find the local maximum or minimum point using the second derivative test
9.3 a) Express the relationship between cost, revenue and profit. *Problems such as

cost, revenue and profit functions
9.3 b) Compute average cost, and marginal cost, marginal revenue and marginal
profit

Method 2 (Second Derivative Test)
d2y

Find the stationary points, and evaluate dx2 or f "( x) of the stationary point.

Sign of Nature of Extremum Concavity

i) d 2 y  0 (positive) Minimum point Concave upward
dx2

ii) d 2 y  0 (negative) Maximum point Concave downward
dx2

iii) d2y =0 Test failed. Therefore, the first derivative test
dx2 must be used to determine the nature of the
point.

Page 9 of 20

AM015/ 9. Application of Differentiation
Example 1
Locate the stationary points on the curve y = 4x3 + 3x2 – 6x – 1 and determine the
nature of each one.

Example 2
Examine the following function for any stationary points and determine their nature.

f (x) = x4 − 9 x2
42.

Page 10 of 20

AM015/ 9. Application of Differentiation
Example 3
Given y = 3 − 4x3 , find if exist the maximum and minimum point using second
derivative test.

Example 4
P is a point of on the curve whose equation is 3y = x3 – 6x2 + 9x +1.
a) Given the coordinates of P is (2, 1), show that it is not a stationary point
b) Find the equation of the tangent at P

Exercise
1. Find the local maximum and local minimum for this function,

f ( x) = x3 – 6x2 + 9x +1 . Use the second derivative test when it applies.
2. Find all the stationary points of the curve f ( x) = x4 − 4x3 +10 and determine their

nature by using second derivative test.
Page 11 of 20

AM015/ 9. Application of Differentiation
9.3 Application of Differentiation in Economics and Business
Relationship between Cost, Revenue and Profit
Cost, Average Cost, Marginal Cost and Minimum Cost

COST

If x is the number of units of a product produced in some time interval, then C ( x) is

the cost function which is the total cost to produce x units of product. The total cost
consists of the fixed cost and the variable cost.

C ( x) = fixed cost + variable cost

The fixed cost is the cost incurred when no unit of product is produced,
C(0) = fixed cost

AVERAGE COST, C(x) = C(x)
x

The average cost is the cost to produce one unit of the product.
MARGINAL COST, C '(x) = d C(x)

dx
The marginal cost can be interpreted as the extra cost incurred by a firm/company
as an extra unit of output is produced at production level of x units.
MINIMUM COST BY USING SECOND DERIVATIVE TEST
● To minimize cost at the level of production xo, the following conditions must be
satisfied.

C '( xo ) = 0, C "(xo )  0

● The minimum cost is given by C ( xo ).

Page 12 of 20

AM015/ 9. Application of Differentiation
Example 5

Given the cost function C ( x) = 2x3 – 3x2 – 12x . Find the average cost and marginal

cost function.

Example 6

A television manufacturer produces x sets per week so that the total cost of

production is C ( x) = x3 −195x2 + 6600x +15000 . Find how many television sets

must be manufactured per week to minimize the total cost.

Page 13 of 20

AM015/ 9. Application of Differentiation

LECTURE 4 OF 4

Learning Outcome:
9.3 a) Express the relationship between cost, revenue and profit. *Problems such as

cost, revenue and profit functions
9.3 b) Compute average cost, and marginal cost, marginal revenue and marginal

profit
9.3 c) Compute maximum revenue, minimum cost and maximum profit by using

second derivative test. *Relate to problems involving taxes. **Include: Finding
quantity that maximize profit or revenue and minimize cost

Revenue, Marginal Revenue and Maximum Revenue

REVENUE

p(x) is the price or demand function which is the unit price offered for x units of product
demanded.

The total revenue, R(x) represents the total revenue for selling x units of product.

Total Revenue = price x quantity

R(x) = p(x) x

MARGINAL REVENUE, R '(x) = d R(x)
dx

The marginal revenue can be interpreted as the extra revenue from an extra unit sold
at number of units demanded x.

MAXIMUM REVENUE BY USING SECOND DERIVATIVE TEST

● To maximize revenue at the level of production xo, the following conditions must
be satisfied.

R '( xo ) = 0, R " (xo )  0

The maximum revenue is given by R( xo ).

Page 14 of 20

AM015/ 9. Application of Differentiation
Example 1
The demand function in RM, for x units of computer component is given by
p(x) = 320 − 0.1x. Find

a) the revenue function.
b) the number of unit of the computer component that need to be sold.
c) the maximum revenue.

Example 2
Innovation Sdn. Bhd. makes toys. The demand function in RM is p(x) = 80 − 0.004x ,
where x is the number of toys made. Find the maximum revenue and the number of
toys that need to be sold. Hence, find the price at the maximum revenue.

Page 15 of 20

AM015/ 9. Application of Differentiation
Profit, Marginal Profit and Maximum Profit
PROFIT
A profit function is derived from the difference between the total revenue function R(x)
and total cost function C(x), denoted by

Total Profit = Total revenue – Total cost

(x)= R(x) –C(x)

MARGINAL PROFIT,  '(x) = d (x)
dx

The marginal profit can be interpreted as the extra profit earned from an extra unit
produced and sold at units of production x.
MAXIMUM PROFIT BY USING SECOND DERIVATIVE TEST
● To maximize profit at the level of production xo, the following conditions must be
satisfied.

 '(x0 ) = 0 ,  ''(x0 ) 0
The maximum profit is given by (x0 ) , and its units selling price to obtain the
maximum profit is p(xo), where p(x) = R(x)

x

Page 16 of 20

AM015/ 9. Application of Differentiation
Example 3
The revenue function (in RM) for selling x kg of butter cake is given by

R( x) = 30x – x2. The total cost to produce x kg of butter cake is given by the cost
function C ( x) = 10x + 20 . Find the demand function and profit function.

Example 4
The demand for an item produced by Delta Sdn. Bhd. is given by p + 0.2x = 100
where p is the price per unit and x is the quantity demanded. The total cost, C(x) of

producing x units of the items is given by C ( x) = 800 + 30x where x is the level of

output. Find
a) The total revenue function.
b) The total profit function.
c) The total profit when 100 units are sold.

Page 17 of 20

AM015/ 9. Application of Differentiation
Example 5
The revenue and cost functions in RM of a product are given as

R( x) = 600x – x2
C ( x) = 200 + 6x + 2x2

a) Find profit function. Hence, determine the quantity to maximize the profit.
b) Currently the company is producing 100 units of the product. Should the

company increase or decrease production?

Page 18 of 20

AM015/ 9. Application of Differentiation
Example 6
A company manufactures and sells x transistor radios per week. If the weekly cost
and price demand equations are

C ( x) = 5000 + 2x and
p( x) = 10 − 0.001x ; 0  x  10000

Find the following for each week;
a) The maximum revenue
b) The maximum profit, the production level that will realize the maximum
profit, and the price that the company should charge for each radio to
realize the maximum profit.

The government has decided to tax the company RM2 for each radio produced.
Taking into account this additional cost how many radios should the company
manufacture each week in order to maximize its weekly profit? What is the maximum
weekly profit? How much should the company charge for the radios to realize the
maximum weekly profit?

Page 19 of 20

AM015/ 9. Application of Differentiation
Exercise
The market research department of a company recommends that the company
manufacture and market a new radio. After suitable test marketing, the research
department presents the following price demand equation:

x =10 000 – 1 000 p
Where x is the number of radios retailer are likely to buy at RMp per radio. The
financial department provides the following cost function:

C ( x) = 7000 + 2x

Find
a) the price-demand equation and the domain of the function.
b) the marginal cost function.
c) quantity of product when the profit is maximum.
d) total profit and total revenue when the profit is maximum.

Page 20 of 20

AM015/ 9. Applications of Differentiation

CHAPTER 9: APPLICATIONS OF DIFFERENTIATION

SUBTOPIC
9.1 Tangent and Normal
9.2 Extremum Problems
9.3 Application of Differentiation in Economics and Business

TUTORIAL 1 OF 7

Learning Outcome:
9.1 a) Find the gradient of the tangent and normal at a point on a curve.
9.1 b) Find the equations of the tangent and normal to a curve.

*Exclude: Parametric, logarithmic and exponential functions

1. Find the equation of the tangent and the normal line on the curves at the given points

below;

a) y  x3  9x; x  3 b) 1 x2  y  5x  2 ;(1, 2)
23

c) y  3 x3  2 x2  1 x  2; x  0 d) y2  x  2  0;1, 1
234

2. The tangent to the curve y  px2 1 at the point 1, q is parallel to the line y  6x  2.

Find the values of p and q.

3. Find the equation of the tangent and normal to the curve y  3  2x  x2 at the point where

the curve meets the y-axis.

4. The curve y  hx2  kx  2 passes through the point (3,5) and the tangent to the curve at
the same point is parallel to the line y  x  2. Find the values of h and k .

5. The equation of a curve is given by 3 y  2x3 y  3 . Find the equation of the tangent to

the curve at y=1.

6. A curve Q is described by the equation x2  4xy  ( y2  27)
a) Find dy in terms of x and y.
dx

b) A point M(x,y) is lies on the curve Q and the tangent to the curve at M is parallel

with y-axis. Find the coordinate of M where x >0

7. A curve is given by the equation y2  x3. Find the equation of the tangent to the curve at

 the point t2,t3 in terms of t.

8. Find an expression in terms of x, y and a for the gradient at any point on the curve

x3  y3  3ay2 with a as a constant. Hence find the equation of the normal to the curve at

the point where y  1 a
3

1|Page

AM015/ 9. Applications of Differentiation

ANSWER FOR TUTORIAL 1 APPLICATIONS OF DIFFERENTIATION

1. (a) y  18x 54 , 18y  x 3 (b) y  6x 8, 6y  x 11

(c) 4y  x 8; y  4x  2 (d) 2y  x 3, y  2x 1

2. p = 3 , q = 4

3. y  2x 3, 2y  x  6

4. h   2 , k  3
3

5. 7y= 18x + 25
6. a) dy  (x  2 y) b) M(3,-6)

dx 2x  y

7. 2y  3tx  t3

8. dy  x2 15x  14a
dx
y2a  y , 12 y

TUTORIAL 2 OF 7

Learning Outcome:
9.2 a) Find the stationary points. *Introduce stationary points of a curve
9.2 b) Classify the type of turning points using the first derivative test.

*Using change of from positive to negative or vice versa

1. Find the stationary points on the curve for the given function and determine their nature

by using the first derivative test.

a) y  x2  6x  8 b) y  x2  x  3

c) y  x3  3 x2  6x  2 d) y  3x4 16x3  24x2  3
2

2. Given the equation of a curve is y  3x2  2x3. Determine the maximum and the

minimum points of the curve by using the first derivative test.

3. Find the local maximum and minimum point of the curve y  6x3  9x2  2 , hence find

the interval where the curve is increasing and decreasing.

4. Given f  x  1 x3  1 x2  6x  8 . Find the intervals where the function f is decreasing or

32
increasing, the relative maximum and minimum.

5. Given the function f  x  x x  22.

a) Determine the increasing and decreasing intervals.
b) Find the coordinates of the points for relative maximum and minimum.

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AM015/ 9. Applications of Differentiation

ANSWER FOR TUTORIAL 2 APPLICATIONS OF DIFFERENTIATION

1. (a) 3,1min (b)  1 , 13  max (c)  1,5 1  max,2, 8 min (d) 0,3min,
 2 4   2 

2,19 no extremum,

2. 1,1max , 0,0min

3. (0,2) local minimum point , (1,5) local maximum point , ,0 and 1,decreasin g

0,1increasin g

4. (-  ,-3 ), (2,  )inc ; (-3,2) dec  3, 43  max ,  2  min
 2   3 
2,

5. (a)  , 2  ,  2,  inc;  2 , 2  dec (b)max  2 , 32  ; min: 2,0
 3   3   3 27 

TUTORIAL 3 OF 7

Learning Outcome:
9.2 c) Find the local maximum or minimum point using the second derivative test

1. Classify the extremum points for each of the following functions by using the second

derivative test.

a) f (x)  5  4x  x2 d) f (x)  x3 x  4

b) f (x)  x3 12x 16 e) f (x)  x2  3 , x  0
c) f (x)  x2 (x  2) x

f) f (x)  2x3 15x2  36x

2. Determine the nature of the stationary point for a curve y  4x2  1 .
x

3. Find the local maximum or minimum point using the second derivative test for the curve

y  9x(3x2 1) where x<0.
4. Find the stationary point on the curve y  x4  2 and determine the nature of the point.

ANSWER FOR TUTORIAL 3 APPLICATIONS OF DIFFERENTIATION

1. a) 2,9 max point d) 0,0 no extremum point, 3, 27 min point

b) 2,32 max point, 2,0 min point e) 1.145,3.931 min point :

c)  0, 0 max po int,  4 ,  32  min point f)  2, 28 max point, 3, 27 min point
 3 27 

2. (0,2), min point

3.  1 , 3 min po int
 2

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AM015/ 9. Applications of Differentiation

4.   1 , 2 max point
 3

TUTORIAL 4 OF 7

Learning Outcome:
9.2 c) Find the local maximum or minimum point using the second derivative test

1. Find the local maximum and minimum point of the curve y  4x3  6x2  3 .
4

2. Determine the maximum point and minimum point of the curve y  x3  x2  2x  4 by
32 3

using the second derivative test.

3. Find the stationary points for the curves below and determine the local maximum and
minimum points.

 a) y  4x3  9x2 12x 13 b) y   x 1 x  22 c) y  x2 1 x2

4. The curve f (x)  ax2  bx  c has a maximum point at (2,10). Find the values of a, b

and c if f passes through point (0,2).

5. Given f (x)  x2 4

8

x3

a) Show that f '(x)   2(x2 16) by using the quotient rule. Hence, find all the

11

3x 3

stationary points.

b) Using the second derivative test, determine the relative minimum or relative

maximum for the curve f(x) at all stationary points.

ANSWER FOR TUTORIAL 4 APPLICATIONS OF DIFFERENTIATION

1 (0,-0.75) min point, (1,1.25) max point

2. (1, − 5), min point (-2,2), max point

2

3. a) (-2,41) max,  1 ,9 3 min b) (0,4) max, (2,0) min c)  1 , 1  max ,
 2 4   2 4 

  1 , 1  max, (0,0) min
 2 4 

4. a= - 2, b= 8, c= 2
5. (a) Shown. SP’s are (4,0.298) and (-4,0.298).

(b) The points (4,0.298) and (-4,0.298) are relative maximum.

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AM015/ 9. Applications of Differentiation

TUTORIAL 5 OF 7

Learning Outcome:
9.3 a) Express the relationship between cost, revenue and profit. *Problems such as cost, revenue

and profit functions
9.3 b) Compute average cost, and marginal cost, marginal revenue and marginal profit
9.3 c) Compute maximum revenue, minimum cost and maximum profit by using second

derivative test. *Relate to problems involving taxes. **Include: Finding quantity that
maximize profit or revenue and minimize cost

1. Given D x  8  0.00025x the price–demand equation and the cost equation are
C  x  600  7x . Express in term of x

a) The revenue function and the profit function
b) The average cost function

2. The total revenue function and total cost function of a company that produced car

components are R(x)  0.2x2  80x and C(x)  0.1x2  40x  6000 respectively

where x is the number of unit components produced. Find
a) the total profit function.
b) the demand function.
c) the profit obtained when 1000 unit components are sold.
d) the number of unit components to be produced to minimized the cost. Hence, find the

minimum cost and the price per unit components.

3. An electronic company is producing microchips for local market. The company finds that

the average cost function C(x) and the demand function p(x) where x is the quantity of

microchips are given by C(x)  20x 10000 and p(x) 100 0.01x . Find
x

a) the total cost function and fixed cost
b) the total revenue function
c) the total profit function
d) the price per microchip when the profit was maximized. Hence, calculate the

maximum profit.

4. The price-demand equation and the average function for a company is given by

p(q)  50 and C(q)  0.5  1000 respectively. Find
q q

a) the revenue function and the cost function
b) the quantity should be produce in order to realize the maximum profit
c) the price per unit (p) that should be charged.

5|Page

AM015/ 9. Applications of Differentiation

ANSWER FOR TUTORIAL 5 APPLICATIONS OF DIFFERENTIATION

1 (a) R x  8x  0.00025x2 ,   x  0.00025x2  x  600
(b) C  x  600  7

x

2 a)  (x)  0.1x2 120x  6000
b) p(x)  0.2x  80
c) (1000)  RM214000.00

d) x  200 , number of units to be produced to minimize the cost is 200 units,

C(200)  RM 2000.00 , minimum cost will be RM 2000.00
p(200)  RM120.00 , price per unit will be RM 120.00

3 (a) C(x) =20x+10000, fixed cost=RM 10000
(b) R(x)= 100x-0.01x2

(c)  (x)  0.01x2  80x 10000

(d) RM 60, maximum profit is RM150000

4 (a) R q  50 q , C q  0.5q 1000 (b) 2500 unit (c) RM1.00

TUTORIAL 6 OF 7

Learning Outcome:
9.3 a) Express the relationship between cost, revenue and profit. *Problems such as cost, revenue

and profit functions
9.3 b) Compute average cost, and marginal cost, marginal revenue and marginal profit
9.3 c) Compute maximum revenue, minimum cost and maximum profit by using second

derivative test. *Relate to problems involving taxes. **Include: Finding quantity that
maximize profit or revenue and minimize cost

1. A manager of the washing machine manufacturer finds that the total cost function Cq
(in RM) and the total revenue function Rq (in RM), of producing and selling q units of

washing machine per week is given as Cq  300q  500 and Rq  400q  q2

respectively. Find the
a) Marginal cost, marginal revenue and demand functions.
b) Profit function and number of units per week that should be produced in order to
maximize profit.
c) Revenue, profit and selling price per unit at maximum profit

2. A computer repair shop states the cost and revenue functions such that the cost function:

C  x  4x2  40x  300 and the revenue function: R x 100x  3x2 . Let x be the

number of computers, find
a) The marginal cost, marginal revenue and demand function.
b) Profit function and the number of computers should be repaired to maximize the profit.
c) The total revenue, total profit and price at maximum profit.

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AM015/ 9. Applications of Differentiation

3. The average cost function and the demand function for a company are given as

C  x  3x 150  2500 and p  300  2x respectively, where x is the number of

x

items produced.
a) Find the cost function, revenue function and profit function.
b) Determine the number of items that should be produced to maximize the
profit.
c) Find the price per unit when profit is maximized.
d) Calculate the increase in cost when the number of items produced is
increased from 32 to 35 units.

4. The revenue function, (in RM) for the sales of x units of a product is R x  ax2  bx ,

where a and b are constants. Given that the sales that maximize revenue is 300 units and
the maximum revenue is RM 27,000.

a) Find the values of a and b.
b) Find the demand function of the product. Hence, find the price of a unit of the

product when the revenue is maximum.

c) Given the cost function C  x  0.05x2  9x  2500 . Find the maximum profit.

ANSWER FOR TUTORIAL 6 APPLICATIONS OF DIFFERENTIATION

1. (a) C'q  300 , R'q  400  2q , Dq  400  q

(b) 50

(c) RM 17500, RM 2000, RM 350

2. (a) C'(x)  8x  40, R'(x) 100 6x , p(x) = 100 – 3x

(b) ∏ (x) = 140x – 7x2 – 300, x = 10 (c) RM 700, RM 400 , RM 70

3. (a) C  x  3x2 150x  2500 , p x  300  2x R x  300x  2x2

(b) 45 (c) RM 210 (d) RM153

4. a) a=0.3, b=180; b) p x  0.3x 180 , price per unit is RM90; c) RM 23015.00

7|Page

AM015/ 9. Applications of Differentiation

TUTORIAL 7 OF 7

Learning Outcome:
9.3 a) Express the relationship between cost, revenue and profit. *Problems such as cost, revenue

and profit functions
9.3 b) Compute average cost, and marginal cost, marginal revenue and marginal profit
9.3 c) Compute maximum revenue, minimum cost and maximum profit by using second

derivative test. *Relate to problems involving taxes. **Include: Finding quantity that
maximize profit or revenue and minimize cost.

1. A company sells x television sets per month. The monthly cost and price–demand

equations are given respectively by C  x  72000  60x and p(x)  200  x ;

30

0  x  6000 .

a) Find the maximum revenue.
b) Find the maximum profit, the production level that will realize the maximum profit,

and the price that should be charged for each television set.
c) If the government decides to tax the company RM5 for each set it produces, how

many units should the company sell each month in order to maximize the profit?
What is the price for each set?

2. A cakes company finds that the total cost function and the demand function are

C(x) 100010x and D(x) 100 x respectively, where x is the number of cakes

produced.
a) Construct the total revenue function and find the number of cakes that should be
produced in order to maximize revenue.
b) Determine the profit function and the number of cakes that should be produced
in order to maximize profit. Hence, find the maximum profit.
c) Compute the maximum profit if each cake is taxed at RM5.

3. Basit Enterprise determined the total cost of producing x units of products A, in RM, is

C(x)  2x3 18x2  48x  650 for year 2013.

a) Determine the production level of product A when the cost is minimum hence
calculate the total cost per week.

b) After 5 years, there is an additional tax RM6.00 charged for each product A.
i) Determine the new cost function for product A.
ii) It is known that the revenue function for product A is

R(x)  5 x3  7x2  546x  500. Find the profit function and the profit gained or
3

lost if the cost as in b i).

8|Page

AM015/ 9. Applications of Differentiation

ANSWER FOR TUTORIAL 7 APPLICATIONS OF DIFFERENTIATION

1. a) RM300,000 b) RM75000, 2100 sets, RM130 each

c) RM64 687.50, 2025 sets, RM132.50 each
2. (a) R(x) = 100x – x2, number of cakes that maximize the revenue is 50

(b) (x)  x2  90x 1000 , number of cakes that maximize the profit is 45, maximum

profit is RM1025.00

(c) 42.5 units of cakes to maximize the profit, maximum profit is RM806.25
3. (a) x  4 ; C(4)  RM682 , (b)

i) C(x)  2x3 18x2  54x  650, ii) (x)   1 x3  25x2  600x 150 ; Profit loss
3

RM4650.00

EXTRA EXERCISES APPLICATIONS OF DIFFERENTIATION

1. Find the gradient of the tangent line to the curve x2  y2  10 at x 1, where y  0 .

2. Find the equation of tangent to the curve y  x3  2x2  x  1 at x  2

3. Find the equation of the tangent to the curve x2  3y2  6y at the point (1, ½)

4. Find the equation of the tangent to the curve y  3x3  4x2  2x 10 at the points where

the curve cuts the y- axis.

5. Find the equation of the normal line for the curve = − 3 + 3 2 + 4 at x=1

6. Find the equation of normal to the curve y  2x3  4x2  x  6 at point 2,4

7. Given a tangent line to the curve y  15 x  2x2 is perpendicular to the straight line
7

y  7x  23 . Find the equation of that tangent line.
6

8. Find the equation of the tangent and normal to each of the following curves at the given
points.

a) y  2x2  4x  3; x  2 d) y  x3  3x2  4x  4; x  1

b) y  2  5x  x2; x  3 e) y  1 2x2 ; x 1

c) 4y  x3; x  2 f) y  x3  6x2 11x; x  3

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AM015/ 9. Applications of Differentiation

9. The average revenue function of x units of product is R(x)  mx  n, where m and n are

constants.
a) Find the revenue and marginal revenue functions in terms of m and n.
b) If the maximum revenue is RM500 and its maximize the revenue at x=10 units,
find the values of m and n

10. The management of a company estimates that the cost (in RM) to produce x units of a

certain product is given by C(x)  0.015x2 10x  300. The revenue (in RM) generated
after selling x units of this product is R(x)  60x  0.01x2 . Find

a) the maximum profit.
b) the selling price per unit in order to maximize the profit.
c) the level of production so that the average cost is minimum.

11. A company produces chairs for the local market. The company found that the average

cost function and the demand function in a year are C(x)  50x  50000 and
x

p(x)  300 0.05x respectively, where x is the number of chairs produced.

a) Find the cost function, revenue function, and profit function.
b) Calculate the number of chairs that will maximize the profit. Hence, find the

maximum profit.
c) Determine the selling price for a unit of chair when the profit is maximum.

ANSWER FOR EXTRA EXERCISES APPLICATIONS OF DIFFERENTIATION

1. dy   1
dx 3

2. y  19x  23

3. 6y=4x – 1
4. y = 2x – 10
5. 3y  x 19

6. y   1 x  34 d) y  5x 1, 5y  x  21
99

7. y   1 x  15
7 44

8. a) y  4x 11; 4y  x 10

b) x  y 11, y  x 5 e) y 12x 3, 12y  x 109

c) y  3x  4, 3y  x 8 f) y  2x, 2y  x 15

9. a) Rx  x mx  n , R'x  2mx  n b) m=5, n=100

10. (a) RM24700 (b) RM50 (c) 141 units
11. a) C(x)  50x 50000 R(x)  300x  0.05x2 (x)  250x  0.05x2  50000

b) 2500 chairs, RM 262500
c) RM 175

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