[Solved] Calculus question

Assignment Prompt & Instructions

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**files (1)**

– ## CALCULUSIREVISONPAPER2.pdf

**## CALCULUSIREVISONPAPER2.pdf**

UNIVERSITY EXAMINATION

INSTRUCTION ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER
TWO QUESTIONS.

QUESTION ONE (30MARKS)

a) Evaluate the following limits

i) 







+→ tt
t

t sin
2lim

0
(2marks)

ii) 






 −−+
→ x

xx
x

11lim
0

(3marks)

b) Use first principle of differentiation to find the rate of change of y(x) with respect to x

given. ( )
x

xxy
31+

= (5marks)

c) For a firm under perfect competition it is given that P=sh. 19,

27285
3

)( 2
3

+++= qqqqC where P represents price per unit, q units of output and

𝐶𝐶(𝑞𝑞) total cost. Find
i) Quantity produced at which profit will be maximum
ii) The amount of maximum profit (7marks)
d) Show that 24 2)( xxxf −= , ]2,2[− satisfies the hypothesis of Rolle’s theorem, find

all x in the interval that satisfy the conclusion of the theorem (4marks)

e) Find
dx
dy given

i) ( )x
x
y 1cos−= (3marks)

ii) 643 2 +−= ttx and 416 ty −= (3marks)

iii) xxy sin= (3marks)

QUESTION TWO (20MARKS)

a) identify and classify the extreme values of the function ( )xxxxf 3632)( 23 −−=
(6marks)

b) Determine the values of a and b for which the function g(x) is continuous everywhere
on real number line

𝑔𝑔(𝑥𝑥) = �
5 𝑖𝑖𝑖𝑖 𝑥𝑥 ≤ 3

𝑎𝑎𝑥𝑥 − 𝑏𝑏 𝑖𝑖𝑖𝑖 3 < 𝑥𝑥 < 4
11 𝑖𝑖𝑖𝑖 5 ≥ 𝑥𝑥

(6marks)

c) Given that
)(
)()(

xv
xuxf = , ( ) 0≠xv , show that ( ) ( ) ( ) ( )

( )( )2

')('
xv

xvxuxvxu
dx

xdf −
= hence find

dx
dy given

x
xy

cos1
sin
+

= (8marks)

QUESTION THREE (20MARKS)

a) Find the equation of the tangent to the curve 132 22 =++ yxyx through the point
(2,1) (3marks)

b) Investigate whether [ ]5,2563)( 2 −−+= xxxf satisfies the mean value theorem
(5marks)

c) Find the rate of change of y(x) with respect to x given
i) ( )323 102ln xyxxyx =+ (4marks)

ii) 231 tx += 2

2

31
2

t
ty

+
−

= (5marks)

iii) )2tan(ln63 3

xexy x −= + (3marks)

QUESTION FOUR(20MARKS)

a) Evaluate ( )xf
x 2
lim
→

given 𝑖𝑖(𝑥𝑥) = � 𝑥𝑥
2 − 4𝑥𝑥 + 6, 𝑥𝑥 < 2

−𝑥𝑥2 + 4𝑥𝑥 − 2, 𝑥𝑥 ≥ 2
(4marks)

b) Discuss the continuity of the function 𝑖𝑖(𝑥𝑥) = � 1
23 2

−

−−

x
xx

5 𝑥𝑥 = 1

𝑥𝑥 ≠ 1

At 𝑥𝑥 = 1 (5marks)
c) The ordering and transportation cost C of the components used in the manufacturing a

certain product is given by 







+
+=

30
200100 2 x

x
x

C 1 ≤ 𝑥𝑥 where C is measured in

thousands of dollars and x is the order size in hundreds. Find the order size that
minimizes the cost (6marks)

d) Gravel is falling in a conical pile at the rate of 100 cubic feet per minute. Find the rate
of change of height of the pile when the height is 10 feet. (assume that the coarseness
of the gravel is such that the radius of the cone is equal to its height) (5marks)

**## CALCULUSIREVISONPAPER2.pdf**

UNIVERSITY EXAMINATION

INSTRUCTION ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER
TWO QUESTIONS.

QUESTION ONE (30MARKS)

a) Evaluate the following limits

i) 







+→ tt
t

t sin
2lim

0
(2marks)

ii) 






 −−+
→ x

xx
x

11lim
0

(3marks)

b) Use first principle of differentiation to find the rate of change of y(x) with respect to x

given. ( )
x

xxy
31+

= (5marks)

c) For a firm under perfect competition it is given that P=sh. 19,

27285
3

)( 2
3

+++= qqqqC where P represents price per unit, q units of output and

𝐶𝐶(𝑞𝑞) total cost. Find
i) Quantity produced at which profit will be maximum
ii) The amount of maximum profit (7marks)
d) Show that 24 2)( xxxf −= , ]2,2[− satisfies the hypothesis of Rolle’s theorem, find

all x in the interval that satisfy the conclusion of the theorem (4marks)

e) Find
dx
dy given

i) ( )x
x
y 1cos−= (3marks)

ii) 643 2 +−= ttx and 416 ty −= (3marks)

iii) xxy sin= (3marks)

QUESTION TWO (20MARKS)

a) identify and classify the extreme values of the function ( )xxxxf 3632)( 23 −−=
(6marks)

b) Determine the values of a and b for which the function g(x) is continuous everywhere
on real number line

𝑔𝑔(𝑥𝑥) = �
5 𝑖𝑖𝑖𝑖 𝑥𝑥 ≤ 3

𝑎𝑎𝑥𝑥 − 𝑏𝑏 𝑖𝑖𝑖𝑖 3 < 𝑥𝑥 < 4
11 𝑖𝑖𝑖𝑖 5 ≥ 𝑥𝑥

(6marks)

c) Given that
)(
)()(

xv
xuxf = , ( ) 0≠xv , show that ( ) ( ) ( ) ( )

( )( )2

')('
xv

xvxuxvxu
dx

xdf −
= hence find

dx
dy given

x
xy

cos1
sin
+

= (8marks)

QUESTION THREE (20MARKS)

a) Find the equation of the tangent to the curve 132 22 =++ yxyx through the point
(2,1) (3marks)

b) Investigate whether [ ]5,2563)( 2 −−+= xxxf satisfies the mean value theorem
(5marks)

c) Find the rate of change of y(x) with respect to x given
i) ( )323 102ln xyxxyx =+ (4marks)

ii) 231 tx += 2

2

31
2

t
ty

+
−

= (5marks)

iii) )2tan(ln63 3

xexy x −= + (3marks)

QUESTION FOUR(20MARKS)

a) Evaluate ( )xf
x 2
lim
→

given 𝑖𝑖(𝑥𝑥) = � 𝑥𝑥
2 − 4𝑥𝑥 + 6, 𝑥𝑥 < 2

−𝑥𝑥2 + 4𝑥𝑥 − 2, 𝑥𝑥 ≥ 2
(4marks)

b) Discuss the continuity of the function 𝑖𝑖(𝑥𝑥) = � 1
23 2

−

−−

x
xx

5 𝑥𝑥 = 1

𝑥𝑥 ≠ 1

At 𝑥𝑥 = 1 (5marks)
c) The ordering and transportation cost C of the components used in the manufacturing a

certain product is given by 







+
+=

30
200100 2 x

x
x

C 1 ≤ 𝑥𝑥 where C is measured in

thousands of dollars and x is the order size in hundreds. Find the order size that
minimizes the cost (6marks)

d) Gravel is falling in a conical pile at the rate of 100 cubic feet per minute. Find the rate
of change of height of the pile when the height is 10 feet. (assume that the coarseness
of the gravel is such that the radius of the cone is equal to its height) (5marks)