Basic Engineering Mathematics, Fifth Edition

Revision Test 14 : Differentiation and integration

This assignment covers the material contained in Chapters 34 and 35.The marks available are shown in brackets at
the end of each question.

1. Differentiate the following functions with respect
   tox.
   (a)y= 5 x^2 − 4 x+9(b)y=x^4 − 3 x^2 − 2
   (4)


2. Giveny= 2 (x− 1 )^2 ,find

dy

dx

(3)

3. Ify=

3

x

determine

dy

dx

(2)

4. Givenf(t)=

√

t^5 ,findf′(t).(2)

5. Determine the derivative of y= 5 − 3 x+

4

x^2

(3)

6. Calculate the gradient of the curvey=3cos

x

3

at

x=

π

4

, correct to 3 decimal places. (4)

7. Find the gradient of the curve
   f(x)= 7 x^2 − 4 x+2 at the point (1, 5) (3)


8. Ify=5sin3x−2cos4xfind

dy

dx

(2)

9. Determine the value of the differential coefficient
   ofy=5ln2x−

3

e^2 x

whenx= 0 .8, correct to 3

significant figures. (4)

10. If y= 5 x^4 − 3 x^3 + 2 x^2 − 6 x+5, find (a)

dy

dx

(b)

d^2 y

dx^2

(4)

11. Newton’s law of cooling is given byθ=θ 0 e−kt,
    where the excess of temperature at zero time is
    θ 0 ◦C and at timetseconds isθ◦C. Determine
    the rate of change of temperature after 40s, cor-
    rect to 3 decimal places, given thatθ 0 = 16 ◦Cand
    k=− 0. 01 (4)

In problems 12 to 15, determine the indefinite integrals.

12. (a)

∫

(x^2 + 4 )dx (b)

∫

1

x^3

dx (4)

13. (a)

∫ (

2

√

x

+ 3

√

x

)

dx (b)

∫

3

√

t^5 dt (4)

14. (a)

∫

2

√ 3

x^2

dx (b)

∫ (

e^0.^5 x+

1

3 x

− 2

)

dx

(6)

15. (a)

∫

( 2 +θ)^2 dθ

(b)

∫ (

cos

1

2

x+

3

x

−e^2 x

)

dx (6)

Evaluate the integrals in problems 16 to 19, each, where

necessary, correct to 4 significant figures.

16. (a)

∫ 3

1

(t^2 − 2 t)dt (b)

∫ 2

− 1

(

2 x^3 − 3 x^2 + 2

)

dx

(6)

17. (a)

∫π/ 3

0

3sin2tdt (b)

∫ 3 π/ 4

π/ 4

cos

1

3

xdx (7)

18. (a)

∫ 2

1

(

2

x^2

+

1

x

+

3

4

)

dx

(b)

∫ 2

1

(

3

x

−

1

x^3

)

dx (8)

19. (a)

∫ 1

0

(√

x+ 2 ex

)

dx (b)

∫ 2

1

(

r^3 −

1

r

)

dr

( 6 )

In Problems 20 to 22, findthe area bounded bythe curve,

thex-axis and the given ordinates. Assume answers are

in square units.Give answers correct to 2 decimal places

where necessary.

20. y=x^2 ; x= 0 ,x=2(3)


21. y= 3 x−x^2 ; x= 0 ,x=3(3)


22. y=(x− 2 )^2 ; x= 1 ,x=2(4)


23. Find the area enclosed between the curve
    y=

√

x+

1

√

x

, the horizontal axis and the ordi-

natesx=1andx=4. Give the answer correct to

2 decimal places. (5)

24. The forceFnewtons acting on a body at a dis-
    tancex metres from a fixed point is given by
    F= 2 x+ 3 x^2. If work done=

∫x 2

x 1

Fdx, deter-

mine the work done when the body moves

from the position whenx=1m to that when

x=4m. (3)