Higher Engineering Mathematics

Assign-08-H8152.tex 23/6/2006 15: 10 Page 329

G

Differential calculus

Assignment 8

This assignment covers the material contained

in Chapters 27 to 31.

The marks for each question are shown in

brackets at the end of each question.

1. Differentiate the following with respect to the
   variable:

(a)y= 5 + 2

√

x^3 −

1

x^2

(b)s=4e^2 θsin 3θ

(c)y=

3ln5t

cos 2t

(d)x=

2

√

(t^2 − 3 t+5)

(13)

2. Iff(x)= 2. 5 x^2 − 6 x+2 find the co-ordinates at
   the point at which the gradient is−1. (5)


3. The displacementscm of the end of a stiff spring
   at timetseconds is given by:
   s=ae−ktsin 2πft. Determine the velocity and
   acceleration of the end of the spring after
   2 seconds ifa=3,k= 0 .75 andf=20. (10)


4. Find the co-ordinates of the turning points on the
   curve y= 3 x^3 + 6 x^2 + 3 x−1 and distinguish
   between them. (7)


5. The heat capacityCof a gas varies with absolute
   temperatureθas shown:

C= 26. 50 + 7. 20 × 10 −^3 θ− 1. 20 × 10 −^6 θ^2

Determine the maximum value ofCand the

temperature at which it occurs. (5)

6. Determine for the curvey= 2 x^2 − 3 xat the point
   (2, 2): (a) the equation of the tangent (b) the
   equation of the normal (6)
   7. A rectangular block of metal with a square cross-
   section has a total surface area of 250 cm^2. Find
   the maximum volume of the block of metal. (7)
   8. A cycloid has parametric equations given by:
   x=5(θ−sinθ) andy=5(1−cosθ). Evaluate

(a)

dy

dx

(b)

d^2 y

dx^2

when θ= 1 .5 radians. Give

answers correct to 3 decimal places. (8)

9. Determine the equation of (a) the tangent, and
   (b) the normal, drawn to an ellipsex=4 cosθ,

y=sinθatθ=

π

3

(8)

10. Determine expressions for

dz

dy

for each of the

following functions:

(a)z= 5 y^2 cosx (b)z=x^2 + 4 xy−y^2 (5)

11. Ifx^2 +y^2 + 6 x+ 8 y+ 1 =0, find

dy

dx

in terms of

xandy. (3)

12. Determine the gradient of the tangents drawn to
    the hyperbolax^2 −y^2 =8atx=3. (3)


13. Use logarithmic differentiation to differentiate

y=

(x+1)^2

√

(x−2)

(2x−1)^3

√

(x−3)^4

with respect tox. (6)

14. Differentiatey=

3eθsin 2θ

√

θ^5

and hence evaluate

dy

dθ

, correct to 2 decimal places, whenθ=

π

3

(9)

15. Evaluate

d

dt

[√t

(2t+1)

]

whent=2, correct to

4 significant figures. (5)