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Question 1. [16 marks]

Find each of the following limits:

8 marks

(a) lim

x!7

4x2 + 31x + 21

3x2 + 19x 14

;

(b) lim

x!1

11x6 42x5 + 6

32x2 + 8x4 17x6 :

8 marks

Question 2. [14 marks]

A satellite of mass m = 10 kg is travelling in a straight line for a short time.

The distance in metres covered by the satellite during this time is described

by the function

s(t) = 360t 42t2 360 ln (t + 1)

where t > 0 is the time in seconds.

(a) Find a function that describes the speed of the satellite. 3 marks

(b) What is the distance covered by the satellite by time t = 4 seconds? 1 mark

(c) Find the value of time t when the speed of the satellite is 25 ms1. 3 marks

(d) Find a function that describes the acceleration of the satellite. 3 marks

(e) Find the acceleration of the satellite at t = 1 seconds. 1 mark

(f) Find the time when the satellite`s acceleration is 44 ms2. 3 marks

Question 3. [14 marks]

Find

dy

dx

at the point (0;1), if y7 = (x2 1)3 + 4ex tan1 y + .

2

Question 4. [16 marks]

The total force due to a distributed load acting on a beam from x = a to

x = b is given by

F =

Zb

a

f(x) dx

where f(x) is the force at the point x.

(a) Find the indenite integral of the force 10 marks

f(x) = 3x5 15x

p

x 48e2x +

48

1 + x2

i.e.

R

f(x)dx.

(b) Hence calculate the exact value of the total force if a = 0 and b = 1 6 marks

i.e. evaluate the integral

F =

Z1

0

3x5 15x

p

x 48e2x +

48

1 + x2

dx:

Question 5. [20 marks]

To help nd the velocity of particles requires the evaluation of the indenite

integral of the acceleration function, a(t), i.e.

v =

Z

a(t) dt:

Evaluate the following indenite integrals.

Check your value of the integral in each part by dierentiation.

8 marks

(a)

Z

(2t + 5) cos 4t dt;

12 marks

(b)

Z

t3 (ln t)2 dt.

3

Question 6. [20 marks]

Two ropes are attached to the ceiling at points 6 metres apart. The rope on

the left is 1 metre long and has a pulley at its end. The rope on the right

passes through the pulley and has a weight attached to its end as shown in

Figure 1 (not drawn to scale). The rope on the right is 7 metres long. At

rest the ropes and pulley arrange themselves such that the vertical distance

from the ceiling to the weight is maximized.

6 m

x m

pulley

1 m

weight

Figure 1: Pulley system.

(a) Express the vertical distance from the ceiling to the weight as a func- 4 marks

tion of x.

(b) Using Calculus nd the value of x (in metres) that gives the maximum 16 marks

vertical distance from the ceiling to the weight.

What is the maximum vertical distance?

Check your value of x by substitution into the derivative.

Total: 100 marks