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

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 ms􀀀1. 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 ms􀀀2. 3 marks
Question 3. [14 marks]
Find
dy
dx
at the point (0;􀀀1), if y7 = (x2 􀀀 1)3 + 4ex tan􀀀1 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 inde nite integral of the force 10 marks
f(x) = 3x5 􀀀 15x
p
x 􀀀 48e􀀀2x +
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 􀀀 48e􀀀2x +
48
1 + x2

dx:
Question 5. [20 marks]
To help nd the velocity of particles requires the evaluation of the inde nite
integral of the acceleration function, a(t), i.e.
v =
Z
a(t) dt:
Evaluate the following inde nite integrals.
Check your value of the integral in each part by di erentiation.
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

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