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Question 1. [16 marks]
Find each of the following limits:
4x2 + 31x + 21
3x2 + 19x 14
11x6 42x5 + 6
32x2 + 8x4 17x6 :
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]
at the point (0;1), if y7 = (x2 1)3 + 4ex tan1 y + .
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
where f(x) is the force at the point x.
(a) Find the indenite integral of the force 10 marks
f(x) = 3x5 15x
x 48e2x +
1 + x2
(b) Hence calculate the exact value of the total force if a = 0 and b = 1 6 marks
i.e. evaluate the integral
x 48e2x +
1 + x2
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.
Evaluate the following indenite integrals.
Check your value of the integral in each part by dierentiation.
(2t + 5) cos 4t dt;
t3 (ln t)2 dt.
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.
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