QUESTION 1:

Find each of the following limits:

(a) lim_x→-9 (5x² + 43x - 18)/(2x² + 23x + 45)

(b) lim_x→∞ (13x⁷ - 2 + 42x⁸)/(5x⁴ - 23x⁸ + x⁷)

QUESTION 2:

A rocket of mass m = 5000 kg is travelling in a straight line for a short time. The distance in metres covered by the rocket during this time is described by the function

s(t) = 840t - 5t² - 420 ln (2t + 1)

where t ≥ 0 and time is given in seconds.

(a) Find a function that describes the speed of the rocket.

(b) What is the speed of the rocket at the time t = 10 seconds?

(c) Find all values of time t (if any) when the speed of the rocket is 483 ms^-1.

(d) Find a function that describes the acceleration of the rocket.

(e) Find the acceleration of the rocket at t = 2 seconds.

(f) Find all values of time t (if any) when the rocket's acceleration is 95 ms^-2.

QUESTION 3

A function is defined implicitly by the equation

(x³ + 2)⁴ - y⁴ + 7ln y cos x - x = 15

(a) Find an expression for dy/dx at the point (x, y).

(b) Using part (a) evaluate the derivative at the point (0, 1).

QUESTION 4

The total force (in Newtons) due to a distributed load acting on a beam from x = a to x = b is given by

F = _a∫^b.f(x)dx

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

(a) Find the indefinite integral of the distributed load

f (x) = 2x³ + 14 x²√x -2e^-4x + 8π cosπx

i.e. ∫ f(x)dx.

(b) Hence calculate the exact value of the total force if a = 0 and b = 1 i.e. evaluate the integral

F = ₀∫¹(2x³ + 14x²√x - 2e^-4x + 8π.cosπx/2)dx

QUESTION 5

To help find the velocity of particles requires the evaluation of the indefinite integral of the acceleration function, a(t), i.e.

v = ∫a(t).dt.

Evaluate the following indefinite integrals.

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

(a) ∫(4t³ - 2t)lnt.dt

(b) ∫(t²(e^t - e^-t)dt

QUESTION 6

Towns A and B are located near a straight river as shown in Figure 1 (not drawn to scale). A pumping station is to be built on the river's edge with pipes extending straight to the two towns. Your task is to find where the pumping station should be built (i.e. what is the value of x) to minimize the total length of the pipe required to connect both towns to the pumping station.

Figure 1: Proposed pumping station.

(a) Express the total length of the pipe required as a function of x.

(b) Using Calculus find the value of x (in kilometres) that gives the minimum total length of the pipes needed.

Check your value of x by substitution into the derivative. What is the minimum total length of the pipes required?

(c) Confirm that you have found the minimum length using an appropriate test.