School of computing, science and Engineering

Engineering Mathematics E1 CRN: 31886

Phase Test 2

Date: Tuesday 1/4/2014 Time:10:00-11:30 hrs

Instructions to Candidates

Formulae sheets are appended to this paper

Approved electronic calculators may be used.

1.(a)Differentiate the following functions with respect to x:

(i),(4 marks)

(ii),(4 marks)

(iii).(4 marks)

(b)A curve is given by the parametric equations:

.

Determine expressions for in terms of t.

(8 marks)

2.(a)If show that

.(12 marks)

(b)The base radius of a cone, r, decreases at a rate of 0.1cm/sec while the perpendicular height, h, increases at a rate of 0.2cm/sec. Calculate the rate at which the volume, v, changes when r = 1.5cm and h = 2.5cm.(8 marks)

3.(a)Solve the following hyperbolic function, giving your answer correct to four decimal places:

.(6 marks)

(b)Prove that .(8 marks)

(c)Use the hyperbolic substitution to determine the following definite integral:

.

(6 marks)

4.(a)If show that:

.

(6 marks)

(b)Use the double angle formula to evaluate the integral;

(6 marks)

(c)Express in partial fractions and hence evaluate:

(8 marks)