Faculty of Engineering Undergraduate Modular Scheme Session 2007/2008 Semester 2 Level 3 SUBJECT:

EG0822: Technology Mathematics 2

DATE:

13 May 2008

TIME:

15.30 – 17.30 Instructions to Candidates This paper contains SIX questions Answer FOUR questions only All questions carry equal marks

CANDIDATES ARE PERMITTED TO BRING ONE APPROVED CALCULATOR INTO THIS EXAMINATION Candidates are reminded that the major steps in all arithmetical calculations are to be set out clearly. Number of Pages: 1 – 3 +Formula Sheet: 1 page

1.

2.

Differentiate the following with respect to x: (a)

y 12x 7

(2 marks)

(b)

f (x)

6 2 x x2

(3 marks)

(c)

y 4x 3 3x

(d)

f ( x) e3 x sin4x

(e)

y

(a)

(b)

6

(6 marks) (5 marks)

ln 2x x

(and simplify your answer)

(9 marks)

The distance x metres moved by a car in a time, t seconds, is given by: 15 2 x 3t 3 t 4t 3 2 (i)

By differentiating with respect to t, find expressions for the velocity of the car in m/s and its acceleration in m/s2. (5 marks)

(ii)

Find the initial velocity of the car and its acceleration after 2 seconds. (3 marks)

(iii)

Find the values of t for which the car is stationary.

(5 marks)

The velocity at time, t seconds, of an object travelling in a straight line, is v m/s where: v 6 cos(4t 0.6) Find the distance (s) travelled by the object during the time interval between t = 1 and t = 3 seconds. Give your answer to 3 significant figures. (7 marks)

(c)

A particle is moving in a straight line with an acceleration of 8 m/s2. If the particle is stationary when t = 0.5 seconds, find the general expression for its velocity at time t seconds. (5 marks)

Continued…

1

3.

Evaluate the following integrals: (a)

2x 3

3

x 8 dx 7

(b)

(x x

(c)

(3

(4 marks)

)dx

(4 marks)

t 2e t )dt

(4 marks)

2

(d)

(sin3 2 cos3) d

(7 marks)

0

(e) 4.

(a)

1 1

(3e 2 y

3 )dy e2y

(6 marks)

A curve is described by the equation:

y x 4 2x 2 1

(b)

(i)

Find the gradient of the curve at the point (2,9).

(5 marks)

(ii)

Hence find the equation of the tangent to the curve at the same point. (4 marks)

(iii)

State the gradient of the normal at the same point.

(2 marks)

A rectangular piece of land of area 25m2 is to be marked out and a perimeter fence built around it (i)

Sketch a diagram of the piece of land showing its dimensions. (1 mark)

(ii)

Show that the perimeter (P) of the rectangle can be expressed in terms of one of its dimensions (x) as:

P 2x (iii)

50 x

(3 marks)

Given that x can vary, use differentiation to find the dimensions of the piece of ground that requires the minimum length of perimeter fence. You need to justify that your answer leads to the minimum perimeter length. (10 marks) Continued…

2

5. (a)

Given that

8x 14 A B ( x 1)( x 3) ( x 1) ( x 3)

Find the values of A and B. Hence, show that

(5 marks)

8x 14 dx ln1125or7.03 ( x 1)( x 3)

6

4

to 3 significant figures. (b)

Use the substitution u 2x 2 1 to evaluate the integral:

6.

(a)

(b)

(8 marks)

2

x (2x 2 1)dx

(12 marks)

0

(i)

Sketch the curve given by y x 2 and draw in the line y x 6 . Determine the coordinates of the points at which the curve and the line intersect. (6 marks)

(ii)

By integrating, find the area of the region bounded between the curve y x 2 and the line y x 6 . Give your answer to 3 significant figures. (7 marks)

(i)

Use Simpson’s Rule to estimate

4 0

x 3 dx (use 4

subdivisions). (8 marks) (ii)

Use standard integration techniques to evaluate the same integral. (4 marks)

END OF EXAMINATION PAPER

3

FORMULA SHEET Standard Derivatives

y or f ( x )

ax n e ax ln ax sinx cos x (ax b)n sin(ax b) cos(ax b)

dy df or f ' ( x ) or y' or dx dx anxn1 where a is a constant ae ax 1 x cos x sinx na(ax b)n1 a cos(ax b) a sin(ax b)

Standard Integrals

y or f ( x )

ydx

ax n

axn1 C n 1 1 ax e C a

e ax

1 x sinx cos x

or

f ( x)dx

ln x C

sin(ax b)

cos(ax b)

Chain Rule Product Rule

Quotient Rule

cos x C sinx C 1 cos(ax b) C a 1 sin(ax b) C a

dy dy dt . dx dt dx d du dv (uv ) v u dx dx dx du dv v u d u dx 2 dx dx v v

or

fg f g fg

or

f f g fg g2 g

Simpson’s Rule Area

1 (Width of interval) {(first + last ordinate) + 4(sum of even ordinates) + 2(sum of remaining odd ordinates)} 3

The distance travelled s, by an object moving with velocity v between times a and b is given by b

s vdt a