Higher Engineering Mathematics

Assign-06-H8152.tex 19/7/2006 17: 3 Page 247

D

Graphs and Vector geometry

Assignment 6

This assignment covers the material contained

in Chapters 19 to 22.

The marks for each question are shown in

brackets at the end of each question.

1. Sketch the following graphs, showing the rele-
   vant points:

(a)y=(x−2)^2 (c)x^2 +y^2 − 2 x+ 4 y− 4 = 0

(b)y= 3 −cos 2x(d) 9x^2 − 4 y^2 = 36

(e)f(x)=

⎧

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎩

− 1 −π≤x≤−

π

2

x −

π

2

≤x≤

π

2

1

π

2

≤x≤π

(15)

2. Determine the inverse off(x)= 3 x+ 1 (3)


3. Evaluate, correct to 3 decimal places:
   2 tan−^11. 64 +sec−^12. 43 −3 cosec−^13. 85
   (3)


4. Determine the asymptotes for the following
   function and hence sketch the curve:

y=

(x−1)(x+4)

(x−2)(x−5)

(8)

5. Plot a graph ofy= 3 x^2 +5 fromx=1tox=4.
   Estimate, correct to 2 decimal places, using 6
   intervals, the area enclosed by the curve, the
   ordinates x=1 and x=4, and thex-axis by
   (a) the trapezoidal rule, (b) the mid-ordinate rule,
   and (c) Simpson’s rule. (12)


6. A circular cooling tower is 20 m high. The inside
   diameter of the tower at different heights is given
   in the following table:
   Height (m) 0 5.0 10.0 15.0 20.0
   Diameter (m) 16.0 13.3 10.7 8.6 8.0

Determine the area corresponding to each diam-

eter and hence estimate the capacity of the tower

in cubic metres. (6)

7. A vehicle starts from rest and its velocity is
   measured every second for 6 seconds, with the
   following results:
   Timet(s)0123456
   Velocity 0 1.2 2.4 3.7 5.2 6.0 9.2
   v(m/s)
   Using Simpson’s rule, calculate (a) the distance
   travelled in 6 s (i.e. the area under thev/tgraph)
   and (b) the average speed over this period. (6)


8. Four coplanar forces act at a pointAas shown
   in Fig. A6.1 Determine the value and direc-
   tion of the resultant force by (a) drawing (b) by
   calculation. (10)

4N

A

5N 45 °^45 °

8N

7N

Figure A6.1

9. The instantaneous values of two alternating
   voltages are given by:
   v 1 =150 sin (ωt+π/3) volts and
   v 2 =90 sin (ωt−π/6) volts
   Plot the two voltages on the same axes to scales
   of 1 cm=50 volts and 1 cm=

π

6

rad.

Obtain a sinusoidal expression for the resultant

v 1 +v 2 in the formRsin (ωt+α): (a) by adding

ordinates at intervals and (b) by calculation

(13)

10. Ifa= 2 i+ 4 j− 5 kandb= 3 i− 2 j+ 6 kdeter-
    mine: (i)a·b(ii)|a+b|(iii)a×b(iv) the angle
    betweenaandb (14)