EXTENSION 1 MATHEMATICS Exercises and Answers - University of Sydney

BRIDGING COURSE IN MATHEMATICS

EXTENSION 1 MATHEMATICS Exercises and Answers

S. Britton and J. Henderson

School of Mathematics and Statistics Mathematics Learning Centre

Copyright c 2005 School of Mathematics and Statistics, University of Sydney. This book was produced using LATEX and Timothy Van Zandt's PSTricks. The authors thank John McCloughan for his assistance in setting some of the answers and diagrams. February, 2011. Re-typeset in 2014 for the Mathematics Learning Centre by Leon Poladian.

Contents

1 Exercises

1

1.1 Functions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Polynomials and rational functions . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.8 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.10 Counting and Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.11 Combinations and the binomial theorem . . . . . . . . . . . . . . . . . . . 16

2 Answers to exercises

18

2.1 Functions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Mathematical induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Polynomials and rational functions . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Solving equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.8 Inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.9 Applications of calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 Counting and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Combinations and the binomial theorem . . . . . . . . . . . . . . . . . . . 32

A Some standard integrals

35

B The Greek alphabet

36

i

Chapter 1 Exercises

1.1 Functions I

1. Find the natural domains of the functions defined by:

x (i ) f (x) = x2 + 2x + 1

t-1 (iii ) f (t) = t2 + 1

(v ) g(x) = 3 2x - 7

11 (ii) f (x) = +

x x+2 (iv ) f (x) = 2x - 7

(vi ) f (x) = 1 - 4 - x2

Find the ranges in parts (iv) and (v).

2. In each of the following find an explicit formula for the composite functions f (g(x)) and g(f (x)). Also find the natural domain of each composite function.

(i ) f (x) = 3x + 2 and g(x) = -4x - 6

(ii )

f (x)

=

1

and

g(x)

=

x2

-1

x

(iii ) f (x) = x - 1 and g(x) = x2

x2 - 1

(iv ) f (x) = x2 + 1 and g(x) = x + 1

(v ) f (x) = x2 - 1 and g(x) = |x|

3. Differentiate the following expressions.

(i ) (a) x2

(e) x

(b) 12x - x2 (f) 1

x

1 (c) x2

x2 - 1 (g)

x

3x + 1 (d) 2x - 3

1 (h) x3 + x

(ii ) (a) (x2 - 4)(x4 + 3) (d) ex(x + 3)

(g) x3ex sin x (j) x

1+ x

(b) x2 sin x

x+3 (e) ex

1 + tan x (h) 1 - tan x

(c) x2 tan x

sin x (f)

x 1 + sin x (i)

cos x

1

2

CHAPTER 1. EXERCISES

4. (i ) Find dy/dx, when y = 24x + 3x2 - x3. Prove that y has a maximum value of 80 when x = 4. When x = -5, y again has a value of 80. Explain this.

(ii ) Write down the gradient of the function 4x2 + 27/x. Hence find the value of x for which the function is a maximum or a minimum. Which is it?

(iii ) Find the equations of the tangent and normal to the curve y = 2x2 - 4x + 5 at (3, 11).

(iv ) Prove that the curves y = x2, 6y = 7 - x3 intersect at right angles at the point (1, 1).

(v ) It is found that the cost of running a steamer a certain definite distance, at an average speed of V knots, is proportional to

V + V 3/100 + 300/V,

the first two terms representing the cost of power and the third term the costs, such as wages, which are directly proportional to the time occupied. What is the most economical speed?

(vi ) Soreau's formulae for the supporting thrust V and the horizontal thrust H of the air on a plane surface making a small angle with the direction of motion are

H = kv2(a2 + b), V = kv2,

where v is the velocity of the plane and k, a and b are constants. For what value of is the ratio H/V a minimum?

1.2. FUNCTIONS II

3

1.2 Functions II

1. Use the chain rule to find dy in the following cases. dx

(i ) y = cos(sin x)

(ii )

(iv ) y = x2 sin 1

(v )

x

(vii ) y = tan x2 + tan2 x

y = sin(cos x) y= x

2+x

(iii ) y = (sin x)3 - (cos x)3

(vi) y = x + x

2. Given the equations below, find dy at the indicated points using implicit differentidx

ation.

(i ) x3 + y3 = 6xy at (3, 3).

(iii )

2

x3

2

+ y3

=

4

at

(-3 3, 1).

(ii ) y2 = x3(2 - x) at (1, 1). (iv ) x2y2 + xy = 2 at (1, 1).

3. The height of an isosceles triangle of constant base 10 cm is increasing at the rate of 0.2 cm/sec. How fast is the area increasing? How fast is the perimeter increasing when h = 10 cm?

4. A hemispherical tank of radius r metres is filled to the brim with water. As the water evaporates, the volume of water decreases. When the water depth is h metres, the volume of water in the tank is given by

V = 1 h2(3r - h). 3

For a tank of radius 3 metres, find the rate of change of volume when the height is 2 metres and dropping at the rate of 1 centimetre per hour.

5. The volume of a sphere is decreasing at the rate of 6 cm3/sec. When the radius is 10 cm, how fast is the surface area decreasing?

6. Find the Cartesian equations (equations involving x and y only) corresponding to these parametric equations, and identify the type of curve. (i ) x = 6t - 1, y = 2 - t (ii ) x = 5t, y = t2 + t3 (iii ) x = 2 - cos , y = 4 - sin (iv ) x = 2 cos t, y = 16 sin t (v ) x = 2 - 6 cos t, y = 1 + 3 sin t

7. Find parametric equations for the following curves. Give the corresponding values of the parameter. (i ) The cubic y = (x - 1)3 (ii ) The "sideways" parabola x = y2 (iii ) The circle with centre (1, 3) and radius 2 (iv ) The ellipse 16(x - 1)2 + 9(y - 3)2 = 1

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