Essential Mathematical Skills (Iqbalkalmati.blogspot.com)

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UNSW

Press

book

ublished by

University of New South Wales Press Ltd

University of New South Wales

UNSW Sydney NSW 2052

AUSTRALIA

www.unswpress.com.au

O Steven Ian Barry and Stephen Alan Davis 2002

First published 2002

This book is copyright. Apart from any fair dealing for

purposes of private study research criticism or

review as permitted under the Copyright Act no part

may be reproduced by any process without written

permission. Inquiries should be addressed to the pub-

lisher.

National Library of Ausbalia

Cataloguing-in-Publicationenby:

Barry Steven Ian.

Essential mathematical skills for engineering science

and applied mathematics.

Includes index.

ISBN 0 86840 565 5.

1. Mathematics. 2. Mathematics roblems

exercises etc. I. Davis Stephen. II. Title.

rinter

BPA


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ONTENTS

reface

Algebra and Geometry

1 1 Elementary Notation

2 Fractions

1 3 Modulus

1 4 Inequalities

1 5 Expansion and Factorisation

1 5 1 Binomial Expansion

1S 2 Factorising Polynomials

1 6 Partial Fractions

1 7 Polynomial Division

1 8 Surds

1 8 1 Rationalising Surd Denominators

1 9 Quadratic Equation

10 Summation

1 11 Factorial Notation

1 12 Permutations

1 13 Combinations

14 Geometry

1 14 1 Circles

1 15 Example Questions

2

Functions and Graphs

2 1 The Basic Functions and Curves

2 2 Function Properties

2 3 Straight Lines

4 Quadratics

5 Polynomials

6 Hyperbola

2 7 Exponential and Logarithm Functions

2 8 Trigonometric Functions

9 Circles

10 Ellipses

2 1 1 Example Questions


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Transcendental Functions

1 Exponential Function

2 IndexLaws

3 3 Logarithm Rules

4 Trigonometric Functions

5 Trigonometric Identities

6 Hyperbolic Functions

7 Example Questions

4

Differentiation

4 1 First Principles

2 Linearity

4 3 Simple Derivatives

4 Product Rule

5 Quotient Rule

6 ChainRule

4 7 Implicit Differentiation

8 Parametric Differentiation

4 9 Second Derivative

4 10 Stationary Points

4 11 Example Questions

5

Integration

1 Antidifferentiation

2 Simple Integrals

3 The Definite Integral

5 4 Areas

5 Integration by Substitution

6 Integration by Parts

7 Example Questions

Matrices

1 Addition

2 Multiplication

3 Identity

4 Transpose

5 Determinants

5 1 Cofactor Expansion

6 Inverse

6 1 Two by Two Matrices

6 2 Partitioned Matrix

6 3 Cofactors Matrix

7 Matrix Manipulation

8 Systems of Equations

9 Eigenvalues and Eigenvectors

6 10Trace

11 Symmetric Matrices


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ONTENTS vii

6.12 Diagonal Matrices

6.13 Example Questions

7 Vectors

7.1 Vector Notation

7.2 Addition and Scalar Multiplication

.3 Length

7.4 Cartesian Unit Vectors

7.5 Dot Product

7.6 Cross Product

7.7 Linear Independence


8 Asymptotics and Approximations

.1 Limits

8.2 L H6pital s Rule

8.3 Taylor Series

8.4 Asymptotics


9

Complex Numbers

9.1 Definition

9.2 Addition and Multiplication

9.3 Complex Conjugate

9.4 Euler s Equation

9.5 De Moivre s Theorem

.6 Example Questions

1 Differential Equations

0.1 First Order Differential Equations

10.1.1 Integrable

10.1.2 Separable

0.1.3 Integrating Factor

0.2 Second Order Differential Equations

10.2.1 Homogeneous

10.2.2 Inhomogeneous


11 Multivariable Calculus

11.1 Partial Differentiation

1.2 Grad. Div and Curl

1.3 Double Integrals



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l l l

2 Numerical Skills 5

2.1 Integration 115

2.2 Differentiation 116

2.3 Newton s Method 117

2.4 Differential Equations 118

2.5 Fourier Series 119

2.5.1 Even Fourier Series 120

2.5.2 Odd Fourier Series 121

2.6 Example Questions 122

3 Practice Tests 23

3.1 Test 1 First Year Semester One 124

3.2 Test 2: First Year Semester One 125

3.3 Test 3: First Year Semester Two 126

3.4 Test 4: First Year Semester Two 127

3.5 Test5.SecondYear 128

3.6 Test

:

Second Year 129

4 Answers 3

5 Other Essential Skills 43

Index 46


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TO THE LE TURER

What do you assume your students know? What material do you expect them to have a vague idea

about (say the proof of Taylor s Theorem) and what material do you want students to know thoroughly

(say the derivative of

sinx)?

This book is an attempt to define what material students should have

completely mastered at each year in an applied mathematics, engineering or science degree. Naturally

we would like our students to know more than the bare essentials detailed in this book. However,

most students do not get full marks in their previous courses and a few weeks after the exam will

only remember a small fraction of a course. They are also doing many other courses not involving

mathematics and are not constantly using their mathematical skills. This book can then act as guide to

what material should realistically be remembered from previous courses. Naturally both the material

and the year in which the students see this material will vary from university to university. This book

represents what we feel is appropriate to our students during their degrees.

We invite you to look at our extensive web site:

http://www.ma.adfa.edu.ad sib/EMS.tml

It contains more questions, solutions, practice tests and Maple code. There is a database of questions

in LaTeX and pdf, which you can use to format your own tests and assignments. We are not concerned

that students may access this database; if they can do the questions in the database then they have, in

effect, learned the necessary skills.

If you have any questions or queries please do not hesitate to email us.

Steven Barry and Stephen Davis

School of Mathematics and Statistics

University College, UNSW

Canberra, ACT,

2600

email: s.barry @adfa.edu.au


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P R TI L FR CTIONS 7

EX MPLES

1

Writing

1

A

in the form mplies

x l) x 1) x + l 2 - 1

The constants and

B

can be found two simple ways. First setting

Alternatively the equation co uld be expanded as

and the coefficients of

x 1

and

xO

equated giving

Solving these equations simultaneously gives

A

-112

and

112.

Thus

2

To expand

3x

using partial fractions write

x + 7) x 3)

giving

Setting

x 3

implies

1

and setting

x

7 implies

A

2.

Alternatively equa ting the

coefficients of

gives

These simultaneous equations are solved for and B to give A

2

and

1.

Hence


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EXAMPLE QUESTIONS 29

2 11 EX MPLE QUESTIONS

(Answers are given in Chapter

14)

Circles and ellipses

I f f

(z)

=

z3 1

what is f

(2)?

23. Draw the circle y2 (z 2)2

=

4.

I f f

(z)

=

z3

1

what is

f

(g)?

24. Draw the ellipse

y2

2z2

=

1.

25. Draw the ellipse 4y2 (z 1)2

=

1.

If f (z)

=

z3 1and g(z)

=

(z 1) what is f (g(z))?

26. Where does the ellipse (z 1)2 2y2

=

1 cut the z

I f f (z)

=

z3

1and g(z)

=

(z 1) what is

f

(g(b))?

axis?

I f f (z)

=

z2 nd g(z)

=

sin z find f (g(a)) and df (z)). 27. What is the equation for an ellipse centred on (0,O) with

I f f (z)

=

z2 1 ind

f

(f (z)).

z axis twice as long as the y axis?

28. What is the equation for a circle centred on (1,2) with

I f f (z)

=

(z 1)2 and g(z)

=

z2 1 find f (g(z))

radius

2?

and g( f (XI).

29. What is the equation for a circle centred on (a, 2) with

If f (z) = 1 ind the inverse f z).

radius 3?

2

find the inverse

f

z).

f f

(z)

=

General

z t l

1

30. Wha t type of curve has equation

If f (z) = 1 ind the inverse f z).

Y + ( z - - 2

=

O?

,2

Lines

Draw the line y

=

-2 1 or z E [0, 11.


2y2 (z q2 2

=

O?


2y (z 1)2

=

O?

Whe re is the zero of the line

y

=

z ?


2y ( z 1)

=

O?

Where does the line 2y z 1

=

0 cross the y axis?

Wha t is the slope of the line?

34. _at type of curve has equa tion

Draw

3y + 3

=

O f o r z E

[0,4].

(z 1)

=

O?

Y 1

Quadratics

35. What is the equation of the quadratic below:

Draw the quadratic y

=

z2 2 1 or z E [O 21.

Whe re are the zeros of the curve

y

=

(Z ) ( ~ )?

(For more questions on manipulation of quadratics see

Chapter 1

0

1

2

3

4

Sines and cosines

36. What is the equation of the shape below:

Draw the curve y

=

2 sin 32 from z

=

0 to z

=

T.

-

Draw the curve y

= cos

from z

=

0 to

2

2

z

=

4T.

Draw the curve y

=

cos 22 1 rom z

=

0 to z

=

2 ~ .

-

What is the period of y

=

sin( I)?

What is the period of y = cos 3z?

What is the period of

y

=

sin(3z I)?

2 3


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34

TR NSCENDENT L FUNCTIONS

EX MPLES

4. If In x 2 and In y

5

then to find ln(x3y2)we write

5

Ifl ny 3ln2 x+c then tofin dy write

In y ln ( 2 ~ ) ~c

== y = e x p ( l n ( 2 ~ ) ~

c

k exp(1n

Z X ) ~ ) ,

where eC k

kpx)3.

6. If x In 3 and y In 4 then to find exp(x 2y) write

7.

If

o

Tie- then

8.

If y a then

lny x lna

y e lna.


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5 DIFFERENTI TION


Answers a re given in Chapter

14)

1. Use

line rity

to find

dyldz:

i)

y 3 s i n z - 5 ~ 0 s ~

ii)

y 3ex 2

iii)

y 3 l n z

iv)

y 2 sinh z 3 cosh z

2. Use the

ch in rule

to find

dyldz:

i)

y sin(2z)

ii)

y sin(%

+

z3)

(iii)

y (z

+

4)3

iv) y (z

+

sin z)5

v)

y sin(lnz2)

vi)

y exp(cos2 2)

vii)

y cosh(2z2)

3 Use the

product

rule or the

quotient

rule to find

dyldz:

i) y zex

COS2

ii)

y

2

iii)

y ex sin z

In z

iv)

y

4

v)

y sin z cos z

In z

vi)

y

x

vii)

y z2sin 2

viii)

y cosh z sinh z

(ix) y

4. Find

dyldz

for these more difficult problems:

2

(i) y exp (z cos z

(ii) y ex cos((2z + I ) ~ )

1

iii)

y

sin z

iv)

y

z

5. Use

implicit

differentiation to find

dyldz:

i) Y

=sin(% 1)

ii)

cos(2y) (1 2)l/2

iii)

ln(y) zex

iv) e e3*

+

5

v)

y y3 z2

vi)

y2 + sin y sin z

vii) y ( z + l ) 2 = z

6 Use p r metric differentiation to find dyldz:

i) y(t) cost, z(t ) sin(t2)

ii)

y(t) et , z( t) t2

iii)

y(t) t2,

z( t ) s int

7.

Find the derivative

dT/dt:

where is a constant.

8. For the following functions find the stationary points and

classify them.

i)

y (z )2

ii)

y z 3 6z2+ 9%+ 1

iii)

y 3z4 8z3

+

6z2

iv)

y zeCX

v)

y z2 n(z)

vi)

y sin z + (1 2) cos

2 for

z [-I, 21

vii)

y (z 1)2ex

9. The function

y

f

(z)

is drawn below. Roughly sketch

the function

f l(z) .

exp z2

vi)

y

2


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58

INTEGR TION



14

1. Find

ii) J +

5)

d z

vi) 1 4 o s h z ) x d z

2. Evaluate

ii) l z / 2 in z d z

16

iii) / z

iv)

1

i n 3 z ) d z

3 Find

i)

J

f z ) z where

ii)

_:

f z ) z where

iii)lmz ) z where

4.

Evaluate the following integrals using a substitution.

5. Evaluate the following integrals using integration by px ts .

e2. sin z d z integrate twice).

J z + 1 ) s i n z d z

J z 2 e x d z

l n z d z b y u s i n g u = l n z a n d d v 1.

6 . Find the following integrals using any method.

i) J z

COS

z

ii) J z in z d z


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M TRIX M NIPUL TION

69

EX MPLES

1. Given that ABC I find B?

ABC I

(A- A) BC

A- I

pre multiply both sides by

A-

B (CC- )

A- IC- post multiply both sides by C-

B A- C- simplifying

(CA)- .

2. If A PDP- , then A3 is

A3 (PDP- ) (PDP- ) (PDP- )

PD ( P - ~ P ) P - ~ P D P - ~

P D ~ P - ~ P D P -

since PP-I I

pD3p-I again since PP-I I.

3.

If Av Xv then

-d -d

4. If Av Xv then if A- exists then

-d -d

See Section 6.9 on eigenvalues since this example shows that if A has eigenvalue

A

with eigen

vector

v,

then

A-

has eigenvalue

1/X

for the same eigenvector.

-d


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SYST MS

OF

EQU TIONS

7

the augmented matrix is

This gives the straightforward solution by back substitution of x -61 y 40 11.

2 Consider the system

written as

Ax

b

such that

The matrix

A

has inverse

A I

[ ] s o


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EX MPLE QUESTIONS 75

6 13

EX MPLE QUESTIONS

Answers are given in Chapter

14

1. F in d A B , A B , B A nd th e tr ac e A ):

ii)

iil)

2. F ind A B :

3 Find A t , A t A , A A t

6

0

ii)

; ;

4.

Find the inve rse, if it exists, of the matrices

5. Find the deteiminants of the following matrices.

3 0

4

iii) 0

o

0 0

6

6 Solve the system of equations

7. First showing that anon-trivial solution does indeed exist,

solve

8.

For what values of and

c

do you get

i) one solution,

ii) no solution,

iii) infinite solutions,

for the system

9. Find the eigenvalues and eigenvectors of A :

ii)

A = [ ;

8


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14

Evaluate the sum

2

, 3 and IIuII:

N

i) u -2, -I), v 1 , l )

N N

ii) u 3,4),v 4,3)

N

iii) u -2, I), v -1, -1)

N N

iv) u 3,4,2),v 1,1,1)

N N

v) 2 3,1 ,1 ,O), v l ,O , 1 , l )

N

vi) u = 2 i 3 j k S v = - j - k

N N N N N N N N

vii) u = i j , v = i - 3 j

N N N N

N

For the above vectors verify the triangle inequality that

112

I 1

5

11211

ll ll.

In the diagram below write down the two vectors u and

N

v

in algebraic foim then find and draw the vector u v .

N N N

Evaluate the sum u v and

1121

v 1 :

N N N N

i) 2 3, 2, -11, v -1 , -2 , l )

N

ii) u 1,0,9),v -2, -2, -2)

N N

iii) 4, -4, -3), v 8,7,1)

N N

Find u .v ,u v and cos where is the angle between

N N N N

t h e u a n d v :

N N

i) 1,2, I), v -1,3,1)

ii) -3,2, -11, v 6,1 ,1)

N N

iii) u 2,3 ,0) ,v 4,1, -2)

N N

iv)

u

O,O,O),

v

1,4 ,3)

N N

For the previous question v erify that w u v s

N N N

orthogonal at right angles) to both

u

and

v .N N

Deteimine whether the following vectors x e inex ly

independent

Find a number

c

so that

1,2,c)

is orthogonal to

2,1,2).

Find the vector which goes from the point 1,3 ,1) o the

point 2,5,3).What is the length of this vector?

Show that the line through thep oints 1,1,1) and 2,3,4)

is perpendiculx to the line through the points 1,0 ,0)

and 3, -1,O).

Show that a b

x

c) can be written as

Verify the above equation using the vectors

a = l , 1 , 2 ) , b = 1 , 0 , 1 ) , ~ = 0 , 1 , 1 ) .


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9

ASYMPTOTICS

AND

APPROXIMATIONS



14)

1.

Find the limits as x f the following functions.

(i) (x 2) (X 3)

(ii) x cos x

(iii) ex xe-

sin(% )

(iv)

(x )

x3 x2 1

() x2 3% 2

x2 1

(vi)

21%

2.

Use L HBpital s Rule t o find the following limits.

sinx x

(i)

lim

+O 23

x 3 - 3 x + 2

(ii) lim

x+1 23 1

x3 22 1

(iii) lim

x+1 23 1

xcosx inx

(iv) lim

X + O x3

3 Find the first two non-zero teims in the Taylor Series for

the following fun ctions as x 0.

(i) cos x

sin x

(

1)

x2 1

(iil)

11%

(iv) xex

cosx 1

(v)

x

(vi) sinhx

cosh x

1

(vii)

2 2

Find the leading order behaviour for the following

functions as x

w

x 2 + x

(i) 3x2 22 1

(ii)

Jx3 2% 1

x3 22

(iii)

3x2 V G T i

x2 1

(iv)

11%

sinh(3x)

(vi)

e4

xe-x

(vii)

inh x

x e-

(viii)

xe-x

ex

(ix)

inh x

Find the first three non-zero teims of the Taylor series

as

x

or the functions f

(x) = ex

and

g(x) = sinx

and hence find the Maclaurin series for

(i) ex sin x

(ii) e2

(iii) sinx2

(iv) ex3

Find the first two non-zero teims for the Taylor series

about x = a / 2 for

(i) sin x

(ii)

cos x

Find the first three teims in the Maclaurin series for e- '

and hence find an approximation fo r

1

e-X' dx

where is a small number. (Expa nd the exponential and

then integrate.)


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EX MPLE QUESTIONS 95

9 6

EX MPLE QUESTIONS


14)

1.

Write the following in the fo im

a bi.

(i) (2 3i) (7 9i) 3 13i)

(ii)

(1 ) (5 ) (6 6i)

(iii)

5(8 9i) (12 4%)

(iv) 3(2 ) 2i (l )

(v)

(7 6i)'

(vi)

(1 )3

(vii) (8 2i) (3 5i)

(viii)

(1 ) (1 )

(ix)

(2 3i) (2 3i)

(x)

(1 4i) (2 )

6 2i

(xi)

7i

3 - i

(xiv)

(2 ) (5 2i)

2. Find

x, y

if

(i)

2% 3yi = (7 )(l )

(ii)

(x2 6) + x i

=

3(2 3i)(l )

3 Write the following in polar fo im.

(i)

z

= -1

(ii)

z = -i

(iii)

z

=

3i

(iv) z = -1 i

(v)

z = l + i

(vi)

z

= 1

(vii)

z

=

-2 2i

5

(viii)

z

=

)

2

(ix) z = 1 i

(x) -

3i

(xi)

z = 5 + 5 i

( x i )

z = & + i

(xiii)

z

= -4 d5

4. Write the following in the foim

z

=

x y.

(i)

2ei /'

(ii)

3ei /4

a

(iii) cis

6

l l a

(iv) cis

(v)

ei /3

5. Find

zlz2

and

z:

if

a

-5a

(iv)

zl

= cis

z2

=

3

6

6 Use De M oivre s theorem to find all

z

where

(i)

z2 = i

(ii)

2 4

= 1

(iii)

z3

=

4@(-1+ i)

(iv) z2 = -i

(v)

z6

= -1

7.

Use Euler s theorem to show

(i)

cosh ix

=

cos x

(ii) sinh ix = i sin x

(iii) cos(x y) = cos x cos y in x sin y

(iv)

sin(% y)

=

sin x cos y cos x sin y

8.

Use De M oivre s theorem to show

(i)

z z- = 2 cos

o

(ii)

cos 2%

=

cos2 x in2 x

(iii)

sin 2%

=

2 sin x cos x

9.

Show that the solution to

where

a, b, c

are all real, must be of the foim

10. Show that

i(2 2i)6

=

2


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GRAD DIV AND CURL

109

EXAMPLES

1 I f f 2x2y z3 th n

2 I f f x xyz th n

a f a f a f

f

( a x a y , a,

(1 yz , xz , xy)

and

xyz 3y sinx

( [sin x] -[3y] i [ s i n ] -[xyz] j

a~V L a~Y

i

c o s x - x y ) j - x z k

(0,xy cos x , - 2 2 .

4 1f

f

x2

Y 2

Y z 2 h n

5 I f f x y z2

th n V

(1,1,2z)

and

v 2.

6

I f v

( x , , z x ) th n

V

v 2 andV (0 , 1 , l ) .


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122 NUMERICAL SKILLS


Answers are given in Chapter 14)

Find an approximation to the integral

11. Numerically solve the the equation

using an interval size of h

=

0.2.

with

y(0)

=

0

to find an approximation to

y(0.4)

using

h

=

0.2.

Approximate the integral

12. Find the Fourier series of y

=

x 1 over the interval

1 ex dx

x [-a, a] .

13. Find the Fourier sine series of

using an interval size of h

=

0.5.

- 2 < x < o

A function f ( x ) s defined by the set of points

f

=

{ 0 5 x

<

2.

Hence find an approximation for the integral

L X I ax.

15. Find the Fourier series for

For the function f ( x )

=

s inhx find an approximate

value for

f l ( l )

using a discretisation

h

=

0.1

and the

i

1, -2 < x

<

-1

central differencing rule. f ( x ) = 0 , - 1 < x < 1

1, 1 < x < 2 .

A function f ( x ) s given by the set of points

O , O ) ,

( L l ) , 2, 3) , 3,513 (4 ,6 ), 537).

16. A double derivative can be approximated by

Find an approximation fo r f' ( x ) t all the points

x

=

0,1,2,3 ,4 ,5 .

f ( x )

=

f l ( x + h ) l ( x h )

2h

Use Newton s m ethod to find a zero of

y =

x

3

start-

If You now replace f l ( x h ) and f l ( x h ) by their

ing with

xo

=

1.

respective definitions, for example

Use Newton s method to find an approximate value for

f l ( x h )

=

f

( x 2 h )

X I

51/3.

(Hint: solve

y

=

x3 5

=

0

with a starting guess

2h

of

xo

=

2.)

derive the approximation

Find the zero to two decimal places of y

=

sin x using

Newton s method starting with xo

=

3.

f x )

=

f ( x 2 h) 2 f ( x ) ( x 2 h)

Numerically solve the differential equation

(2hI2

or by rewriting

H

=

2h

as

f ( x + H ) - 2 f ( ~ ) + f ( x - ~ ) .

f ( x )

=

rr

-

with y(0)

=

1 to find an approximation to y(0.3) using

a step size of

h = 0.1.

17. For the function

f ( x )

defined by the set of points

L y 2 + 1

find estimates of

f l ( l )

and

f1'(3).

dx

with y(0)

=

0 to find an approximation to y(0.6) using

a step size of h

=

0.2.


132/158


133/158


134/158

TEST

:

FIRST

YEAR M ST R

ONE

125

13 2

TEST

:

FIRST YEAR EMESTER ONE

1. Solveformifm(m-4)+4=0.

2. Factorise x3 2x2 x.

3. For what values of x is 1x 21 > l ?

4. Expand ( a + b+ c) (a - b-c).

1

5. Rearrange : 3 to find x.

6. Solve for r if 6r2 6r 1 0.

dy 4

7. Find f y .

dx x2

8. Find fl(x) iff (x) C O S ~ X .

3

x

9. Find fl (x) if f( x)

2 3 .

lo. Find fl (x ) i f f (x) xe"

11. Evaluate x6 dx.

12. Evaluate x.

iX

13. If

In

a 2 and

In b

what is

In

ab2?

14. The equation 2x2

y2

3 0 is what type of curve?

15 Evaluate the sum

2i 1 .

16. I f f (z) z2 and g(x) x 2 what is g(f (a ))?

5

17. Evaluate .

3

x

18. Find the partial fraction form for

(X )($-3) '

19. Use polynomial division to divide x3

1

by x 1.

20. Find the inverse function f -l(x) if f (x) m


135/158

126 PR CTICE TESTS

13 3

TEST : FIRST YEAR EMESTER TWO

1

Findx if x(3x 5) = -1.

2. For what values of x is lx 31 < 15?

3. Expand (3x2 42) 2.

4. Rearrange =

-

o find x.

l x

5 Rearrange

-

= to find x.

z

6. Find f'(x ) if f (x)

=

3 cos(x2).

dy

7. Find f y = x2e .

dx

dy

8. Find n terms of x and y if y2 y = sinx2

dx

dy

9. Find n terms of t if y (t)

=

t2 and x (t)

=

sin t.

dx

10. Evaluate /x'2dx.

11. Evaluate

I

2 dx.

12. I f f (z)

=

22 and g(z)

=

1/z2 what is g(f ( z))?

13. Simplify exp(1n x 2 In y) .

14. Find the determinant

17. Find (1,2,3 ) (1,1,1 ).

18. Write 2 sinh x 2 cosh x in terms of exponentials and simplify.

19. What does approach as x m.

x + m

20. How many ways are there of choosing a team of 4 people from a group of

8

people (with no

order)?


136/158

TEST

: FIRST

YEAR EMESTER TWO

127

13 4

TEST : FIRST YE R EMESTER TWO

1

Solve form if m m

2 )

- .

2

For what values of t is

I -

1

<

l o?

Expand

( x +

:

.

6 1

Make as one fraction the expression

.

2 6

m

Rearrange the following equation to find m:

t2+

3.

1 - m

Find

f l ( x )

i f f

( x )

1

1 3 ~ ) ~

Find

x l ( t )

f

x ( t ) e C t

sinh

2t.

dy

Find f x y 2

+

y coshx.

d x

1

Find / + 4x 2 dx .

Find llull if u ( 2 , 2 , l ) .

The equation x 2

+

x 2

+

3y 0 is what type of curve?

Find rin x d x .

1

15. Find the partial fraction form of

( x 2 ) '

xe

16. What does

( x )

approach as x cm?

sinh

( x )

4

.

ind the determinant

17. Find / sin x2 1 ) d x by writing u x2 1.

What is the angle between the vectors ( 1 , 2 , 1 )and (2 , 1 , O ?

-1 2

4

0

-3

1

0

-2

18. Find

l2x1 dx.

19. Is

( x ) x

cosh

x

an odd or even function?

20. What is ATA if A [: I


137/158

128

PRACTICE TESTS

13 5

TEST : SECOND YEAR

1. Rearrange

z

to find

y.

1

2. Simplify

a n 2x .

cos2

x

3. Find

x

if

x 2 52 6 = 0 .

dy

5. Find f

y =

sin

3 x 2 .

d x

dy

6. Find f

y = x 2

sinh

22.

d x

1

7. Find /

x .

22 3

8. Find /

eCt dt.

2

9. Find

2 x d E d x .

lo. Find

( 5 , 2 ) ( - 7 , l ) .

11. Find

( 2 , 3 , - 1 ) ( 2 , 0 , 3 ) .

12. Find the eigenvalues of

[

I

dy

Y

13. Solve the differential equation for

y ( x )

f

x x 3

d Y

14. Find

y ( x )

if

-9y.

d x 2

4 2i

15. Find the imaginary component of

- i

16. Write z

= 2 2i

in polar form z

= reie .

17. Find the inverse of the matrix

k ;

I

18. Find the first two non zero terms of the Taylor series for

y =

sin

x

about

x = 0 .

cosx

- 1

19. What does

approach as

x O?

x 2

20. Iff

( x , ) = x 2 3 y 4

what is f ?


138/158

TEST

6:

SECOND YEAR

29

13 6 TEST

:

SECOND YEAR

Factorise 2x2 32 2.

Iflnx 7andlny 2findln

f

.

af

Find f f (x, t) eCt sin TX.

a t

Find the imaginary component of z 3ei I2.

Simplify

2x

asx+cm.

4x2 22

For what values of p is 15p 41 > l ?

Simplify cos(n.rr) where n 0,1,2, .

dy

Solve the differential equation xy.

dx

Solve the differential equation y 6y' 5y 0 for y (x).

Find / cos

TX

dx.

By letting sin x find dx.

Findu

x

v i f u (1,0,1) andv (0,1,1/2).

Find the eigenvalue of [ : corresponding to the eigenvector v (1,2) .

Find the first three terms of the Taylor series for exp(x2) about x 0.

I f f (x, y) x2 3Y2 ind the slope in the direction (1, l ) when at the point (0, l ) .

I f f (x, y) x2 3Y2 ind the direction of maximum slope at the point (0 , l ) .

Find the eigenvector of

[

: corresponding to eigenvalue 1.

If v (z x, xy, 22) what is V v?

Iff x2

y

wha t i sv2f?


139/158

his



140/158

CH PTER

4

NSWERS

Chapter

Algebra and Geometry

x2 32 15

ii)

(x 2)(x 3)

x2 7 2

iil)

4

2

x

vi)

+ 1

i) d >

512

ii)

d <

-8

iii)

5

<

x

<

15

iv)

z > 5 o r z

5-11

v)

a > - 3 o r a < - 5

vi)

-3

<

x

<

5

i)

x2 9

ii)

9 x2 - 2 4 x + l 6

iii) x3 xZy xy2 y3

iv)

3x3 2x2 27 18

v)

x3 122' 48 64

i)

x4 8x3 24 ' 32 16

ii) xs 8x7 28x6 56x5 70x4 56x3

28 ' 82

1

iil) 21

5.

- 3 1 3 1

i)

2 2 - 2 2 2 - 4

1

3

ii)

- 1

x + 2

1 1

iil)

+ 3 x + 2

2

1

iv)

+ 4 2 - 2

1

ii)

3

iil) 3

3

v

iv)

i) y ( x + ~ ) ( x + )

ii)

y (x 5)(x 1)

iii)

y (x 5)(x 1)

iv) y (x 5)(x 1)

v)

y (22 1)(x 1)

i) x -2, -2

ii) x

-6, -1

iil) x -4,3

iv)

x -2,

v) x -4,

- 1 0 - 1 - 0

vi)

x

2 ' 2


141/158

132 NSWERS

2

i) (x 1)

ii) (x 1)

iii) x2 3 1

10.

i) 720

ii) 30

iii)

15

iv) 27

Chapter 2

Functions and Graphs

1.

9

2. 93 1

3. (x 1)3 1

4. b 1)3 1

5.

f (g(a)) sin2(a),g(f (x)) sinx2

6 f (f (x) ) (x2 1)' 1

7 f(g(x)) ( x ~ 212, g( f( x) ) (X 114 1

1

8

f - l ( x )

- 1

1

9 f - l (x ) 1

x


142/158


143/158

134 ANSWERS

i)

0 x/4: cos 0

I/ ,

s i nO = l / & , t a nO = l , s e c O = & .

ii)

0 13x16: cos 0 a / 2 ,

sin 0 112, ta n 0

116,

ec 0 2 / a .

iii)

0 2x13: cos 0 -112,

sin 0 a / 2 , a n 0 - , sec 0 -2.

iv)

0 -5~13:cos 0 112,

sin 0 a / 2 , a n 0 a ec 0 2.

v)

0 5x14: cos 0

-I/ ,

sin 0 -I/&, ta n 0 1, sec 0

- .

1 1 cos-

4

12

12. t 170000

13. o

Chapter

Differentiation

3cosx+ 5s inx

3ex 2%

3

x

2 cosh x 3 sinh x

5(x

+

sin x ) ~1

+ cos

x)

xp(cos2 x)2 sin x cos x

4% inh(2x2)

inx cosx

2-

x2 x3

(cos x

+

sin x)ex

4 i)

exp(% os x2)(cos x2 2%' sin x2 )

ii)

ex(cos(2x + 1)' 4(2x 1) sin(2x + 1)')

2

iii)

(2

+

x2)3/2

cos

x sin x

iv)

2

(X

112

(X+ 113

v)

cos(x2

+

exp(x3

+

x))

x

(2%

+

(3%'

+

1) exp(x3

+

5))

dy cos(x 1) cos(x 1)

i)

dx 2~ 2 J s i n ( x 7

dy

iii)

ex (x

+

l)y ex (x

+

1) exp(xex)

dx

3e3 3e3

iv) y

x eg 5

+

e3

du 2%

v)

x

1

+ 3y2

dy cosx

vi)

dx 2y +c os y

dy

1 - Y

vii)

d x x + 1 - 2 y

6

d~ sin t

i)

dx 2tcos(t2)

(..

d y - e t

11

dx

dy 2t

i i )

x cost

8

i) x 2 minimum.

ii)

x 3

minimum,

x

1maximum.

iii)

x 0

minimum,

x

1 nflection.

iv)

x

1

maximum.

v)

x e-'/'

minimum.

vi)

x 0

maximum,

x

1minimum.

vii)

x

-1 maximum,

x

1minimum.

9.


144/158

NSWERS 135

Chapter

Integration

7

i)

+ c

2x2

x6

ii) 1 2 1 n 1 x 1 + + c

6

e7x

e-14x

iil)

14

+ c

iv) + + + + c

10 11 12 13

1

v) cosh 2% c

2

vi) 4sinhx -ex c

i) 1

-e-5

ii) 0

iii)

16 1n 2

2

iv)

3

vi) 7(1 -3)

5

i)

1

ii)

12

1

iil)

2

1

iv)

2

1

i)

--e- 2

2

+ c

ii)

? ln1x2+4x+51+c

iii)

In lnyl c

1

iv)

( 7 x 4 1)3/2 c

21

vi) 2efi c

vii)

1 cos(?T2))

viii) e-3 e-2

5.

i) -2

1

ii) n(3)

2

1

iii)

e 2 2 sin x os x) c

5

iv) - ( x + l ) c o s x + s i n x + c

v)

e x(x2 - 2x+ 2) + c

vi)

xl nx x c

6

i)

-In Icoszl c

ii) ( 2 - x 2 ) c o ~ x + 2 x s i n x + c

iii) 9 e ~ / ~ ( x3) c

2 2

iv)

-u5/' -u3/' c

5 3

v) arcsinh x c

Chapter

Matrices

ii) A :]

A -57

8

-44

.A

-42

I

-1

-59

trace A)

5

22

I


145/158

136

NSWERS

iii) ; ;

ii) At

=

[ :

I

AtA

A

85 35 ]

35 41

AAt = 3:

1 ::

4 2 - 2 0 7 4

i) - 38

ii) -36

iil)

0

8

i) 2112

ii) = 2112, c

iii)

=

2112, c

=

Chapter

Vectors

vii) 2 4 5

< r 45

i) u . v = 6

. .

x

v =

( - 1 , - 2 , 5 )

. .

cost

=

ii) u . v = - 1 7

. .

x

v =

( 3 , 3 , - 1 5 )

. .

17

cost

=

dad

iii) u .

=

11

. .

u x v

= ( - 6 , 4 - 1 0 )

. .

11

c o s t =


146/158


147/158

138

NSWERS

Chapter

9

Complex Numbers

(i) 6 25i

(ii)

1 2 + 6 i

(iii) 82 33i

(iv) 8

(v)

13 84i

(vi) -2 2i

(vii)

34(1 )

(vii1) 2

(ix)

13

(x) 6 7i

26 38.

(xi)

z

53 53

( x i )

-i

(xiii)

$(1

37 9

(xiv)

45 145%

2.

(i) x = 4 , y = 2

(ii)

x = 3

3

(i) cis

a

(ii) cis3a/2

a

(iii)

3

cis

2

3a

(iv) cis

7a

(vi) cis

4

5 a

(vii) 2 4 5 is

1 a

(viii)

5

cis

a

(ix)

2

cis

3

(x)

2v%

cis

2x13

(xi) 5 4 5 is

4

a

( x i ) 2 cis

6

7a

(xiii)

2 4 5

is

(i) 2i

3

(ii) 1 )

45

1

(iii)

( a )

2

1

(iv)

( 4 )

(i) ei /4 1 i)

ei5?r/4

1

+ i )

(ii)

1,

f i

(iii) 2ei?r/4 2eill?r/12 2e-i5?r/12

(iv)

&(I ) , &(-I +i )

(v) cis

(i(a/6 ka/3)), k

0, . .

5

or

z = f i , i ( a & i ) , $ ( - a f i )

Chapter 10

Differential Equations

1

(ii)

+ c

x

(iii) y = x - l n 1 x + l I + c

1

(iv)

y l n x c

5

(v) y = 3 ~ ~ 0 ~ ~ - 3 s i n x + c

1

2. (i)

x = y 4 c

4

(iii)

y cx 1

(iv)

-3e-'9 2e3


148/158

NSWERS 139

6 i)

y

=

s i n % +ccosx

Y(X)= c1e4x c2e-4x

y(x) = e-2x(cle-fix c2efix)

Y(X)= c1e2x/3 c2e-x/4

y(x) = cl cos 32 c2 sin 32

Y(X)= c1e-4x c2xe-4x

y(x)

=

~ l e - / ~ c2ex/4

Y(X)

=

c1 c2e-5x/2

fi

Y(X)= e-x/3cl

cos

3

fi

+e-x/3~2sin x

3

y(x) = ~ 1 e ~ ~c2xe5

3

y(x) = ~ 1 e ~ ~c2e-2x

y(x)

=

6 42 x2 cle- c2xe-

1

y(x)

=

c lc o sx + c2 si n x- c o s 3 x

8

1

y(x)

=

e 2 cl cos x c2 sinx

5

24 18

y(x) = cos 22 in(2x) cle-

25 25

c ~ x e - ~

1 1

y(x) = c l e x + c 2 e - x + x e x - e x - x 2 - 2

2 4

hapter

Multivariable alculus

i)

= 2 y

= 2 y + 2 x

8 f = 2

BxBy

ii) = y cos xy

f =

xcosxy,

BY

= cosxy -xysinx y

iii)

= 22 in(% Y)

f = in(% y)

BY

=

os(x y)

iv) = ex-

f = -ex-Y

BY2f

ex-a,

BxBy

v) = 3yx2

f = % 3

BY

2f %2

BxBy

2.

i)

(2x+2z,-2y,2x)

ii)

sec2(xyz)(yz,xz, XY)

iil) 112

2 f l

vi)

22 4y 62

vii) 2 + 2 + 2 = 6

viii) 2(yz)2 2(xz)2 2(xy)2

ix)

P O

2% 2 ~ )

x)

PY

2, 2%)


149/158

140

NSWERS

i) 4 , 3 )

ii) 5

iii)

f 3 , - 4 )

iv)

fi

i)

- 1 , - 1 )

ii) fi

iil)

f 1 , - 1 )

iv) d5

4 t 3 t 2

i)

54

..

1)

iii)

416

4 4

x 2

+

y 2 d x d y

05

3

x + y d y d x

20

Proof by expansion.

Chapter 2

Numerical Skills

3.

L

x ) dx 25 . 5

16.

Easily proved.

17. f l l ) 2 , f l 1 3 ) - 2

Chapter

3

Practice Tests

Test

1:

Section 13.1

1

9. x - -

2x2

10.

c o sx - x s i n x

11. 1

12. x - y

13.

Parabola.

14.

2

15. + 1

a

16. 18

17. 15

18. A - 1 , 1

19. x 4 + 4 x 3 + 6 x 2 + 4 x + 1


150/158

NSWERS 141

Test 2: Section 13.2

12.

l n x

2

14. Ellipse


2 cos x2

8

2y

2t

9.

os t


2 - 2

4.

f i

t2 3

5.

2

6

6

1 3x)3

7. e-t(2 cosh 2t inh 2t)

sinh x y2

8

2yx

10. Parabola

11. 2

12.

-5

19. odd


151/158


152/158

CH PTER

5

OTHER ESSENTI L SKILLS

Use these pages to write any other essential mathematics skills your lecturers think are appropriate


153/158

44

OTHER ESSENTI L SKILLS

his



154/158

OTHER ESSENTI LSKILLS 45

his



155/158

ndex

absolute value, 3

algebra, 1

angle, 35

antiderivative, 5 1

approximations, 87

area

integration, 55

numerical, 115

asymptotics, 87

augmented matrix, 70

binomial expansion, 5

calculus

differentiation, 41

fundamental theorem, 53

integration, 5 1

Cartesian unit vectors, 80

central difference, 116

chain rule, 45

characteristic equation

differential equations, 100

eigenvalues, 73

circle, 27

cofactors matrix, 64,68

combinations, 13

complex number, 9 1

conjugate, 92

De Moivre s theorem, 94

Euler s equation, 93

imaginary, 92

polar form, 93

real, 92

conjugate, 92

cosh, 38

derivative, 43

integral, 52

cosine, 14, 26

derivative, 43

integral, 52

cross product, 82

curl, 108

De Moivre s theorem, 94

definite integral, 53

denominator, 2

surd, 10

derivative, 4 1

determinant, 63

diagonal matrix, 74

differential equations, 97

characteristic equation, 100

first order, 97

homogeneous, 100

inhomogeneous, 102

integrable, 97

integrating factor, 99

numerical, 118

particular solution, 102

resonance, 104

second order, 100

separable, 98

undetermined coefficients, 103

differentiation, 41

chain rule, 45

double, 47

first principles, 41

implicit, 46

linearity, 42

maxima, 48

numerical, 116

parametric, 47

partial, 107

product rule, 43

quotient rule, 44

second, 47


156/158

IN EX 147

divergence, 108

domain, 18

dot product, 80

double derivative, 47

double integrals, 111

eigenvalues, 73

eigenvectors, 73

ellipse, 28

equilateral triangle, 15

Euler s equation, 93

even

Fourier series, 120

function, 20

expansion, 4

binomial, 5

exponential, 3 1

derivative, 43

function, 25

general, 3 1

integral, 52

factorial, 12

factorising, 4

polynomial, 6

first order differential equations, 97

first principles, 41

forward difference, 116

Fourier series, 1 19

even, 120

odd, 121

fractions, 2

partial, 6

function, 17

circle, 27

cosine, 26

domain, 18

ellipse, 28

even, 20

exponential, 25, 3 1

hyperbola, 24

hyperbolic, 38

inverse, 19

line, 21

logarithm, 25, 3

odd, 20

parabola, 22

polynomial, 23

properties, 18

quadratic, 22

range, 18

shifting, 20

sine, 26

tangent, 26

trigonometric, 35

zeros, 19

fundamental theorem of calculus. 53

Gaussian elimination, 70

gradient, 108

graphs, 17

homogeneous differential equations, 100

hyperbola, 24

hyperbolic functions, 38

hypotenuse, 14

identity matrix, 61

imaginary

complex number, 92

implicit differentiation, 46

index laws, 32

inequalities, 3

inflection points, 48

inhomogeneous differential equations, 102

integrable differential equations, 97

integrating factor, 99

integration, 5 1

area, 55

by parts, 57

definite, 53

double, 111

linearity, 52

numerical, 1 15

substitution, 56

inverse

cofactors matrix, 68

function, 19

matrix, 65

isosceles triangle, 15

L H6pital s Rule, 88

length

vector, 79

limit, 87


157/158

line, 21

linearity

differentiation, 42

integration, 52

logarithm, 33

derivative, 43

function, 25

general, 3 1

integral, 58

natural, 33

Maclaurin series, 88

matrix, 59

addition, 59

augmented, 70

characteristic equation, 73

cofactors, 64, 68

determinant, 63

diagonal, 74

eigenvalues, 73

eigenvectors, 73

identity, 61

inverse, 65, 68

manipulation, 68

multiplication, 60

partitioned, 66

row operations, 66

symmetric, 74

systems of equations, 70

trace, 74

transpose, 62

maxima, 48

minima, 48

modulus, 3,4

multivariable calculus, 107

natural logarithm, 33

Newton s method, 117

notation, 1

numerator, 2

numerical methods, 115

area, 1 15

central difference, 1 16

differential equations, 1 18

differentiation, 116

finding zeros, 1 17

integration, 1 15

Newton s method, 1 17

odd

Fourier series, 121

function, 20

parabola, 22

parameterised curves, 47

parametric

circle, 27

ellipse, 28

parametric differentiation, 47

partial differentiation, 107

partial fractions, 6

particular solution, 102

partitioned matrix, 66

Pascal s triangle, 5

permutations, 13

polar form, 93

polynomial

division, 9

factorising, 6

function, 23

product rule, 43

Pythagoras theorem, 14

quadratic, 11, 22

quadratic equation, 11

quotient rule, 44

range, 18

rationalised surd, 10

real

complex number, 92

numbers, 1

reciprocal trigonometric functions, 36

resonance, 104

right angled triangles, 14

roots

quadratic, 11

row operations, 66

second derivative, 47

second order differential equations, 100

separable differential equation, 98

simplification, 4

sine, 14,26

derivative, 43


158/158

IN EX 149

integral 52

sinh 38

derivative 43

integral 52

slope 41

stationary points 48

substitution

integration 56

summation 12

surd 10

symmetric matrix 74

systems of equations 70

tangent 14 26 41

Taylor series 88

trace 74

transcendental functions 3 1

transpose 62

triangle

equilateral 15

isosceles 15

right angled 14 15

trigonometric

functions 35

identities 36

Documents

Essential Mathematical Skills (Iqbalkalmati.blogspot.com)