essential mathematical skills (iqbalkalmati.blogspot.com)
TRANSCRIPT
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
1/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
2/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
3/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
4/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
5/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
6/158
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
7.8 Example Questions
8 Asymptotics and Approximations
.1 Limits
8.2 L H6pital s Rule
8.3 Taylor Series
8.4 Asymptotics
8.5 Example Questions
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
0.3 Example Questions
11 Multivariable Calculus
11.1 Partial Differentiation
1.2 Grad. Div and Curl
1.3 Double Integrals
11.4 Example Questions
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
7/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
8/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
9/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
10/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
11/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
12/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
13/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
14/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
15/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
16/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
17/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
18/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
19/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
20/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
21/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
22/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
23/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
24/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
25/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
26/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
27/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
28/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
29/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
30/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
31/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
32/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
33/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
34/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
35/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
36/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
37/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
38/158
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.
31. Wha t type of curve has equation
2y2 (z q2 2
=
O?
32. Wha t type of curve has equation
2y (z 1)2
=
O?
Whe re is the zero of the line
y
=
z ?
33. Wha t type of curve has equation
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
39/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
40/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
41/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
42/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
43/158
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.
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
44/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
45/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
46/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
47/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
48/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
49/158
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
50/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
51/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
52/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
53/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
54/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
55/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
56/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
57/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
58/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
59/158
5 DIFFERENTI TION
4 11 EX MPLE QUESTIONS
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
60/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
61/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
62/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
63/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
64/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
65/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
66/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
67/158
58
INTEGR TION
5 7 EX MPLE QUESTIONS
Answers a re given in Chapter
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
68/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
69/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
70/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
71/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
72/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
73/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
74/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
75/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
76/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
77/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
78/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
79/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
80/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
81/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
82/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
83/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
84/158
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
85/158
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
86/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
87/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
88/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
89/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
90/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
91/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
92/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
93/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
94/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
95/158
7 8 EX MPLE QUESTIONS
Answers a re given in Chapter
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 ) .
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
96/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
97/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
98/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
99/158
9
ASYMPTOTICS
AND
APPROXIMATIONS
8 5 EX MPLE QUESTIONS
(Answers are given in Chapter
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.)
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
100/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
101/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
102/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
103/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
104/158
EX MPLE QUESTIONS 95
9 6
EX MPLE QUESTIONS
(Answers are given in Chapter
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
105/158
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
106/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
107/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
108/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
109/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
110/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
111/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
112/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
113/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
114/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
115/158
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
116/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
117/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
118/158
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 ) .
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
119/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
120/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
121/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
122/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
123/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
124/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
125/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
126/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
127/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
128/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
129/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
130/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
131/158
122 NUMERICAL SKILLS
12 6 EX MPLE QUESTIONS
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.
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
132/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
133/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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)?
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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 ?
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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?
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
139/158
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
142/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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.
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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 =
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
146/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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%)
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
150/158
NSWERS 141
Test 2: Section 13.2
12.
l n x
2
14. Ellipse
Test 3: Section 13.3
2 cos x2
8
2y
2t
9.
os t
Test 4: Section 13.4
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
151/158
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
152/158
CH PTER
5
OTHER ESSENTI L SKILLS
Use these pages to write any other essential mathematics skills your lecturers think are appropriate
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
153/158
44
OTHER ESSENTI L SKILLS
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
154/158
OTHER ESSENTI LSKILLS 45
his
page intentionally left blank
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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
-
8/9/2019 Essential Mathematical Skills (Iqbalkalmati.blogspot.com)
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