1
Ch7/MATH1013/YMC/2012-13/1st
Chapter 7. Matrices and Determinants
7.1. Matrix Arithmetic and Operations
This section is devoted to developing the arithmetic of matrices. We will see some of the dierences
between arithmetic of real numbers and matrices.
T1S/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 1 Suggested solution (Oct 8 - 12)
1. (a) From the graph of f (x) = x , the range of f is the set Z of integers.
(i) The graph of f .
MATH1013 University Mathematics II
Tutorial 5 Solution
1. (a) First, we have f (0) = B which is well-dened as B > 0. Then we compute f (x) =
A
A
, and so f (0) = which is also well-dened as B > 0. Hence the linear
2 Ax + B
2B
approximation P1 (x) of f (x)
MATH1013
University Mathematics II
Tutorial 7 Solution
1. (a)
x cos x dx =
x d sin x
= x sin x
sin x dx
= x sin x + cos x + C
(b)
e
1
x=e
e
ln x dx = [x ln x]1
e
=e
x
1
x d ln x
x=1
1
dx
x
e
=e
dx
1
e
= e [x]1
= e (e 1) = 1
2. (a)
In =
xn eax dx =
1 n a
MATH 1804 University Mathematics A
Suggested Solutions to Past Exam Problems
Chapter 3. Limits and Continuity
March 3, 2012
Keywords:
(left/right-hand) limit, sandwich/squeezing theorem
continuity, intermediate value theorem
December 2011.
2. (a) For f
MATH 1804 University Mathematics A
Suggested Solutions to Past Exam Problems
Chapter 4. Dierentiation and Its Applications
March 3, 2012
Keywords:
dierentiability, derivative, product/quotient rule, chain rule, implicit dierentiation
mean value theorem
MATH 1804 University Mathematics A
Suggested Solutions to Past Exam Problems
Chapter 7. Matrices and Determinants
December 20, 2011
Keywords: inverse, transpose, (skew-)symmetric
December 2011.
1.
(i) C
For A to be symmetric and nonsingular, it requires t
A1/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Assignment 1
Due date : Oct 3, 2012 before 17:00.
Please write all your answers on A4-size papers. Staple your work and remember to write down
MATH1013 University Mathematics II
Assignment 2 Solution
1. (a) Notice that the cosine function satises that
1 cos y 1,
for any y R. As long as x = 0, we also have
1 cos
4
1.
x
Then multiply the above inequality by x4 and get
x4 x4 cos
4
x4 .
x
Now, sin
A3/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Assignment 3
Due date : Nov 2, 2012 before 17:00.
Please write all your answers on A4-size papers. Staple your work and remember to write down
A4/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Assignment 4
Due date : Nov 22, 2012 before 17:00.
Please write all your answers on A4-size papers. Staple your work and remember to write dow
MATH1013 University Mathematics II
Assignment 4 Solution
1. Let f (x) = cos(3x). Then, the linear and quadratic approximations of f at x = a are given by
P1 (x) = f (a) + f (a)(x a)
1
P2 (x) = f (a) + f (a)(x a) + f (a)(x a)2
2
If x = /6, then,
f (/6) = 0
MATH1013
University Mathematics II
Dr. Yat-Ming Chan
Department of Mathematics
The University of Hong Kong
First Semester 2012-13
Content Outline
1. Pre-Calculus Topics
Functions and Graphs, Composite and Inverse Functions, Limits and Continuity
2. Single
1
Ch2/MATH1013/YMC/2012-13/1st
Chapter 2. Functions
2.1. Functions and Graphs
The idea of a function expresses the dependence between two quantities, one of which is given and
the other is the output. A function associates a unique output with every input
1
Ch5/MATH1013/YMC/2012-13/1st
Chapter 5. Exponential and Logarithmic Functions
5.1. Review
Exponential Functions
An exponential function is a function of the form
f (x) = ax
where a > 0, a = 1. Note that the exponent x can be any real number here, and t
1
Ch6/MATH1804/YMC/2012-13/1st
Chapter 6. Integration
6.1. The Fundamental Theorem of Calculus
The principal theorem of this section is the Fundamental Theorem of Calculus, which is the central
theorem of integral calculus. It provides a connection betwee
MATH1013/YMC/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II (2012-13 First Semester)
Course Information
Instructor
Dr. Yat-Ming CHAN
Oce: Rm 312, Run Run Shaw Building
Consultation Hours: Mon, Tue, Th
Appendix(Ch2)/MATH1013/YMC/LP/2012-13/1st
Appendix to Chapter 2: Note on Trigonometric Functions
1. Denitions
Sine:
sin =
y
r
Cosine:
cos =
x
r
Tangent:
tan =
y
x
2. Radian Measure
s
=
r
180 = radians
radians, 60 = radians,
6
3
45 = radians, 90 = radians.
SECh8/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Supplementary Exercise on Chapter 8
(NOT to be handed in)
1. Find
1 + 4i
+ (3 + 2i)i13 .
3 2i
2. Let z = a + bi C. Express the following nu
MATH1013
University Mathematics II
Test 2 Solution
1.
(i) Answer: C.
Since lim f (x) = and lim+ f (x) = , so A is true. Since
x0
x0
1
1
f ( x)
lim (f (x) x) = lim
= lim (1 + 2 ) = 1 and
= 0,
lim
x
x x
x
x x
x
so y = x is an oblique asymptote. Therefore B
T1/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 1 (Oct 8 12)
1. Let f : R R be dened by f (x) = x where x denotes the smallest integer greater
than or equal to x. This function is k
MATH 1013 University Mathematics II
Tutorial 2 Solution
f (h) f (0)
= 3 merely means that f is dierentiable at x = 0 with f (0) = 3. Graphically,
h
f has a tangent line of slope 3 at x = 0.)
1. ( lim
h0
(a) Might be true.
Example: f (x) = 3x + 3 satises
T3/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 3 (Oct 22 26)
1. Use lHpitals rule to nd the following limits:
o
x2 + 3 x
x 2x3 x + 1
(3x + 1) x 4
(b) lim
2
x1
x3 1
(a) lim
2. Find
MATH1013 University Mathematics II
Tutorial 4 Solution
1. Since f (x) < 0 for x < 0, f (x) is strictly decreasing when x < 0. Hence f (x) has no local maxima
or local minima when x < 0.
Since f (x) < 0 for 0 < x < 2, f (2) = 0 and f (x) > 0 for x > 2, x =
T5/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 5 (Nov 12 16)
1. Find the linear approximation P1 (x) of the following functions at x = 0:
(a) f (x) = Ax + B where A, B R and A = 0
MATH1013 University Mathematics II
Tutorial 6 Solution
1. (a) Let y = sin x + 1. We have
=
1
(sin x + 1)3
3
d
dy
=
y 2 cos x
=
d
dx
13
y
3
(sin x + 1)2 cos x.
dy
dx
(b)
2
2
(sin x + 1) cos xdx
2
1
(sin x + 1)3
3
0
3
1
1
3
sin + 1 (sin 0 + 1)
3
2
3
81
33
7
T7/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 7 (Nov 26 30)
1. Evaluate the following integrals by integration by parts:
(a)
x cos x dx
e
(b)
ln x dx
1
2. (Reduction Formula)
Let
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Tutorial 8 Solution (Dec 3 - 7)
1. (a) For example, the square of each of the followings is the zero matrix 022 :
01
,
00
Remark If A =
00
,
20
1
1
,
1 1
6
4
.
9 6
ab
WE/MATH1013/2012-13/1st
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Warm-up Exercise (Chapter 1)
(NOT to be handed in)
1. Describe the following sets with the set-builder notation:
(a) The set of all negative o