L02: Proof by Contradiction
and Mathematical Induction
Summer Supplementary Math
Course,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Proof by Contradiction
HKUST CSE Summer Math Course
2
Proof by contradiction
The contra-positive law:
pq (~q)(~p

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 06: Differentiation (Solution)
1. Find each of the following derivatives:
(c)
d
(x5 4x3 + 2x 9).
dx
d 1
( x5 ).
dx x2
d
[(x2 + 2x)( x13 x)].
dx
(a)
d
(x5
dx
(a)
(b)

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 11: Indefinite Integrals (Solution)
1. Evaluate each of the following integrals:
R
(a) (3x2 2x + 1)dx = x3 x2 + x + C.
R
(b) (4x5 3x3 + 9x)dx = 32 x6 43 x4 + 92 x2

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 4: Simple Probability (Solution)
1. A fair coin is tossed five times. What is the probability of obtaining
three heads and two tails?
p(3H2T)
5
.
= 52C52 = 16
2. A

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 07: Differentiation by Chain
Rule(Solution)
1. Find each of the following derivatives:
(a)
d
[(x2
dx
(c)
d2 1
( x2 +2 ).
dx2
(d)
dn n
x
dxn
(a)
2x 3)2 + (4x 7)2 x41

L21: System of Linear
Equations
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Linear system
Consider a system of equations of the
form:
a11x1 a12 x2 a1n xn b1
a x a x a x b
21 1 22 2
2n n
2
,
am1 x1 am 2 x2 a

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 13: Definite Integrals (Solution)
1. Find each of the following indefinite integrals by Riemann definition:
R 20
(a) 10 3x dx.
Rb
(b) a (x + )dx where and are const

L04: Simple Probability
Summer Supplementary Math
Course,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Probabilistic experiment
Many experiments do not yield exactly the
same results when performed repeatedly.
Coin tosses.
Dice tosses.
Experiment

MATH2000 Summer Exercise for new MATH/MAEC students
Set Theory
Set
A set is a collection of distinct objects in which the ordering of the objects is not relevant.
The objects in a set are called the elements of the set.
Sets are usually denoted by capital

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 2: Proof Techniques (Solution)
1. Prove the law of conditional, p q (p) q, by constructing a
truth table.
p q
T T
T F
F T
F F
p
F
F
T
T
pq
T
F
T
T
(p) q
T
F
T
T
2.

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 08: Derivative of Elementary
Functions (Solution)
1. Find each of the following derivatives:
(a) y = x2 + sin x
(b) y = x2 cos x
(c) y = x sin x + 7 x
(d) y = x2 si

L12. Methods of
Integration
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Integration Formula of Trigonometric
Functions
cos x dx sin x c
sin x dx cos x c
sec x dx tan x c
csc x dx cot x c
sec x t

L17: Vector
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Definition of vector
A quantity that can be completely
specified by its numerical value is called a
scalar.
A scalar is merely a real number.
A quantit

L16. Applications of
Definite Integrals
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Finding area by integration
Recall that the definite integral is defined
to calculate the area under a curve.
Recall: When f

L23: Vector Space
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Vectors Space
Vector space is a useful abstract
mathematical structure.
Space is similar to set concept, i.e.
a collection of objects.
Set can be

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 12: Methods of Integration
(Solution)
1. Evaluate each of the following trigonometric integrals:
(Solutions of (n)(o)(p)(q) are left to students)
R
(a) sec 2x tan 2

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 10: Limit by LHospitals Rule
(Solution)
1. Find each of the following limits by using LHospitals Rule:
(ln x)2
x
ax bx
limx0 x for
limx0+ [x3 ln x].
(a) limx
(b)
(c

L05: Limit
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Example 1
Let f ( x) ( x 2)( x 1) .
( x 1)
We find that f(x)=x+2 when x1.
And f(x) is undefined when x=1.
y
y
3
( x 2)( x 1)
( x 1)
Empty circl

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 18: VectorMore Operations
(Solution)
1. The position vectors of the points A, B, C are ~i + 2~j + 2~k, 2~i ~j + 2~k,
2~i 2~j ~k respectively.
(a) Find AB and AC.
(b

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 15: Integration by Parts and
Improper Integrals (Solution)
1. Evaluate each of the following integrals:
R
(a) xn ln x dx where n 6= 1 is a real number.
R3
(b) 2 xex

MATH2000 Summer Exercise for new MATH/MAEC students
Matrices and Determinants
Matrix
A rectangular array of mn numbers in the form
a11 a12 a1n
a11 a12
a
a21 a22 a2 n
21 a22
or
a
m1 am 2 amn
am1 am 2
a1n
a2 n
amn
1 1 2 3
is a 24 mat

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 16: Applications of Definite Integrals
(Solution)
1. Assume the displacement s(t) of a car is a function of time t. And
2
the acceleration of the car is a(t) = ddt2

L06. Differentiation
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Rate of change
In many different fields, the concept of
rate of change is commonly applied.
For example, speed is the rate of change

L22: Linear Independence
+ Matrix Transformation
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Linearly combination
k1a1+k2a2+knan is called the linear
combination of a1, a2, , an where k1,
k2, , kn are arbitrar

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 22: Vector Space (Solution)
1. Let M3,2 be the set of all 3x2 matrices with entries in a scalar field K
with the usual operations of matrix addition and scalar mult

L10: Limit by
LHospitals Rule
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Indeterminate forms
A limit
f ( x)
x p g ( x)
lim
is said to be in
indeterminate form of
g ( x) 0
and xlim
.
p
A limit
f ( x)

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 14: More on Definite Integrals
(Solution)
1. Evaluate each of the following definite integrals:
R3
(a) 0 x x + 1dx.
R 2 2x
(b) 1 eex 1 dx.
R2
(c) 1 lnxx dx.
R8
(d)

L15. Integration by Parts
and Improper Integrals
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Integration by parts
Theorem (Integration by parts):
If u and v are functions of x then
u dv uv v du.
Summer Math C

Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 09: Applications of Dierentiation
1. Suppose f (x) is a dierentiable function with f (1) = 2, f (2) = 2,
f (1) = 3 and f (2) = 3. And suppose y = f (x) is plotted o

Introduction to the course
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Course objective
Math background is important in
understanding the many concepts in
computer science.
Students who have not tak

L07. Differentiation by
Chain Rule
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
HKUST CSE Summer Math Course
1
Differentiable
Recall that when y=f(x):
f ( x0 x) f ( x0 )
dy
lim
.
dx x x x0
x
0
We call f(x) to be differentiable at

L19: Matrix
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Definition of matrix
Formally, a matrix A of size (order) mn
is a rectangular array of numbers with m
rows and n columns.
a11 a12
a
a
21
22
A
am1 am 2

L14. More on Definite
Integrals
Summer Math Course
for Direct Entry Students,
CSE Department, HKUST.
Summer Math Course
1
Method of substitution
The method of substitution can still be
used for evaluating definite integrals, but
we have to update the lowe