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