2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 23: Basis and Dimension (Solution)
1. Determine whether or not each of the following form a basis of R3 :
(a) [1, 1, 1], [1, 0, 1];
(b) [1, 2, 3], [1, 3, 5], [
2011 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
(
2011 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 plot
2011 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 =
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 07: Dierentiation 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
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 06: Dierentiation (Solution)
1. Find each of the following derivatives:
(c)
d
(x5 4x3 + 2x 9).
dx
d1
5
( x ).
dx x2
d
1
[(x2 + 2x)( x3 x)].
dx
(a)
d
( x5
dx
(a
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 05: Limit (Solution)
1. Find the limit in each of the following case or prove that the limit does
not exist:
(a) limx0 f (x) where f (x) =
10, if x = 0;
20, if
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 4: Simple Probability (Solution)
1. A fair coin is tossed ve times. What is the probability of obtaining
three heads and two tails?
p(3H2T)
5
C
= 5252 = 16 .
2
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 3: Functions and Counting
(Solution)
1. Given f : R cfw_1 R cfw_1 such that f (x) =
bijective.
x1
,
x+1
prove that f is
x1 , x2 R cfw_1, if f (x1 ) = f (x2 ),
2011 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.
pq
TT
TF
FT
FF
p
F
F
T
T
pq
T
F
T
T
(p) q
T
F
T
T
2
2011 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)
(a)
sec 2x ta
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 13: Denite Integrals (Solution)
1. Find each of the following indenite integrals by Riemann denition:
(a)
(b)
20 x
3 dx.
10
b
(x + )dx
a
where and are constant
2011 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 eld K
with the usual operations of matrix addition and scalar m
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 21: System of Linear Equation
(Solutions)
1. Solve the following systems of linear equations.
x1 + 2 x2 + x3 = 3
3x1 x2 3x3 = 1
(a)
2x1 + 3x2 + x3 = 4
x1 + 2
2011 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 + 2j + 2k , 2i j + 2k ,
2i 2j k respectively.
(a) Find AB and AC .
(b)
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 17: VectorBasic Operations
(Solution)
1. By using triangle law, prove:
(a) P Q = OQ OP .
(b) AB + BC + CD + DE = AE .
(c) AB + BC + CD + DE + EA = 0.
(a) O
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 16: Applications of Denite Integrals
(Solution)
1. Assume the displacement s(t) of a car is a function of time t. And
2s
the acceleration of the car is a(t) =
2011 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:
xn ln x dx where n = 1 is a real number.
(a)
(d)
3
xex
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 14: More on Denite Integrals
(Solution)
1. Evaluate each of the following denite integrals:
3
(a) 0 x x + 1dx.
(b)
(c)
(d)
2
1
2
1
8
3
e2x
dx.
ex 1
ln x
dx.
x
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 1: Basic Set Theory (Solution)
1. Assume the universal set U = cfw_1, 2, 3, 4, 5, 6, 7, 8, 9. Denote the sets
A = cfw_1, 2, 3, 4, 5, B = cfw_1, 3, 5, 7, 9, and
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 23: Basis and Dimension
1. Determine whether or not each of the following form a basis of R3 :
(a) [1, 1, 1], [1, 0, 1];
(b) [1, 2, 3], [1, 3, 5], [1, 0, 1], [
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 10: Limit by LHospitals Rule
1. Find each of the following limits by using LHospitals Rule:
(ln x)2
.
x
x bx
limx0 a x for
limx0+ [x3 ln x].
(a) limx
(b)
(c)
s
2011 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 plot
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 08: Derivative of Elementary
Functions
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 sin x ln
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 07: Dierentiation by Chain Rule
1. Find each of the following derivatives:
(a)
d
[(x2
dx
2x 3)2 + (4x 7)2 x41 5 ].
3
d
3
5 + 8)].
(b) dx [(x 4x)( x 2)( 3x
(c)
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 06: Dierentiation
1. Find each of the following derivatives:
(a)
(b)
(c)
d
(x5 4x3 + 2x 9).
dx
d1
5
( x ).
dx x2
1
d
[(x2 + 2x)( x3 x)].
dx
2. Find each of the
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 05: Limit
1. Find the limit in each of the following cases or prove that the limit does
not exist:
(a) limx0 f (x) where f (x) =
10, if x = 0;
20, if x = 0.
(b