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
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:
(
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,
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
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)
d
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
(
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 ex
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
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
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 tab
2011 Summer Math Course for Direct Entry Students,
Computer Science Department, HKUST.
Tutorial 11: Indenite Integrals (Solution)
1. Evaluate each of the following integrals:
(a)
(3x2 2x + 1)dx = x3 x
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:
(Soluti
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
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
wi
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
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
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 +
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 o
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:
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
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
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
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
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,
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)
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
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
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) l