Q1. Verify the following identities of sets:
C (A B) = (C A) (C B) ,
C (A B) = (C A) (C B) .
Answer. For the 1st identity: (). Suppose
x C (A B), then x C or x A B. When
x C, x (C A) (C B) is obvious.
MATH1851 (13/14) Assignment 1 (Due date: 27 Sept 2013
5pm)
19-09-2013
Please submit your solution to the assignment box located on 4th oor of Run
Run Shaw building. DO NOT COPY. Late submission will n
Q1. (Ex2.1 Q33) Does the graph of
x2 sin 1
x
f (x) =
0
x=0
x=0
have a tangent at the origin? Give reasons for
your answer.
Answer. We calculate if the slope of f exists
at 0, if so, we can nd the tang
Basic techniques in Calculus
06-09-2013
Set 1 To show two sets are equal, A = B, you need to prove in two steps:
If x A, then . x B; and
if x B, then . x A.
Tri 0 sin2 a + cos2 a = 1
Tri 1 sin(a + b)
MATH1851 (13/14) Tutorial 3
10-09-2013
1. (Ex2.1 Q33) Does the graph of
f (x) =
1
x2 sin x
0
x=0
x=0
have a tangent at the origin? Give reasons for your answer.
2. (Ex2.2 Q3) Let g(t) = t1 , nd g (1),
MATH 1851 First and second ODEs: Worked Examples 2
(1) Are the functions exp(x) and xexp(x) linearly independent?
We consider a 2 X 2 determinant where the first row consists of the functions
exp(x) a
MATH 1851 First order differential equations: Worked Examples 1
(1) Consider the differential equation
dy/dx = (y2 x2)/(2xy) .
Homogeneous equation
Since g(x, y) = (y2 x2)/(2xy) is homogeneous of degr
MATH 1851 Finding particular solutions, undetermined coefficients, variation of
parameters and resonance: Worked Examples 4
(Again prime of = derivative with respect to the independent variable t or x
MATH1851 Constant coefficients and Cauchy-Euler equations, simple
undetermined coefficients: Worked Examples 3
A, B, C1, C2are constants and primes () = derivatives with respect to the
independent var
MATH1851 (13/14) Tutorial 4
19-09-2013
1. (Ex7.2 Q18) Let y = (x2 ln x)4 , nd dy/dx.
2. (Ex7.2 Q63) Use logarithmic dierentiation to nd dy/d, where
y=
+5
.
cos
3. (Ex7.3 Q111) Find y where y = (x +
Q1. Let y = (x2 ln x)4, nd dy/dx.
Answer.
1
y = 4(x2 ln x)3 (2x ln x + x2 )
x
= 4(x2 ln x)3 (2x ln x + x) .
1
Q2. Use logarithmic dierentiation to nd
dy/d, where
y=
+5
.
cos
Answer. Take log on both
MATH1851 (Part I) - Tutorial class 6
Partial fractions, FTC, MVT (integral version)
F. Tsang
October 13, 2017
University of Hong Kong
Partial fraction decomposition
Fact:
Any polynomial g(x) is a prod
MATH1851 (Part I) - Tutorial class 2
Continuity, Differentiability
F. Tsang
September 8, 2017
University of Hong Kong
Limits - How to find lim f (x)? (Evaluation techniques)
xa
4. Sandwich Theorem: If
Chapter 3. Applications of derivatives
Topics
1. Extreme Values
2. The Mean Value Theorem & its consequences
3. L Hpitals rule
4. Monotonicity and the First Derivative
5. Concavity and the Second Deri
MATH 1851
Calculus and Ordinary
Differential Equations
Dr. W.F. Lee
Location: Room 528A, Haking Wong Building
Phone: 2219 4813, E-mail: [email protected]
Outline of ODEs
Motivation and Basic Concepts
Fir