MA2181 Notes 4 Partial Differentiation
1. Introduction
Quantities of interest in real problems usually depend on more than one variable.
Illustrations
(i) The height of the ground at map coordinates ( x, y ) might be expressed mathematically as
Height = h
MA1200 (CB1)
Additional Exercises on Chapter 5
1.
Evaluate log 5 100 log 5 4 log 6 12 log 6 18 7 log 7 8
2.
The functions F (x ) and G (x) are defined by
(ii)
(iii)
G 1 ( x) ,
( F G )( x) .
The functions F and G are defined as follows:
1
F ( x) (e x e x
MA1200, CEl, Quiz 1 (2015/16, SemA)
Name:
Total: 40 points. Time: 30 min.
1. (15 points) Given that the equation of a conic section is
wmz + 4y2 — 24y + 20 = 0.
3.) Using completing square, identify the type of the conic section;
b) Hence, sketch the grap
MA1200, om, Mock Test 1 (2015/16, SemA)
Name:
Total: 100 points. rI‘ime: 60 min.
1. (25 points) Given that the equation of a conic section is
3:24-43? «~24y+20=0.
a) Using completing square, identify the type of the conic section;
b) Hence, sketch the gra
City University of Hong Kong
Department of Mathematics
MATHEMATICS PLACEMENT TEST FOR MA1200
Sample Questions & Solutions
1
1.
The general solution of the equation tan x = 1 is
A.
B.
x = n
C.
x = n + ( 1)
D.
2.
x = n +
x = 2n +
The graph of
4
4
, nZ .
,
City University of Hong Kong
Department of Mathematics
MATHEMATICS COMBINED PLACEMENT TEST FOR MA1200 & MA1201
Sample Questions & Solutions
1
Section A
1.
The general solution of the equation tan x = 1 is
A.
B.
x = n
C.
x = n + ( 1)
D.
2.
x = n +
x = 2n
MA1200
Practice Exercise for Ch. 4 Trigonometric Functions and Inverse Trigonometric Functions
1.
(a) Convert the following angles to radians.
(i) 48
(ii)
120
(b) Convert the following angles to degree.
123
(i) rad
(ii)
rad
6
180
(iii)
315
(iii)
2
rad
5
t
MA1200
1.
Practice Exercise 5
Exponential and Logarithmic Functions
Determine whether each of the following is true or false. If it is false, give a counterexample.
i. If N is positive, then as N increases, log5 N increases. (T / F)
ii. If a, b are positi
MA1200
Chapter 4
Calculus and Basic Linear Algebra I
Trigonometric Functions and Inverse Trigonometric Functions
1
Trigonometric Functions
1.
In elementary trigonometry, the trigonometric functions are defined as the ratios of sides of a
right-angled tria
MA1200
Chapter 1
1
Calculus and Basic Linear Algebra I
Coordinate Geometry and Conic Sections
Review
In the rectangular/Cartesian coordinates system, we describe the location of points using coordinates.
y
P2(x2, y2)
P(x, y)
O
P1(x1, y1)
x
The distance d
MA1200
Chapter 3
Calculus and Basic Linear Algebra I
Polynomials and Rational Functions
Review
A function f x an x n an 1 x n 1 a0 (where the a' i s are real numbers and n is a non-negative
integer) is called a polynomial function. If a n 0 , n is the deg
MA1200
Chapter 5
1
Calculus and Basic Linear Algebra I
Exponential and Logarithmic Functions
Exponential Functions
Definition:
The exponential function f with base b is defined by
f x b x
or
y bx ,
where b is a positive constant other than 1 ( b 0 and b 1
MOCK EXAMINATION SET B
Question-Answer Book
Time allowed: 2 hours
INSTRUCTIONS
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Write your Candidate Number in the space provided on
Page 1.
Attempt ALL tasks in Part A (Tasks 1-4), and for Part B,
attempt EITHER those in Sectio
Derivatives
derivative of exponential function
de x
e x+h e x
eh 1
= lim
= e x lim
= ex
h0
h0
dx
h
h
Let e h 1 = t then h = loge (1 + t) and
t
1
eh 1
= lim
= lim
1 = 1
t0
t0
h0
h
loge (1 + t)
loge (1 + t) t
lim
dax
ah 1
= ax lim
= ax loge a
h0
dx
h
ah 1
1
MA1200 (CB1)
Additional Exercises on Chapter 1
Semester A, 2015-16
1.
A straight line passes through the point (5, 2) and has equal x -intercept and y intercept. Find the equation of the straight line.
2.
The straight line
3.
The equation of a conic is y
MA1200 (CB1)
1.
2.
3.
Additional Exercises on Chapter 7
(a)
dy
for each of the following.
dx
y e 2 x cos 3x
(b)
y sin 2 x 5 sec3 x
(c)
y x2 x2 1
(d)
y loge
(e)
y2
(f)
y x 3x
Find
Semester A, 2015-16
1 x
1 x
x
For each of the following, find
dy
in terms of
MA1200 (CB1)
Additional Exercises on Chapter 6
sin( 2 x)
.
x 0
x
sin
2
1.
Evaluate the limit lim
2.
Evaluate the limit lim
3 sin 2 x tan x
( x 3 5) x 2
x 0
3.
Evaluate the limit lim
1 cos3 x
x0
4.
Evaluate the limit lim
x
5.
6.
7.
8.
9.
Semester A, 20
MA2181
Chapter 1
Eigenvalues and Eigenvectors
1. Introduction
If A is an 3 3 matrix and x is a vector in R 3 , then there is usually no general geometric relationship
r
between the vector x and the vector Ax , see figure (a). However, there are often cert
MA2181
Chapter 3
Higher Order Ordinary Differential Equations
3.1 Second-order Linear Ordinary Differential Equations
The general form of a second-order linear ordinary differential equation is:
a x y b x y c x y d x
(non-homogeneous ODE)
or
a x y b x y
The Laplace Transform
1
Definition and the Inverse Laplace Transform
Let f t be a function defined for t 0 . We define the Laplace transform of f t , denoted by F s or
L f , to be the integral F s L f e st f t dt (provided this integral exists).
0
Illustr
FOURIER SERIES
Periodic Functions
Definition. A function f t is periodic if there is a positive number T such that
f t T f t for all t,
and f t is defined for all real t . The number T is called a period of f t .
Periodic functions occur in many areas of
MA2181
Chapter 2
First Order Ordinary Differential Equations
2.1 DEFINITION OF A DIFFERENTIAL EQUATION
A differential equation is an equation in which one or more of the derivatives or differentials of one or
more unknown functions occur.
Illustration
The
Basic Calculus and Linear Algebra
Applications of Derivatives
Additional Examples
2.1
Find the equation of the tangent to the curve x 3 y y 2 0 at the point (4, -1).
Solution
Differentiate both sides of the equation x 3 y y 2 = 0 with respect to x:
1 3
dy
Semester A, 2015-16
MA1200
Calculus and Basic Linear Algebra I
Chapter 4
Dr. Emily Chan
Page 1
Semester A, 2015-16
MA1200
Calculus and Basic Linear Algebra I
Chapter 4
Trigonometric Functions
In elementary trigonometry, the 3 basic trigonometric functions
Semester A, 2015-16
MA1200
Calculus and Basic Linear Algebra I
Chapter 7
Dr. Emily Chan
Page 1
Semester A, 2015-16
MA1200
Calculus and Basic Linear Algebra I
Chapter 7
Brief table of derivatives of some elementary functions with respect to
Function
= ()
Sem est er A, 2 0 1 5 - 1 6
M A1 2 0 0
Ca l cu l u s a n d Ba si c Li n ea r Al g eb r a I
D r . Em i l y Ch a n
Sem est er A, 2 0 1 5 - 1 6
Ch a pt er 2
Pa g e 1
M A1 2 0 0
Ca l cu l u s a n d Ba si c Li n ea r Al g eb r a I
Ch a pt er 2
Set Notation
A s
MA1200 Exercise for Chapter 6 Limits, Continuity and Differentiability
Limits
1. Evaluate the following limits:
2.
x2 1
x 2 x 3 x
(a)
lim
(c)
lim
(b) lim
x
m sin mx n sin nx
x 0
tan mx tan nx
x + x4 x2 1
2x 2 1 x 4 1
( m n)
Evaluate the following limit
Sem est er A, 2 0 1 5 - 1 6
M A1 2 0 0
Ca l cu l u s a n d Ba si c Li n ea r Al g eb r a I
D r . Em i l y Ch a n
Ch a pt er 6
Pa g e 1
Sem est er A, 2 0 1 5 - 1 6
M A1 2 0 0
Ca l cu l u s a n d Ba si c Li n ea r Al g eb r a I
Ch a pt er 6
Limit of a funct