MA 1505 Mathematics I
Tutorial 9 Solutions
1. Let z = 22 x2 y 2 . Then
zx = x(4 x2 y 2 )1/2 and zy = y (4 x2 y 2 )1/2 .
Substitute z = 1 into x2 + y 2 + z 2 = 4 gives
x2 + y 2 + 1 = 4
x2 + y 2 = 3
3.
which is the equation of a circle of radius
This means

Limits
MA1505
Mathematics I
Chapter 1
Outline
1.
Definition of Limits.
2.
Basic Results on Limits
3.
One-sided Limits
Limits
In the concept of limits, we are interested in the
behaviour of the values of f ( x ) when x get closer
and closer to some number

Probability
But to us, probability is the very guide of life.
Bishop Joseph Butler
1
I NTRODUCTION
Statistical methods are used to evaluate information in uncertain situations
and probability plays a key role in that process. Remember our definition of
st

Stats Starts Here
But where shall I begin? asked Alice.
Begin at the beginning, the King said gravely, and go on
till you come to the end: then stop.
Lewis Carroll
1
W HAT IS S TATISTICS ?
What are statistics?
Statistics (plural) are particular calculat

CSlOlOE
NATIONAL UNIVERSITY OF SINGAPORE
SCHOOL OF COMPUTING
EXAMINATION FOR
Semester 1 AY2013/2014
CSlOlOE PROGRAMMING METHODOLOGY
Nov / Dec 2013 Time Allowed: 2 Hours
1.
INSTRUCTIONS TO CANDIDATES
This examination paper consists of FOURTEEN (14) quest

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 27 October 31 October 2014
Tutorial 9
String Functions and Structures
1. You have been introduced to the four string functions: strlen, strcpy, strcat and
strcmp. In this exercise, you will impl

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 3 November 7 November 2014
Tutorial 9 Suggested Answers
Recursion
1. Given a real number x and a non-negative integer k, we would like to compute the
exponent xk .
(a) Write a recursive function

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 20 October 24 October 2014
Tutorial 8 Suggested Answers
Character Strings and Pointers
1. Study the following program fragment.
int i1, i2;
int *p1, *p2;
i1
p1
i2
p2
=
=
=
=
5;
&i1;
(*p1)/2 + 10

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 13 October 17 October 2014
Tutorial 7 Suggested Answers
Simulation via Random Number Generation
1. A circle is inscribed in a square as shown in the following diagram. Let c be the area
c
of the

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 27 October 31 October 2014
Tutorial 9 Suggested Answers
String Functions and Structures
1. You have been introduced to the four string functions: strlen, strcpy, strcat and
strcmp. In this exerc

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 6 October 10 October 2014
Tutorial 6 Suggested Answers
Problem Solving with Arrays
1. (a) Given a non-decreasing array of n elements and a key value, write a function
findIndex to return the ind

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 15 19 September 2014
Tutorial 4
Functions as Procedures
1. Just like how printing number sequences is the best way to test your understanding of
looping constructs, printing patterns is another

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 15 19 September 2014
Tutorial 4 Suggested Answers
Functions as Procedures
1. Just like how printing number sequences is the best way to test your understanding of
looping constructs, printing pa

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 29 September 3 October 2014
Tutorial 5
Arrays
1. What is printed by the following program? You are advised to hand-trace the program
to obtain the answer before running the program to verify.
#i

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 29 September 3 October 2014
Tutorial 5 Suggested Answers
Arrays
1. What is printed by the following program? You are advised to hand-trace the program
to obtain the answer before running the pro

CS1010E Programming Methodology
Semester 1 2014/2015
Week of 8 12 September 2014
Tutorial 3 Suggested Answers
Value-Returning Functions
1. In the following parts, you are required to dene functions to nd the maximum of
a set of integer values. Note that s

Probability
But to us, probability is the very guide of life.
Bishop Joseph Butler
1
I NTRODUCTION
Statistical methods are used to evaluate information in uncertain situations
and probability plays a key role in that process. Remember our definition of
st

ST2334
Probability and Statistics
Academic Year 2016/2017
Semester I
A IMS & O BJECTIVES
Course Information
This module teaches students some fundamental concepts of probability, such as how to calculate the probability A will happen given
the knowledge t

Chapter 2
Differentiation
Outline
n
Derivative
q
q
n
Definitions
Rules of Differentiation
Other Types of Differentiation
q
q
q
Parametric Differentiation
Implicit Differentiation
Higher Order Derivatives
n
Maxima and Minima
q
q
q
q
n
Derivative Test
q
q
q

Other Types of Differentiation
Cartesian equation - An equation connecting x and y
y = x + 4x
3
y=x + x
2
Parametric equations
1. x = 2t
y = t2 +1
2. x = sin + 2
3. x = 1 + et
y = cos 5
y = e 2t
x2 + y 2 = 9
Other Types of Differentiation
n
Parametric Dif

Increasing functions
Let f be a function defined on an interval I .
y
y = f ( x)
f (u )
f (v )
0
v
u
x
f is increasing on I if u > v f (u ) > f (v).
Bigger x value, bigger f ( x) value
y increases as x increases
Decreasing functions
Let f be a function de

Points of Inflection
A point c is a point of inflection of the function f
if f is continuous at c and there is an open interval
containing c such that the graph of f changes from
concave up (or down) before c to concave down (or
up) after c.
Question: How

Indeterminate Forms
Indeterminate Forms
Let f and g be continuous at x = a.
Suppose f (a ) = 0 and f (b) = 0.
Then the limit
f ( x)
0
lim
is of the form
x a g ( x )
0
f (a) 0
because
=
g (a) 0
f ( x) 0
lim
=
x a g ( x )
0
0
0
Indeterminate form
Use LHospi

Chapter 3
Integration
Overview
n
Integral
q
q
Indefinite Integral
Definite Integral
n
Fundamental Theorem of Calculus
n
Various Integration Techniques
q
q
Integration by Substitution
Integration by Parts
Overview
n
Application of Integration
q
q
Area betw

Fundamental Theorem of Calculus (Part II)
(II) If F is an antiderivative of f on [a, b], then
b
a
f ( x) dx = [ F ( x)]
b
a
= F (b) F (a )
Example
2
Evaluate 0
y
sin x dx.
1
2
0
sin x dx = [ cos x ]
2
0
= (cos 2 cos 0)
=0
y = sin x
0
-1
p
x
2p
Example
Eva

Application of Integration
Area between two curves
y = g(x)
A
y = f (x)
a
b
x
b
A = a ( g ( x) f ( x) dx
Top curve bottom curve
Area between two curves
Consider g ( x) f ( x)
y
y
g ( x)
g ( x)
f ( x)
0
y
a
f ( x)
b
x
0
y
g ( x)
a
x
b
x
g ( x)
f ( x)
0
b
a

Chapter 4
Sequences & Series
Overview
n
Infinite Sequences
q
q
n
Limits of Sequences
Sequences & Functions
Infinite Series
q
q
q
Partial Sums
Geometric Series
Ratio Test
Overview
n
Power Series
q
q
q
n
Convergence of Power Series
Radius of Convergence
Dif

ST2334
Probability and Statistics
Academic Year 2016/2017
Semester I
David Chew
Department of Statistics and Applied Probability
email: david.chew@nus.edu.sg
Typesetted using the MiKTEX sytem.
Contents
Contents
iii
Course Information
v
The Moral of the St

1
Stats Starts Here
But where shall I begin? asked Alice.
Begin at the beginning, the King said gravely, and go
on till you come to the end: then stop.
Lewis Carroll
1
Two main parts of Statistics:
W HAT IS S TATISTICS ?
What are statistics?
Statistics

ST2334
Probability and Statistics
Academic Year 2016/2017
Semester I
Course Information
A IMS & O BJECTIVES
This module teaches students some fundamental concepts of probability, such as how to calculate the probability A will happen given
the knowledge t