MA1102R
Calculus
AY 2014/2015 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
MA1102R Calculus
AY 2014/2015 Sem 1
Version 1: December 11, 2014
Written by
Lee Kee Wei
Audited by
Henry Morco
Contributors
Question 1
f (x)
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Topic: Part II of Chapter 4
1. Show that if f 00 is positive throughout an interval [a, b], then f 0 has at most one zero
in [a, b].
2. Suppose that f is differentiable on R and that
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Tutorial 5 (26/09 30/09)
Tutorial Part I
This part will be discussed during the tutorial session. Solution of selected questions
(indicated by ) from this part will be provided to dem
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Tutorial 7 (10/10 14/10)
Tutorial Part I
This part will be discussed during the tutorial session. Solution of selected questions
(indicated by ) from this part will be provided to dem
The Role of Logic in Teaching Proof
Author(s): Susanna S. Epp
Source: The American Mathematical Monthly, Vol. 110, No. 10 (Dec., 2003), pp. 886-899
Published by: Mathematical Association of America
Stable URL: http:/www.jstor.org/stable/3647960 .
Accessed
On Mathematical Induction
Author(s): Leon Henkin
Source: The American Mathematical Monthly, Vol. 67, No. 4 (Apr., 1960), pp. 323-338
Published by: Mathematical Association of America
Stable URL: http:/www.jstor.org/stable/2308975
Accessed: 16-08-2014 07:1
On Mathematical Induction
Author(s): J. W. A. Young
Source: The American Mathematical Monthly, Vol. 15, No. 8/9 (Aug. - Sep., 1908), pp. 145-153
Published by: Mathematical Association of America
Stable URL: http:/www.jstor.org/stable/2969864
Accessed: 16-
Logic from A to G
Author(s): Paul R. Halmos
Source: Mathematics Magazine, Vol. 50, No. 1 (Jan., 1977), pp. 5-11
Published by: Mathematical Association of America
Stable URL: http:/www.jstor.org/stable/2689742 .
Accessed: 30/08/2014 23:21
Your use of the J
Fundamental concepts of Mathematics
Chin CheeWhye
Department of Mathematics
National University of Singapore
3
Lecture 12
29 September 2014
Announcements
Homework 4:
Due 13 October 2014 Monday at 12pm
Write your Name and Student ID Number at the top of th
Opinion
Mathematics Is a Quest
for Truth
The question of how to teach mathematics in primary
and secondary education has appeared in newspaper
opinion pages regularly during the last few years. The
topic is a complicated one and does not allow for easy
or
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
Tutorial 6 (03/10 07/10)
MA1102R Calculus
Tutorial Part I
This part will be discussed during the tutorial session. Solution of selected questions (indicated
by ) from this part will be provided to dem
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Chapter 3 Chapter 4 Part I
1. Using the definition of derivative, find the derivatives of the given functions. Then
find the values of the derivatives as specified.
1
(a) f (x) = ;
x
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Tutorial 8 (17/10 21/10)
Tutorial Part I
This part will be discussed during the tutorial session. Solution of selected questions
(indicated by ) from this part will be provided to dem
MA1102R
Calculus
AY 2009/2010 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
solutions prepared by Boyan,Tay Jun Jie
MA1102R Calculus
AY 2009/2010 Sem 1
Question 1
(a) By LHpitals Rule,
o
lim
x 4
sec2 x 2 tan x
2 sec2
MA1102R
Calculus
AY 2008/2009 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to
Sean Lim Wei Xinq
He Jinxin
MA1102R Calculus
AY 2008/2009 Sem 1
Question 1
(a) By LHospitals Rule we have
lim
x7
x+23
1
= li
MA1102R
Calculus
AY 2007/2008 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Lee Yung Hei, Joseph Nah
MA1102R Calculus
AY 2007/2008 Sem 1
Question 1
2x
22
1
=
= .
2
2
x2 3x
32
3
(a) Since lim x2 4 = 0
MA1102R
Calculus
AY 2013/2014 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
MA1102R Calculus
AY 2013/2014 Sem 1
Version 1: October 25, 2014
Written by
Henry Jeerson Morco
Audited by
Chua Hongshen
Contributors
Questio
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Topic: Chapter 8
1. Solve the following differential equations.
dy
= x2 y, y > 0.
(a)
dx
p
dy
(b)
= 2x 1 y 2 , 1 < y < 1.
dx
dy
(c)
= y cos2 y, y > 0.
dx
(d)
dy
x2 + y 2
=
, x > 0.
d
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Topic: Chapters 5, 6 (Part I)
dy
of the following functions.
dx
Z sin x
Z tan x
dt
cos(t2 ) dt,
,
(b) y =
(a) y =
2
1
+
t
2
x
0
1. Find
dy
d2 y
2. Find
and 2 if y =
dx
dx
Z
Z
0
(c) y
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Extra Problems Set 7 (Revision)
1. For the functions f (x) =
x2 1, g(x) =
(ii) g f and their domains.
1
, find the composite functions (i) f g,
x2
2. For each of the following, evalua
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2016/2017
MA1102R Calculus
Topic: Chapters 6 (Part II), 7
1. Evaluate the following indefinite integrals.
Z
9 w2
(a)
dw,
w2
Z
5 dx
3
, x> ,
(c)
2
5
25x 9
Z
(e) t2 e4t dt,
Z
(g) (ln x)2 dx,
Z
(i)
e 3x+9 dx
Z
Next, use the tangent function to represent the secant, and differentiate to obtain
the usual formula for the derivative of the secant. For the derivatives of the sine and
cosine, observe that cos x = 1/(secx) and sin x = tan x cos x for x (n + %)7r.
Diff
LESSON 4: INTEGRATIONS AND CONDITIONAL STATEMENTS
1. Riemann Sum
1.1. The Sum. The command sum is used to evaluate definite or indefinite sum of expressions.
Let f be an expression with summation index i. Then sum(f,i=m.n) gives the definite
sum of f(i) f
LESSON 3: DIFFERENTIATION AND PROCEDURE
1. Implicit Differentiation
We have introduced the command diff in Lesson 2, which is used to differentiate a
function in the form y = f (x). However, sometimes we are given an equation f (x, y) = 0
instead. For exa
The Concept of 'Rigorous Proof'
Author(s): Jean Dieudonn
Source: The Mathematical Gazette, Vol. 80, No. 487, Centenary Issue (Mar., 1996), pp. 204-206
Published by: The Mathematical Association
Stable URL: http:/www.jstor.org/stable/3620351 .
Accessed: 28
Fundamental concepts of Mathematics
Chin CheeWhye
Department of Mathematics
National University of Singapore
Lecture 19
3 November 2014
Announcements
Last lecture for the course:
6 November 2014 (this Thursday).
Accounting for scores:
You will receive an
Fundamental concepts of Mathematics
Chin CheeWhye
Department of Mathematics
National University of Singapore
Lecture 5
25 August 2014
Announcements
Midterm test 1:
18 September 2014 (Thursday of Week 6)
Tutorials:
Tutorial start today: 25 August 2014
Clar
Fundamental concepts of Mathematics
Chin CheeWhye
Department of Mathematics
National University of Singapore
Lecture 3
21 August 2014
Announcements
Midterm test 1:
18 September 2014 (Thursday of Week 6)
Tutorials:
Tutors:
Mr LIM Wei Quan
Mr CHUA Nam Chew