CHAPTER 4: APPLICATIONS OF
DIFFERENTIATION
4.1
Extreme Values of Functions
(Reference: TC, 4.1)
Local (Relative) versus Absolute (Global) Extrema
Definition
Let f be a function and D be its domain.
1. f has a local maximum (or relative maximum) value at c
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2017/2018
Tutorial 1 (28/08 01/09)
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
CHAPTER 1: LIMITS
1.1
A First Encounter of Limits
(Reference: TC, 2.2, 2.4, 2.6)
Intuitive Definition
A function f is said to approach the limit L as x approaches
a if f (x) gets arbitrarily close to L for all x sufficiently close (but not equal) to a.
No
CHAPTER 3: DERIVATIVES
3.1
The Derivative of a Function
(Reference: TC, 3.1, 3.2, 3.3, 3.4)
Tangent Line
Definition
The slope of the curve y = f (x) at the point P (a, f (a) is the number
f (a + h) f (a)
,
h0
h
m = lim
provided the limit exists. The tange
MA1102R CALCULUS
(2017/2018, Semester 1)
Lecture Notes (Group 1)
Goh Say Song
Department of Mathematics
National University of Singapore
Reference:
[TC] G. B. Thomas, M. D. Weir and J. Hass, Thomas Calculus, 13th edition, Pearson,
2016.
CHAPTER 0: FUNCTIO
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2017/2018
MA1102R Calculus
Homework Assignment 1
IMPORTANT: Please write your name, student card number and tutorial group
number on the answer script, and submit during either Group 1s lecture on 21st Septembe
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2017/2018
MA1102R Calculus
Tutorial 4 (18/09 22/09)
Tutorial Part I
This part will be discussed during the tutorial session. Partial solution of selected questions (indicated by ) from this part will be provide
NATIONAL UNIVERSITY OF SINGAPORE
SEMESTER 1, 2017/2018
MA1102R Calculus
Tutorial 3 (11/09 15/09)
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, 2017/2018
Tutorial 2 (04/09 08/09)
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
MS3127
Engineering Mathematics 1
Chapter 8: Newtons Method
Objectives :
1.
Apply Newtons Method to solve equations.
8.1
Introduction
Solving an equation is to find the values of the unknown that satisfy the equation. So
far, you have learnt how to solve:
MS3127
Engineering Mathematics 1
Chapter 11: Matrices
Objectives :
1.
2.
3.
4.
5.
Define a matrix as an ordered rectangular array of numbers.
Define special matrices such as the unit matrix, row and column matrices,
square matrix etc.
Add, subtract and mu
MS3127
Engineering Mathematics 1
Chapter 7: Application of Differentiation
Objectives :
1.
2.
3.
Solve problems involving related rates of change .
Find the stationary points on a curve and distinguish between maximum and
minimum turning points and points
MS3127
Engineering Mathematics 1
Chapter 4: Trigonometry
Objectives :
1.
2.
3.
4.
5.
Use the compound angle formulae and double angle formulae to solve
problems without evaluation of the angle State fundamental identities.
Apply the factor formulae to cha
MS3127
Engineering Mathematics 1
Chapter 5: Hyperbolic Functions
Objectives :
1.
4.
Define the hyperbolic sine and cosine in terms of the basic exponential
functions
Plot the graphs of sinh x and cosh x.
Derive the hyperbolic functions identities through
MS3127
Engineering Mathematics 1
Chapter 9: Integration
Objectives :
1.
2.
Determine indefinite and definite integrals of functions which lead to
algebraic, logarithmic, exponential and trigonometric functions.
Perform numerical integration by the applica
MS3127
Engineering Mathematics 1
Chapter 1 : Frequency Distribution
Objectives :
1.
4.
Understand statistics as a methodology that is concerned with the collection,
presentation, analysis and interpretation of data.
Appreciate the difference between descr
MS3127
Engineering Mathematics 1
Chapter 10: Determinants
Objectives :
1.
2.
3.
4.
Define a determinant as an ordered square array of numbers.
Define the minor and co-factor of an element.
Evaluate a third-order determinant.
Use Cramers rule to solve simu
MS3127
Engineering Mathematics 1
Chapter 2: Measures of Central Tendencies
Objectives :
1.
2.
Determine the central tendency of a frequency distribution by finding the
mean, mode and median.
Construct a box plot.
2.1
Measures of central tendencies
A measu
MS3127
Engineering Mathematics 1
Chapter 3 : Measures of Dispersion
Objectives :
1.
2.
Determine the dispersion of a frequency distribution by finding the range and
standard deviation.
Determine the coefficient of variation of a frequency distribution.
3.
1
2
When 1.00 dm3 of aqueous [Ni(NH3)6]2+ solution was added to an equal volume of sodium cyanide
solution, colour changes are observed.
Given that 0.6 moles of [Ni(NH3)6]2+, 1.0 moles of CN- and 0.6 moles of [Ni(CN)4]2- were mixed together,
and when equi
National University of Singapore
Department of Mathematics
MA2101 Linear Algebra II
Tutorial 6
1. Let V be an n-dimensional vector space over a field F and with a
basis B = (v1 , . . . , vn ).
Show that the coordinate map : V Fcn (v 7 [v]B ) is an isomorp
National University of Singapore
Department of Mathematics
Tutorial 9 HW5
MA2101 Linear Algebra II
1. Let A M4 (F ) and let J Mn (F ) be a Jordan canonical form of A
(you may assume A = J to do the questions below).
Find J and the characteristic and minim
National University of Singapore
Department of Mathematics
Tutorial 3 HW2
MA2101 Linear Algebra II
In this Tutorial, you may use the fact from linear algebra 1: n column
6 0.
vectors are L.I. in Fcn iff the matrix A they form has |A| =
1a. Given vectors v
National University of Singapore
Department of Mathematics
Tutorial 1 HW1
MA2101 Linear Algebra II
1. For the sets below and with operatins given, determine whether or
not they are vector spaces over R. For those that are not, list up all
Axioms in the de
Tutorial 7 HW4
MA2101 Linear Algebra II
Common Test (covering 1-6; Tut 1-5) be held on Mon 4:10pm5:10pm, 16th October 2017 at UT-AUD3 during the lecture time,
at lecture venue. Students who are absent from the test without an MC will be given
0 mark for t
National University of Singapore
Tutorial 5 H3
MA2101 Linear Algebra II
1. Determine whether or not the maps below are linear transformations. If yes, prove it; if no, give a counter-example.
[Suggested Answers: Y, N, Y, N, N, N.]
T1 : R[x] R[x]
f (x) = a
AY2015-16 Sem 2
MA1102R
MA1102R Calculus
Tutorial 9 Solutions
Basic Problems
Fundamental Theorem of Calculus 2 (FTC2)
1(c)
sin 2 x
h(x) =
e 3xt
3
dt
0
sin 2 x
h ( x)
3
e 3x t dt e 3 x
sin 2 x
0
et
3
dt
0
sin 2 x
h' ( x ) 3e 3 x
et
3
dt e 3 x e (sin 2 x
AY2015-16 Sem 2
MA1102R
MA1312 Calculus with Applications
Tutorial 8 Solutions
Basic Problems
2(a)
I
16 9 x 2
dx
x2
3
4
2
x
3
dx
x2
2
Let x
4
sin . Then,
3
dx
4
cos d
3
2
2
4
4
4
2
2
x 1 sin cos
3
3
3
sin
3x
4
Hence,
4
cos
4
3
I 3
cos d
2
3
4
2
AY2015-16 Sem 2
MA1102R
MA1102R Calculus
Tutorial 4 Solutions
Discussion Problems
1
F (x) R
iff
x
For (1),
x 2 2x 8
0
x
8
2 0 - (1)
x
and
x0
- (2)
( x 1) 2 7
0
x
Dividing both sides of the above inequality by ( x 1) 2 7
(which is positive for all real x)