1} Which ofthe following statements about unanticipated inﬂation is true?
-9- It redistributes purchasing power in the economy.
- Its effects are spread evenly throughout the economy so that no one gains or loses from inﬂation.
- It reduces average purcha
LESSON 4: INTEGRATIONS AND CONDITIONAL STATEMENTS
1. Riemann Sum
1.1. The Sum. The command sum is used to evaluate denite or indenite sum of expressions.
Let f be an expression with summation index i. Then sum(f,i=m.n) gives the denite
sum of f(i) for i f
Hydrogenation of fats the manufacture of margarine
Margarine and butter are emulsions formed between fats and water. Fats are known as triglycerides.
They are formed by the reaction of three fatty acid molecules including glycerol which contains three
-OH
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Addition -pathways a n d p r o d u c t s
The addition reactions we have studied so far all proceed by
CHAPTER
1
Probability
1.1 Introduction
The idea of probability, chance, or randomness is quite old, whereas its rigorous
axiomatization in mathematical terms occurred relatively recently. Many of the ideas
of probability theory originated in the study of
https:/en.wikipedia.org/wiki/Alkene#Physical_properties
Synthesis
Industrial methods
Alkenes are produced by hydrocarbon cracking. Raw materials are mostly natural gas
condensate components (principally ethane and propane) in the US and Mideast and naphth
CM1417 Semester 1, 2015-16
Assignment 1
Select one of the topics given in the next page. You have to register through IVLE for the
topic. Maximum number of students for each topic is 8. The registration is by first come
first served basis.
Registration Ti
LESSON 1: BASIC COMMANDS AND CURVE PLOTTING
1. Introduction
Maple is a powerful software for doing mathematics.
In this course, we would like to
introduce students this powerful software. We are going to learn how to use certain basic
commands of Maple by
LESSON 5: ORDINARY DIFFERENTIAL EQUATIONS AND
REPETITION
1. Ordinary Differential Equations
Recall that an ordinary dierential equation is a equation that contains functions of
one independent variable and its derivatives with respect to that variable. It
MA1102R CALCULUS
(2014/2015, 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. R. Hass, Thomas Calculus, 12th edition, Pearson,
2010.
CHAPTER 0: FUNC
CHAPTER 1: LIMITS
1.1
A First Encounter of Limits
(Reference: TC, 2.2, 2.4, 2.6)
Intuitive Denition
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 suciently close (but not equal) to a.
Notatio
CHAPTER 2: CONTINUOUS FUNCTIONS
2.1
Continuity
(Reference: TC, 2.5)
Intuition
A function y = f (x) is continuous if its graph can be drawn with one
continuous motion of the pen.
Denition
Let [c, d] be the domain of a function f .
1. f is continuous at an
CHAPTER 3: DERIVATIVES
3.1
The Derivative of a Function
(Reference: TC, 3.1, 3.2, 3.3, 3.4)
Tangent Line
Denition
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 tangent
CHAPTER 7: TECHNIQUES OF INTEGRATION
7.1
Integration by Substitution
(Reference: TC, 8.3)
Recall (see 5.4) that by letting u = g(x),
f (g(x)g (x) dx =
This often converts the more complicated integral
f (u) du.
f (g(x)g (x) dx into a simpler integral
f (u
CHAPTER 9: FIRST-ORDER DIFFERENTIAL
EQUATIONS
9.1
Separable Dierential Equations
(Reference: TC, 7.4, 9.1)
Dierential Equations
Goal: To apply calculus to solve dierential equations, which are used to model physical
phenomena.
Denition
An ordinary dierent
CHAPTER 6: TRANSCENDENTAL FUNCTIONS
Inverse Functions and Their Derivatives
6.1
(Reference: TC, 7.1)
One-to-One Functions and Their Inverses
Denition
A function f with domain D is one-to-one if for any x1 , x2 D,
x1 = x2
implies f (x1 ) = f (x2 ).
[Equiva
REVISION
Chapter 0: Functions
Domain and range of function.
Composite functions.
Examples of functions: linear functions, polynomials, rational functions, piecewise
dened functions, trigonometric functions, root functions, logarithmic functions,
and ex
CHAPTER 5: INTEGRALS
5.1
The Denite Integral
(Reference: TC, 5.1, 5.2, 5.3)
Approximation of Area Under a Graph
Consider the area under the graph of y =
1 x2 , x [0, 1], in the rst quadrant.
Divide the interval [0, 1] into n equal subintervals.
Use rectan
CHAPTER 4: APPLICATIONS OF
DIFFERENTIATION
4.1
Extreme Values of Functions
(Reference: TC, 4.1)
Local (Relative) versus Absolute (Global) Extrema
Denition
Let f be a function and D be its domain.
1. f has a local maximum (or relative maximum) value at c D
CHAPTER 8: APPLICATIONS OF
INTEGRATION
8.1
Area Between Curves
(Reference: TC, 5.6)
Goal: To use denite integrals to nd areas of regions between curves.
Recall (see 5.1) that for f (x) 0 for all x, the area between y = f (x) and the x-axis
b
from x = a to