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 ga
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 aci
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Addition -pathways a n d p r o d u c t s
The ad
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
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MA1102Rb CALCULUS
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Wang Fei
[email protected]
Department of Mathematics
Office: S14-02-09
Tel: 6516-2937
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Introductio
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 (principal
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 registrati
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.
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
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 variabl
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
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
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
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
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
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
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 an
REVISION
Chapter 0: Functions
Domain and range of function.
Composite functions.
Examples of functions: linear functions, polynomials, rational functions, piecewise
dened functions, trigonometric f
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 th
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