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 5: ORDINARY DIFFERENTIAL EQUATIONS AND
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
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
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Published on 18 July 2002 on http:/pubs.rsc.org | doi:10.1039/9781847557827-00028
Addition -pathways a n d p r o d u c t s
The addition reactions we have studied so far all proceed by
(2014/2015, Semester 1)
Lecture Notes (Group 1)
Goh Say Song
Department of Mathematics
National University of Singapore
[TC] G. B. Thomas, M. D. Weir and J. R. Hass, Thomas Calculus, 12th edition, Pearson,
CHAPTER 0: FUNC
CHAPTER 1: LIMITS
A First Encounter of Limits
(Reference: TC, 2.2, 2.4, 2.6)
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.
CHAPTER 2: CONTINUOUS FUNCTIONS
(Reference: TC, 2.5)
A function y = f (x) is continuous if its graph can be drawn with one
continuous motion of the pen.
Let [c, d] be the domain of a function f .
1. f is continuous at an
CHAPTER 3: DERIVATIVES
The Derivative of a Function
(Reference: TC, 3.1, 3.2, 3.3, 3.4)
The slope of the curve y = f (x) at the point P (a, f (a) is the number
f (a + h) f (a)
m = lim
provided the limit exists. The tangent
CHAPTER 7: TECHNIQUES OF INTEGRATION
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
CHAPTER 9: FIRST-ORDER DIFFERENTIAL
Separable Dierential Equations
(Reference: TC, 7.4, 9.1)
Goal: To apply calculus to solve dierential equations, which are used to model physical
An ordinary dierent
CHAPTER 6: TRANSCENDENTAL FUNCTIONS
Inverse Functions and Their Derivatives
(Reference: TC, 7.1)
One-to-One Functions and Their Inverses
A function f with domain D is one-to-one if for any x1 , x2 D,
x1 = x2
implies f (x1 ) = f (x2 ).
Chapter 0: Functions
Domain and range of function.
Examples of functions: linear functions, polynomials, rational functions, piecewise
dened functions, trigonometric functions, root functions, logarithmic functions,
CHAPTER 5: INTEGRALS
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.
CHAPTER 4: APPLICATIONS OF
Extreme Values of Functions
(Reference: TC, 4.1)
Local (Relative) versus Absolute (Global) Extrema
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
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
from x = a to
CM1417 Semester 1, 2015-16
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.
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