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Foreword
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Contents
Index
Advanced Calculus and Analysis
MA1002
Ian Craw
Foreword
These Notes
Acknowledgements
Contents
List of Figures
Introduction.
The Need for Good Foundations
The Real Numbers
Properties of
Inequalities
Intervals
Functions
Neighbourhoods
Absolute Value
The Binomial Theorem and other Algebra
Sequences
Definition and Examples
Examples of sequences
Direct Consequences
Sums, Products and Quotients
Squeezing
Bounded sequences
Infinite Limits
Monotone Convergence
Three Hard Examples
Boundedness Again
Monotone Convergence
The Fibonacci Sequence
Limits and Continuity
Classes of functions
Limits and Continuity
One sided limits
Results giving Coninuity
Infinite limits
Continuity on a Closed Interval
Differentiability
Definition and Basic Properties
Simple Limits
Rolle and the Mean Value Theorem
l'Hôpital revisited
Infinite limits
(Rates of growth)
Taylor's Theorem
Infinite Series
Arithmetic and Geometric Series
Convergent Series
The Comparison Test
Absolute and Conditional Convergence
An Estimation Problem
Power Series
Power Series and the Radius of Convergence
Representing Functions by Power Series
Other Power Series
Power Series or Function
Applications*
The function
e
x
grows faster than any power of
x
The function log
x
grows more slowly than any power of
x
The probability integral
e
d
x
The number
e
is irrational
Differentiation of Functions of Several Variables
Functions of Several Variables
Graphing functions of Several Variables
Continuity
Partial Differentiation
Higher Derivatives
Solving equations by Substitution
Maxima and Minima
Tangent Planes
Linearisation and Differentials
Implicit Functions of Three Variables
Multiple Integrals
Integrating functions of several variables
Repeated Integrals and Fubini's Theorem
Change of Variable -- the Jacobian
Bibliography
Index
About this document ...
Ian Craw 2002-01-07
e
dx
is irrational