Calculus: Concepts and Methods

699 Pages·2001·29.837 MB·English

by  Ken Binmore, Joan Davies

Calculus
KEN BINMORE held the Chair of Mathematics for many years at the London School of Economics,
where this book was born. In recent years, he has used his mathematical skills in developing the
theory of games in a number of posts as a professor of economics, both in Britain and the USA. His
most recent exploit was to head the team that used mathematical ideas to design the UK auction of
telecom frequencies that made 35 billion dollars for the British taxpayer.
JOAN DAVIES has a doctorate in mathematics from the University of Oxford. She is committed to
bringing an understanding of mathematics to a wider audience. She currently lectures at the London
School of Economics.
Dedications
To Jim Clunie Ken Binmore

To my family
Roy, Elizabeth, Sarah and Marion Joan Davies
Calculus
Ken Binmore
University College, London
Joan Davies
London School of Economics
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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© Cambridge University Press 2001
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reproduction of any part may take place without the written permission of Cambridge University Press.
First published 2001
7th printing 2012
Printed and Bound in the United Kingdom by the MPG Books Group
A catalogue record of this publication is available from the British Library
ISBN 978-0-521-77541-0 Paperback
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Contents
Preface
Acknowledgements
1 Matrices and vectors
1.1 Matrices
1.2 Exercises
1.3 Vectors in 2
1.4 Exercises
1.5 Vectors in 3
1.6 Lines
1.7 Planes
1.8 Exercises
1.9 Vectors in n
1.10 Flats
1.11 Exercises
1.12 Applications (optional)
1.12.1 Commodity bundles
1.12.2 Linear production models
1.12.3 Price vectors
1.12.4 Linear programming
1.12.5 Dual problem
1.12.6 Game theory
2 Functions of one variable
2.1 Intervals
2.2 Real valued functions of one real variable
2.3 Some elementary functions
2.3.1 Power functions
2.3.2 Exponential functions
2.3.3 Trigonometric functions
2.4 Combinations of functions
2.5 Inverse functions
2.6 Inverses of the elementary functions
2.6.1 Root functions
2.6.2 Exponential and logarithmic functions
2.7 Derivatives
2.8 Existence of derivatives
2.9 Derivatives of inverse functions
2.10 Calculation of derivatives
2.10.1 Derivatives of elementary functions and their inverses
2.10.2 Derivatives of combinations of functions
2.11 Exercises
2.12 Higher order derivatives
2.13 Taylor series for functions of one variable
2.14 Conic sections
2.15 Exercises
3 Functions of several variables
3.1 Real valued functions of two variables
3.1.1 Linear and affine functions
3.1.2 Quadric surfaces
3.2 Partial derivatives
3.3 Tangent plane
3.4 Gradient
3.5 Derivative
3.6 Directional derivatives
3.7 Exercises
3.8 Functions of more than two variables
3.8.1 Tangent hyperplanes
3.8.2 Directional derivatives
3.9 Exercises
3.10 Applications (optional)
3.10.1 Indifference curves
3.10.2 Profit maximisation
3.10.3 Contract curve
4 Stationary points
4.1 Stationary points for functions of one variable
4.2 Optimisation
4.3 Constrained optimisation
4.4 The use of computer systems
4.5 Exercises
4.6 Stationary points for functions of two variables
4.7 Gradient and stationary points
4.8 Stationary points for functions of more than two variables
4.9 Exercises
5 Vector functions
5.1 Vector valued functions
5.2 Affine functions and flats
5.3 Derivatives of vector functions
5.4 Manipulation of vector derivatives
5.5 Chain rule
5.6 Second derivatives
5.7 Taylor series for scalar valued functions of n variables
5.8 Exercises
6 Optimisation of scalar valued functions
6.1 Change of basis in quadratic forms
6.2 Positive and negative definite
6.3 Maxima and minima
6.4 Convex and concave functions
6.5 Exercises
6.6 Constrained optimisation
6.7 Constraints and gradients
6.8 Lagrange’s method – optimisation with one constraint
6.9 Lagrange’s method – general case
6.10 Constrained optimisation – analytic criteria
6.11 Exercises
6.12 Applications (optional)
6.12.1 The Nash bargaining problem
6.12.2 Inventory control
6.12.3 Least squares analysis
6.12.4 Kuhn–Tucker conditions
6.12.5 Linear programming
6.12.6 Saddle points
7 Inverse functions
7.1 Local inverses of scalar valued functions
7.1.1 Differentiability of local inverse functions
7.1.2 Inverse trigonometric functions
7.2 Local inverses of vector valued functions
7.3 Coordinate systems
7.4 Polar coordinates
7.5 Differential operators
7.6 Exercises
7.7 Application (optional): contract curve
8 Implicit functions
8.1 Implicit differentiation
8.2 Implicit functions
8.3 Implicit function theorem
8.4 Exercises
8.5 Application (optional): shadow prices
9 Differentials
9.1 Matrix algebra and linear systems
9.2 Differentials
9.3 Stationary points
9.4 Small changes
9.5 Exercises
9.6 Application (optional): Slutsky equations
10 Sums and integrals
10.1 Sums
10.2 Integrals
10.3 Fundamental theorem of calculus
10.4 Notation
10.5 Standard integrals
10.6 Partial fractions
10.7 Completing the square
10.8 Change of variable
10.9 Integration by parts
10.10 Exercises
10.11 Infinite sums and integrals
10.12 Dominated convergence
10.13 Differentiating integrals
10.14 Power series
10.15 Exercises
10.16 Applications (optional)
10.16.1 Probability
10.16.2 Probability density functions
10.16.3 Binomial distribution
10.16.4 Poisson distribution
10.16.5 Mean
10.16.6 Variance
10.16.7 Standardised random variables
10.16.8 Normal distribution
10.16.9 Sums of random variables
10.16.10 Cauchy distribution
10.16.11 Auctions
11 Multiple integrals
11.1 Introduction
11.2 Repeated integrals
11.3 Change of variable in multiple integrals
11.4 Unbounded regions of integration
11.5 Multiple sums and series
11.6 Exercises
11.7 Applications (optional)
11.7.1 Joint probability distributions
11.7.2 Marginal probability distributions
11.7.3 Expectation, variance and covariance
11.7.4 Independent random variables
11.7.5 Generating functions
11.7.6 Multivariate normal distributions
12 Differential equations of order one
12.1 Differential equations
12.2 General solutions of ordinary equations
12.3 Boundary conditions
12.4 Separable equations
12.5 Exact equations
12.6 Linear equations of order one
12.7 Homogeneous equations
12.8 Change of variable
12.9 Identifying the type of first order equation
12.10 Partial differential equations
12.11 Exact equations and partial differential equations
12.12 Change of variable in partial differential equations
12.13 Exercises
13 Complex numbers
13.1 Quadratic equations
13.2 Complex numbers
13.3 Modulus and argument
13.4 Exercises
13.5 Complex roots
13.6 Polynomials
13.7 Elementary functions