Calculus: Concepts and Methods

by Ken Binmore , Joan Davies

• ISBN13: 9780521775410
• ISBN10: 0521775418
• Format: Paperback
• Copyright: 2002-03-11
• Publisher: Cambridge University Press

Summary

The pebbles used in ancient abacuses gave their name to the calculus, which today is a fundamental tool in business, economics, engineering and the sciences. This introductory book takes readers gently from single to multivariate calculus and simple differential and difference equations. Unusually the book offers a wide range of applications in business and economics, as well as more conventional scientific examples. Ideas from univariate calculus and linear algebra are covered as needed, often from a new perspective. They are reinforced in the two-dimensional case, which is studied in detail before generalisation to higher dimensions. Although there are no theorems or formal proofs, this is a serious book in which conceptual issues are explained carefully using numerous geometric devices and a wealth of worked examples, diagrams and exercises. Mathematica has been used to generate many beautiful and accurate, full-colour illustrations to help students visualise complex mathematical objects. This adds to the accessibility of the text, which will appeal to a wide audience among students of mathematics, economics and science.

Author Biography

Ken Binmore held the Chair of Mathematics for many years at the London School of Economics Joan Davies currently lectures at the London School of Economics

Table of Contents

Preface	xi
Acknowledgements	xiv
Matrices and vectors	1	(50)
Matrices	1	(10)
Exercises	11	(3)
Vectors in R2	14	(6)
Exercises	20	(1)
Vectors in R3	21	(3)
Lines	24	(3)
Planes	27	(8)
Exercises	35	(3)
Vectors in Rn	38	(2)
Flats	40	(3)
Exercises	43	(2)
Applications (optional)	45	(6)
Commodity bundles	45	(1)
Linear production models	46	(1)
Price vectors	46	(1)
Linear programming	47	(1)
Dual problem	48	(1)
Game theory	49	(2)
Functions of one variable	51	(36)
Intervals	51	(1)
Real valued functions of one real variable	51	(1)
Some elementary functions	52	(3)
Power functions	52	(1)
Exponential functions	53	(1)
Trigonometric functions	53	(2)
Combinations of functions	55	(2)
Inverse functions	57	(2)
Inverses of the elementary functions	59	(2)
Root functions	59	(1)
Exponentional and logarithmic functions	60	(1)
Derivatives	61	(2)
Existence of derivatives	63	(1)
Derivatives of inverse functions	64	(1)
Calculation of derivatives	65	(4)
Derivatives of elementary functions and their inverses	65	(1)
Derivatives of combinations of functions	66	(3)
Exercises	69	(2)
Higher order derivatives	71	(1)
Taylor series for functions of one variable	72	(4)
Conic sections	76	(4)
Exercises	80	(7)
Functions of several variables	87	(38)
Real valued functions of two variables	87	(5)
Linear and affine functions	88	(1)
Quadric surfaces	89	(3)
Partial derivatives	92	(4)
Tangent plane	96	(2)
Gradient	98	(2)
Derivative	100	(1)
Directional derivatives	100	(6)
Exercises	106	(4)
Functions of more than two variables	110	(5)
Tangent hyperplanes	111	(3)
Directional derivatives	114	(1)
Exercises	115	(5)
Applications (optional)	120	(5)
Indifference curves	120	(2)
Profit maximisation	122	(1)
Contract curve	122	(3)
Stationary points	125	(24)
Stationary points for functions of one variable	125	(3)
Optimisation	128	(2)
Constrained optimisation	130	(2)
The use of computer systems	132	(3)
Exercises	135	(4)
Stationary points for functions of two variables	139	(3)
Gradient and stationary points	142	(1)
Stationary points for functions of more than two variables	143	(2)
Exercises	145	(4)
Vector functions	149	(30)
Vector valued functions	149	(4)
Affine functions and flats	153	(2)
Derivatives of vector functions	155	(7)
Manipulation of vector derivatives	162	(1)
Chain rule	163	(5)
Second derivatives	168	(2)
Taylor series for scalar valued functions of n variables	170	(4)
Exercises	174	(5)
Optimisation of scalar valued functions	179	(56)
Change of basis in quadratic forms	179	(6)
Positive and negative definite	185	(3)
Maxima and minima	188	(5)
Convex and concave functions	193	(7)
Exercises	200	(3)
Constrained optimisation	203	(5)
Constraints and gradients	208	(2)
Lagrange's method - optimisation with one constraint	210	(7)
Lagrange's method - general case	217	(2)
Constrained optimisation - analytic criteria	219	(1)
Exercises	220	(3)
Applications (optional)	223	(12)
The Nash bargaining problem	223	(1)
Inventory control	224	(3)
Least squares analysis	227	(2)
Kuhn-Tucker conditions	229	(2)
Linear programming	231	(2)
Saddle points	233	(2)
Inverse functions	235	(30)
Local inverses of scalar valued functions	235	(6)
Differentiability of local inverse functions	237	(1)
Inverse trigonometric functions	238	(3)
Local inverses of vector valued functions	241	(6)
Coordinate systems	247	(7)
Polar coordinates	254	(2)
Differential operators	256	(3)
Exercises	259	(4)
Application (optional): contract curve	263	(2)
Implicit functions	265	(16)
Implicit differentiation	265	(1)
Implicit functions	266	(2)
Implicit function theorem	268	(7)
Exercises	275	(2)
Application (optional): shadow prices	277	(4)
Differentials	281	(14)
Matrix algebra and linear systems	281	(1)
Differentials	282	(5)
Stationary points	287	(2)
Small changes	289	(1)
Exercises	290	(2)
Application (optional): Slutsky equations	292	(3)
Sums and integrals	295	(50)
Sums	295	(2)
Integrals	297	(1)
Fundamental theorem of calculus	298	(2)
Notation	300	(2)
Standard integrals	302	(2)
Partial fractions	304	(4)
Completing the square	308	(1)
Change of variable	309	(2)
Integration by parts	311	(2)
Exercises	313	(3)
Infinite sums and integrals	316	(4)
Dominated convergence	320	(3)
Differentiating integrals	323	(2)
Power series	325	(1)
Exercises	326	(2)
Applications (optional)	328	(17)
Probability	328	(1)
Probability density functions	329	(1)
Binomial distribution	330	(1)
Poisson distribution	331	(2)
Mean	333	(1)
Variance	334	(1)
Standardised random variables	335	(1)
Normal distribution	336	(3)
Sums of random variables	339	(2)
Cauchy distribution	341	(1)
Auctions	341	(4)
Multiple integrals	345	(28)
Introduction	345	(2)
Repeated integrals	347	(6)
Change of variable in multiple integrals	353	(7)
Unbounded regions of integration	360	(1)
Multiple sums and series	361	(2)
Exercises	363	(3)
Applications (optional)	366	(7)
Joint probability distributions	366	(1)
Marginal probability distributions	366	(1)
Expectation, variance and covariables	367	(1)
Independent random variables	368	(1)
Generating functions	369	(1)
Multivariate normal distributions	370	(3)
Differential equations of order one	373	(32)
Differential equations	373	(1)
General solutions of ordinary equations	374	(1)
Boundary conditions	375	(1)
Separable equations	375	(7)
Exact equations	382	(2)
Linear equations of order one	384	(3)
Homogeneous equations	387	(2)
Change of variable	389	(3)
Identifying the type of first order equation	392	(1)
Partial differential equations	393	(2)
Exact equations and partial differential equations	395	(2)
Change of variable in partial differential equations	397	(2)
Exercises	399	(6)
Complex numbers	405	(16)
Quadratic equations	405	(1)
Complex numbers	406	(1)
Modulus and argument	407	(2)
Exercises	409	(1)
Complex roots	410	(2)
Polynomials	412	(2)
Elementary functions	414	(2)
Exercises	416	(1)
Applications (optional)	417	(4)
Characteristic functions	417	(2)
Central limit theorem	419	(2)
Linear differential and difference equations	421	(50)
The operator p(D)	421	(2)
Difference equations and the shift operator E	423	(2)
Linear operators	425	(1)
Homogeneous, linear, differential equations	426	(3)
Complex roots of the auxiliary equations	429	(2)
Homogeneous, linear, difference equations	431	(3)
Nonhomogeneous difference equations	434	(10)
Nonhomogeneous differential equations	435	(6)
Nonhomogeneous difference equations	441	(3)
Convergence and divergence	444	(3)
Systems of linear equations	447	(6)
Change of variable	453	(1)
Exercises	454	(4)
The difference operator (optional)	458	(3)
Exercises	461	(1)
Applications (optional)	462	(9)
Cobweb models	462	(5)
Gambler's ruin	467	(4)
Answers to starred exercises with some hints and solutions	471	(78)
Appendix	549	(1)
Index	550