Calculus Concepts and Methods 1st Edition by Ken Binmore, Joan Davies 0521775418 9780521775410
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ISBN 10: 0521775418
ISBN 13: 9780521775410
Author: Ken Binmore, Joan Davies
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.
Table of contents:
1. Matrices and vectors
1.1 Matrices◊
1.2 Exercises
1.3 Vectors in R2
1.4 Exercises
1.5 Vectors in R3
1.6 Lines
1.7 Planes
1.8 Exercises
1.9 Vectors in Rn
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♣
13.8 Exercises
13.9 Applications (optional)
13.9.1 Characteristic functions
13.9.2 Central limit theorem
14. Linear differential and difference equations
14.1 The operator P(D)
14.2 Difference equations and the shift operator E
14.3 Linear operators♣
14.4 Homogeneous, linear, differential equations♣
14.5 Complex roots of the auxiliary equation
14.6 Homogeneous, linear, difference equations
14.7 Nonhomogeneous equations♣
14.7.1 Nonhomogeneous differential equations
14.7.2 Nonhomogeneous difference equations
14.8 Convergence and divergence♣
14.9 Systems of linear equations♣
14.10 Change of variable♣
14.11 Exercises
14.12 The difference operator (optional)♣
14.13 Exercises
14.14 Applications (optional)
14.14.1 Cobweb models
14.14.2 Gambler’s ruin
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Tags: Ken Binmore, Joan Davies, Calculus, Methods