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Calculus: Concepts and Methods

Book Description

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. Cover
  2. Half Title
  3. Dedications
  4. Title Page
  5. Copyright
  6. Contents
  7. Preface
  8. Acknowledgements
  9. 1. Matrices and vectors
    1. 1.1 Matrices◊
    2. 1.2 Exercises
    3. 1.3 Vectors in R2
    4. 1.4 Exercises
    5. 1.5 Vectors in R3
    6. 1.6 Lines
    7. 1.7 Planes
    8. 1.8 Exercises
    9. 1.9 Vectors in Rn
    10. 1.10 Flats
    11. 1.11 Exercises
    12. 1.12 Applications (optional)
      1. 1.12.1 Commodity bundles
      2. 1.12.2 Linear production models
      3. 1.12.3 Price vectors
      4. 1.12.4 Linear programming
      5. 1.12.5 Dual problem
      6. 1.12.6 Game theory
  10. 2. Functions of one variable
    1. 2.1 Intervals◊
    2. 2.2 Real valued functions of one real variable◊
    3. 2.3 Some elementary functions◊
      1. 2.3.1 Power functions
      2. 2.3.2 Exponential functions
      3. 2.3.3 Trigonometric functions
    4. 2.4 Combinations of functions◊
    5. 2.5 Inverse functions◊
    6. 2.6 Inverses of the elementary functions◊
      1. 2.6.1 Root functions
      2. 2.6.2 Exponential and logarithmic functions
    7. 2.7 Derivatives◊
    8. 2.8 Existence of derivatives◊
    9. 2.9 Derivatives of inverse functions◊
    10. 2.10 Calculation of derivatives
      1. 2.10.1 Derivatives of elementary functions and their inverses
      2. 2.10.2 Derivatives of combinations of functions◊
    11. 2.11 Exercises
    12. 2.12 Higher order derivatives
    13. 2.13 Taylor series for functions of one variable
    14. 2.14 Conic sections
    15. 2.15 Exercises
  11. 3. Functions of several variables
    1. 3.1 Real valued functions of two variables
      1. 3.1.1 Linear and affine functions
      2. 3.1.2 Quadric surfaces
    2. 3.2 Partial derivatives
    3. 3.3 Tangent plane
    4. 3.4 Gradient
    5. 3.5 Derivative
    6. 3.6 Directional derivatives
    7. 3.7 Exercises
    8. 3.8 Functions of more than two variables
      1. 3.8.1 Tangent hyperplanes
      2. 3.8.2 Directional derivatives
    9. 3.9 Exercises
    10. 3.10 Applications (optional)
      1. 3.10.1 Indifference curves
      2. 3.10.2 Profit maximisation
      3. 3.10.3 Contract curve
  12. 4. Stationary points
    1. 4.1 Stationary points for functions of one variable◊
    2. 4.2 Optimisation
    3. 4.3 Constrained optimisation
    4. 4.4 The use of computer systems
    5. 4.5 Exercises
    6. 4.6 Stationary points for functions of two variables
    7. 4.7 Gradient and stationary points
    8. 4.8 Stationary points for functions of more than two variables
    9. 4.9 Exercises
  13. 5. Vector functions
    1. 5.1 Vector valued functions
    2. 5.2 Affine functions and flats
    3. 5.3 Derivatives of vector functions
    4. 5.4 Manipulation of vector derivatives
    5. 5.5 Chain rule
    6. 5.6 Second derivatives
    7. 5.7 Taylor series for scalar valued functions of n variables
    8. 5.8 Exercises
  14. 6. Optimisation of scalar valued functions
    1. 6.1 Change of basis in quadratic forms◊
    2. 6.2 Positive and negative definite
    3. 6.3 Maxima and minima
    4. 6.4 Convex and concave functions
    5. 6.5 Exercises
    6. 6.6 Constrained optimisation
    7. 6.7 Constraints and gradients
    8. 6.8 Lagrange’s method – optimisation with one constraint
    9. 6.9 Lagrange’s method – general case♣
    10. 6.10 Constrained optimisation – analytic criteria♣
    11. 6.11 Exercises
    12. 6.12 Applications (optional)
      1. 6.12.1 The Nash bargaining problem
      2. 6.12.2 Inventory control
      3. 6.12.3 Least squares analysis
      4. 6.12.4 Kuhn–Tucker conditions
      5. 6.12.5 Linear programming
      6. 6.12.6 Saddle points
  15. 7. Inverse functions
    1. 7.1 Local inverses of scalar valued functions
      1. 7.1.1 Differentiability of local inverse functions
      2. 7.1.2 Inverse trigonometric functions
    2. 7.2 Local inverses of vector valued functions
    3. 7.3 Coordinate systems
    4. 7.4 Polar coordinates
    5. 7.5 Differential operators♣
    6. 7.6 Exercises
    7. 7.7 Application (optional): contract curve
  16. 8. Implicit functions
    1. 8.1 Implicit differentiation
    2. 8.2 Implicit functions
    3. 8.3 Implicit function theorem
    4. 8.4 Exercises
    5. 8.5 Application (optional): shadow prices
  17. 9. Differentials
    1. 9.1 Matrix algebra and linear systems◊
    2. 9.2 Differentials
    3. 9.3 Stationary points
    4. 9.4 Small changes
    5. 9.5 Exercises
    6. 9.6 Application (optional): Slutsky equations
  18. 10. Sums and integrals
    1. 10.1 Sums◊
    2. 10.2 Integrals◊
    3. 10.3 Fundamental theorem of calculus◊
    4. 10.4 Notation◊
    5. 10.5 Standard integrals◊
    6. 10.6 Partial fractions◊
    7. 10.7 Completing the square◊
    8. 10.8 Change of variable◊
    9. 10.9 Integration by parts◊
    10. 10.10 Exercises
    11. 10.11 Infinite sums and integrals♣
    12. 10.12 Dominated convergence♣
    13. 10.13 Differentiating integrals♣
    14. 10.14 Power series♣
    15. 10.15 Exercises
    16. 10.16 Applications (optional)
      1. 10.16.1 Probability
      2. 10.16.2 Probability density functions
      3. 10.16.3 Binomial distribution
      4. 10.16.4 Poisson distribution
      5. 10.16.5 Mean
      6. 10.16.6 Variance
      7. 10.16.7 Standardised random variables
      8. 10.16.8 Normal distribution
      9. 10.16.9 Sums of random variables
      10. 10.16.10 Cauchy distribution
      11. 10.16.11 Auctions
  19. 11. Multiple integrals
    1. 11.1 Introduction
    2. 11.2 Repeated integrals
    3. 11.3 Change of variable in multiple integrals♣
    4. 11.4 Unbounded regions of integration♣
    5. 11.5 Multiple sums and series♣
    6. 11.6 Exercises
    7. 11.7 Applications (optional)
      1. 11.7.1 Joint probability distributions
      2. 11.7.2 Marginal probability distributions
      3. 11.7.3 Expectation, variance and covariance
      4. 11.7.4 Independent random variables
      5. 11.7.5 Generating functions
      6. 11.7.6 Multivariate normal distributions
  20. 12. Differential equations of order one
    1. 12.1 Differential equations
    2. 12.2 General solutions of ordinary equations
    3. 12.3 Boundary conditions
    4. 12.4 Separable equations
    5. 12.5 Exact equations
    6. 12.6 Linear equations of order one
    7. 12.7 Homogeneous equations
    8. 12.8 Change of variable
    9. 12.9 Identifying the type of first order equation
    10. 12.10 Partial differential equations
    11. 12.11 Exact equations and partial differential equations
    12. 12.12 Change of variable in partial differential equations
    13. 12.13 Exercises
  21. 13. Complex numbers
    1. 13.1 Quadratic equations
    2. 13.2 Complex numbers
    3. 13.3 Modulus and argument
    4. 13.4 Exercises
    5. 13.5 Complex roots
    6. 13.6 Polynomials
    7. 13.7 Elementary functions♣
    8. 13.8 Exercises
    9. 13.9 Applications (optional)
      1. 13.9.1 Characteristic functions
      2. 13.9.2 Central limit theorem
  22. 14. Linear differential and difference equations
    1. 14.1 The operator P(D)
    2. 14.2 Difference equations and the shift operator E
    3. 14.3 Linear operators♣
    4. 14.4 Homogeneous, linear, differential equations♣
    5. 14.5 Complex roots of the auxiliary equation
    6. 14.6 Homogeneous, linear, difference equations
    7. 14.7 Nonhomogeneous equations♣
      1. 14.7.1 Nonhomogeneous differential equations
      2. 14.7.2 Nonhomogeneous difference equations
    8. 14.8 Convergence and divergence♣
    9. 14.9 Systems of linear equations♣
    10. 14.10 Change of variable♣
    11. 14.11 Exercises
    12. 14.12 The difference operator (optional)♣
    13. 14.13 Exercises
    14. 14.14 Applications (optional)
      1. 14.14.1 Cobweb models
      2. 14.14.2 Gambler’s ruin
  23. Answers to starred exercises with some hints and solutions
  24. Appendix
  25. Index