Calculus with Applications and Computing

1 Real numbers.- 1.1 The algebra of numbers; a review.- 1.2 The number line.- 1.3 Infinite decimals.- 1.4 Convergent sequences.- 1.5* Infinite sums.- 1.6 The least upper bound.- Appendix 1.1 Irrationality of $$\sqrt{2}$$ and e.- Appendix 1.2 Floating point representation.- 2 Functions.- 2.1 The notion of a function.- 2.2* Functions of several variables.- 2.3 Composite functions.- 2.4 Sums, products, and quotients of functions.- 2.5 Graphs of functions.- 2.6 Linear functions.- 2.7 Continuous functions.- 2.8 Convergent sequences of functions.- 2.9 Algorithms.- Appendix 2.1 Partial fraction expansion.- 3 Differentiation.- 3.1 The derivative.- 3.2 Rules of differentiation.- 3.3 Increasing and decreasing functions.- 3.4 The geometric meaning of derivative.- 3.5 Maxima and minima.- 3.6 One-dimensional mechanics.- 3.7 Higher derviatives.- 3.8 Mean value theorems.- 3.9* Taylor's theorem.- 3.10* Newton's method for finding zeros of a function.- 3.11 Economics and the derivative.- 4 Integration.- 4.1 Examples of integrals.- 4.2 The integral.- 4.3* Existence of the integral.- 4.4 The fundamental theorem of calculus.- 4.5 Rules of integration and how to use them.- 4.6 The approximation of integrals.- 4.7* Improper integrals.- 5 Growth and decay.- 5.1 The exponential function.- 5.2 The logarithm.- 5.3 The computation of logarithms and exponentiels.- 6 Probability and its applications.- 6.1 Discrete probability.- 6.2 Information theory or how interesting is interesting.- 6.3 Continuous probability.- 6.4 Law of errors.- 6.5 Diffusion.- 7 Rotation and the trigonometric functions.- 7.1 Rotation.- 7.2 Properties of cosine, sine, arcsine, and arctan.- 7.3 The computation of cosine, sine, and arctan.- 7.4 Complex numbers.- 7.5 Isometries of the complex plane.- 7.6 Complex functions.- 7.7 Polar coordinates.- 7.8 Two-dimensional mechanics.- 8 Vibrations.- 8.1 The differential equation governing vibrations of a simple mechanical system.- 8.2 Dissipation and conservation of energy.- 8.3 Vibration without friction.- 8.4 Linear vibrations without friction.- 8.5 Linear vibrations with friction.- 8.6 Linear systems driven by an external force.- 8.7 An example of nonlinear vibration.- 8.8 Electrical systems.- 9 Population dynamics and chemical reactions.- 9.1 The differential equation.- 9.2 Growth and fluctuation of population.- 9.3 Mathematical theory of chemical reactions.- FORTRAN programs and instructions for their use.- P.1 The bisection method for finding a zero of a function.- P.2 A program to locate the maximum of a unimodal function.- P.3 Newton's method for finding a zero of a function.- P.4 Simpson's rule.