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Mathematics is vigorously and brilliantly pursued in our time on a very broad front; yet the authors of this text feel that not enough mathematical talent is devoted to furthering the interaction of mathematics with other sciences and disciplines. This imbalance is harmful to both mathematics and its users; to redress this imbalance is an educational task which must start at the beginning of the college curriculum. No course is more suited for this than the calculus; there students can learn at first hand that mathe matics is the language in which scientific ideas can be precisely formulated, that science is a source of mathematical ideas which profoundly shape the development of mathematics, and last but not least that mathematics can furnish brilliant answers to important scientific problems. Our purpose in writing this text has been to emphasize this relation of calculus to science. We hope to accomplish this by devoting whole connected chapters to single-or several related-scientific topics, letting the reader observe how the notions of calculus are used to formulate the basic laws of science and how the methods of calculus are used to deduce consequences of those basic laws. Thus the student sees calculus at work on worthwhile tasks.
List of contents
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.
