By K. D. Stroyan
Calculus utilizing Mathematica
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Extra resources for Calculus Using Mathematica
1. 1. Linear Approximation of Oxbows This section shows you one dramatic example of non-local linear approximation. The project on CO2 in the Scientific Projects shows you a more serious example where longterm linear approximation really is a plausible idea. The local linearity of calculus is only good for small steps; we hope Mark Twain's wit will help you remember this. The Chapter Cut-offs and Stephen of Mark Twain's Life on the Mississippi contains the following excerpt about ox-bows on the lower Mississippi.
T h e parameter m is t h e slope of t h e line t h a t graphs t h e function. This linear function is increasing if m > 0, decreasing if m < 0 a n d horizontal (or constant) if m = 0. Calculus tells when a nonlinear function y = f[x] is increasing by computing an approximating linear function, the differential. ) It is easy t o tell if the linear function is increasing a n d t h e nonlinear function also increases where t h e approximation is valid... These words are complicated, b u t have an easy graphical interpretation.
23. ) This is a difficult limit to compute analytically, but it turns out to be a key to a complete understanding of the model. 2. 24. ma NoteBook. 15 above. The long term graphs raise many questions such as: How long does the epidemic last? Is there a formula for s[oo] ? These are very difficult questions, but the computer will give you practical answers, if not formulas. Write a summary of your experiments on how s[oc] depends on a, b and s ? 4. Calculus a n d t h e S-I-R Differential E q u a t i o n s Up to this point you should have followed all the mathematical derivations by working the exercises.