8: Appendix Calculus Review

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"Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say." Bertrand Russell \((1872-1970)\)

BEFORE YOU BEGIN OUR STUDY OF DIFFERENTIAL EQUATIONS perhaps you should review some things from calculus. You definitely need to know something before taking this class. It is assumed that you have taken Calculus and are comfortable with differentiation and integration. Of course, you are not expected to know every detail from these courses. However, there are some topics and methods that will come up and it would be useful to have a handy reference to what it is you should know.

Most importantly, you should still have your calculus text to which you can refer throughout the course. Looking back on that old material, you will find that it appears easier than when you first encountered the material. That is the nature of learning mathematics and other subjects. Your understanding is continually evolving as you explore topics more in depth. It does not always sink in the first time you see it. In this chapter we will give a quick review of these topics. We will also mention a few new methods that might be interesting.

8.1: Introduction
THERE ARE TWO MAIN TOPICS IN CALCULUS: derivatives and integrals. You learned that derivatives are useful in providing rates of change in either time or space. Integrals provide areas under curves, but also are useful in providing other types of sums over continuous bodies, such as lengths, areas, volumes, moments of inertia, or flux integrals.
8.2: Trigonometric Functions
Another set of useful functions are the trigonometric functions. They have their origins as far back as the building of the pyramids. Typical applications in your introductory math classes probably have included finding the heights of trees, flag poles, or buildings. It was recognized a long time ago that similar right triangles have fixed ratios of any pair of sides of the two similar triangles. These ratios only change when the non-right angles change.
8.3: Hyperbolic Functions
So, are there any other functions that are useful in physics? Actually, there are many more. However, you have probably not see many of them to date. We will see by the end of the semester that there are many important functions that arise as solutions of some fairly generic, but important, physics problems.
8.4: Derivatives
Now that we know some elementary functions, we seek their derivatives. We will not spend time exploring the appropriate limits in any rigorous way. We are only interested in the results. We expect that you know the meaning of the derivative and all of the usual rules, such as the product and quotient rules.
8.5: Integrals
Integration is typically a bit harder. Imagine being given the last result in Equation 8.4.2 and having to figure out what was differentiated in order to get the given function.
8.6: Geometric Series
Geometric series are fairly common and will be used throughout the book. You should learn to recognize them and work with them.
8.7: Power Series
ANOTHER EXAMPLE OF AN INFINITE SERIES that the student has encountered in previous courses is the power series. Examples of such series are provided by Taylor and Maclaurin series.
8.8: The Binomial Expansion
Another series expansion which occurs often in examples and applications is the binomial expansion.
8.9: Problems

Thumbnail: A straight line tangent to a curve. (CC BY-SA 3.0 Unported; AxelBoldt via Wikicommons)