11: A - Calculus Review - What Do I Need to Know From Calculus?
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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.
- 11.1: 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. In physics, one can look at graphs of position versus time and the slope (derivative) of such a function gives the velocity.
- 11.3: Hyperbolic Functions
- In your calculus classes you have also seen that some relations are represented in parametric form. However, there is at least one other set of elementary functions, which you should already know about. These are the hyperbolic functions. Such functions are useful in representing hanging cables, unbounded orbits, and special traveling waves called solutions. They also play a role in special and general relativity.
- 11.6: Geometric Series
- Infinite series occur often in mathematics and physics. Two series which occur often are the geometric series and the binomial series. we will discuss these next.
- 11.7: 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.
- 11.8: The Binomial Expansion
- Another series expansion which occurs often in examples and applications is the binomial expansion. This is simply the expansion of the expression (a+b)ⁿ in powers of a and b. We will investigate this expansion first for nonnegative integer powers n and then derive the expansion for other values of n.