Today’s soundtrack is Creed: My Own Prison, the band’s debut album, and, funnily enough, the last of their albums that I listened to.
This morning, I’m taking the bull by the horns and starting in on Silvanus P. Thompson’s Calculus Made Easy, which, despite its claim, has me utterly daunted due to the subject matter alone.
In the preface, Martin Gardner (1998) says that since we now have calculators that cost less than a calculus textbook and can solve the majority of the problems in it, our focus should be less on solving the problems that the calculators can solve more efficiently than we can and more on understanding what the calculator is doing while it solves those problems.
Martin states that Thompson’s book is well-balanced for the beginner: not too easy, not too hard. One thing he notes is that Thompson assumes that readers have an understanding of “[t]he two most important concepts in calculus[, which] are functions and limits” (Thompson, 1998, p. 8), so Gardner covers those topics in the preliminary chapters so that we can embark on what he calls “a great intellectual adventure” (Thompson, 1998, p. 9): the philosophy of calculus, “the mathematics of change” (Thompson, 1998, p. 9).
Preliminary Chapter One: What is a Function?
A function is a set of two variables: one independent, one dependent. We usually use x to represent our independent variable (the one that we change), and y to represent the dependent variable (the value that results from our changes). An example of a function “is the dependence of a square’s area on the length of its side” (Thompson, 1998, p. 10), represented as y = x². We can also represent y as being the function of x: thus we could represent our square’s area equation as f(x) = x². If we want to solve for a specific value of x, we replace its value on both sides with that constant. For example, if we know that the side length is 7″, we would calculate f(7) = x², which of course is 7²: 49. So to summarize, if every input (x) has only one possible output (y), we are working with a function. There are many kinds of functions: linear, quadratic, trigonometric, logarithmic, exponential, and so on.
Some functions are dependent on more than one independent variables, such as the area of a triangle (depends on both altitude and base), or the volume of a rectangular prism (length, width, and height are all independent variables).
When we get into volume, we can no longer work with the simple ideas of domain and range (x and y). We must include z, the vertical axis, which turns the whole thing three-dimensional.
To paraphrase an example that Gardner gives on page 17, a child is an independent variable; a father is a dependent variable. A father can have many children, but a child can only have one [biological] father. So we can see that functions are not limited to numbers; functions are everywhere.
That’s it for today! In the next chapter, we’ll be learning about limits.
Steven Mills 