The Calculus Diaries - Jennifer Ouellette [102]
Function. The notation for a function is f(x). Whenever you see this at the start of an equation, you know you’re dealing with a function of some kind: For example, f(x) = x 2 tells you that x 2 is a function. However, just as it’s possible to convey the same meaning using different words, there can be more than one way to write an equation for a function. The function above can be written more generally as f(x) = ax 2, with a denoting “some constant.” It is also common to write f(x) simply as y. In that notation, x is the “independent variable” (it can be anything) and y is the “dependent variable” (it depends on x). This is important to remember when plotting points on a Cartesian grid (see page 268).
The above function describes a parabola. The most general notation is f(x) = ax 2 + bx + c, where x is the independent variable, y is the dependent variable, and a, b, and c are constants. There is also the so-called vertex form: f(x) = a(x − h)2 + k. The vertex of the parabola is the point where it turns, and in this format, (h, k) delineates that point.
Even though these functions seem at first glance to be different from one another, they actually all describe the same thing: a parabola. This variation can be confusing for the beginning calculus student. I found it helpful to view the different formulations for a function as synonyms: different words that describe the same thing. The shifts in the structure are akin to shifting around clauses, subjects, and predicates of sentences in grammar—there are specific rules that kick in whenever you “reword” an equation, just as there are rules of grammar for reworking the structure of a sentence. The overall meaning conveyed remains the same. The true test of mathematical fluency is the ability to see past the symbolic clutter and find the essence of a given equation. That’s why simply memorizing formulas won’t suffice; you have to know what they mean.
Limit. We discussed the concept of the limit in chapter 1. Per Kelley (aka Idiot Guide Extraordinaire), “A limit is the height a function intends to reach [on a graph] at a given x value, whether or not it actually reaches it.” For instance, the limit of f(x) = 2x + 5 as x approaches 3 is 11. In math-ese, that sentence would be rendered thus: . The 3 is the value of x that we are approaching, f(x) represents the function of interest, and 11 is the limit. In this case, the limit is simply the value of the function, but other cases are more subtle.
Sometimes the limit does not exist, most notably when a function at a given value for x does not approach a fixed number, but instead increases or decreases infinitely. The textbook example of this is the function f(x) = sin when x = 0. No general limit exists in that case because the function wriggles back and forth on the graph (see page 266) and never settles on a definite numeric value. Then we can just say that the limit as x approaches 0 does not exist.
Derivative. The common notation for a derivative is . Derivatives arise from ratios, or the difference between two dx points. The top value is the change in position, say, at two different times, while the bottom value is the difference in the time. If you want to take the derivative of f(x) = ax 2, you would write it out like this: .
Integral. The integral is represented by a long S-shaped figure: ∫ .
A handy mnemonic device is to remember that integration is a process of summing (S), hence the elongated S shape is its symbol. Often, when taking an actual integral, there will be numerical values at the top and bottom of the symbol indicating the range over which one is integrating: .
This is known as a definite integral; if there is no specified range, that is called an indefinite integral. If you wanted to undo the work of the derivative on f(x) = ax 2, you would take an integral and write it like this: ∫ax2 dx = .
Exponentials and Logarithms. It’s worth including a short note about exponentials and logarithms, which play an important role in calculus. Like the derivative