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You’ll notice that there is some new notation here: log e. This means the logarithm of Euler’s constant (e). I didn’t discuss Euler’s constant specifically in the text, despite its importance, because, frankly, it muddies the waters of comprehension for those dipping a toe into calculus for the first time. It is an irrational number, like π, which means it goes on forever when written out in explicit form: e = 2.71828 . . . That’s why it is usually just left as e in an equation. The logarithm of e, in case you’re wondering, is 0.43429 . . .

5. Exponential Decay Curve: f(x) = 10−ax

This is another function that pops up frequently in physics, describing the rate at which a cup of coffee cools, for example, or the rate at which our sodden clothes dry out after being drenched on Splash Mountain in chapter 4. It’s exactly the same as the exponential growth curve, but the power to which the base is raised is negative.

Derivative:

Integral:

Note that the derivative and integral of the exponential decay curve also are virtually identical to that of the exponential growth curve, except for the minus sign in the power.

6. Logarithm: f(x) = log(ax)

We didn’t discuss the logarithmic function specifically in the main text, but this is what physicists often use to determine the entropy (disorder) of a physical system, such as a box filled with gas, a black hole, or Carnot’s heat engine in chapter 7. Note that because there is no such thing as a logarithm for a negative number, the curve is not defined for negative values of x. Instead, as x approaches 0 moving from the right, the logarithm goes to minus infinity.

Derivative:

Integral:

∫ log ax dx = x log( ax ) − x + c

7. Sine: f(x) = sin(ax)

This is an example of a periodic function: one whose values repeat over and over, at the same rate and at the same intervals in time. That interval is called the period. We encountered sine waves, or sinusoid curves, in chapter 9 while talking about ocean waves, but the concept can apply to any wavelike phenomenon (light waves, sound waves, gravitational waves) or any process that repeats itself after a fixed period of time (the ticking of a clock, a human heartbeat, the rising of the sun every twenty-four hours).

Derivative:

Integral:

8. Cosine: f(x) = cos(ax)

The cosine is the complement to the sine function, and is also an example of a sinusoid curve, applying to wavelike behavior.

Derivative:

Integral:

9. Catenary (or Hyperbolic Cosine):

This is the curve we discussed in chapter 8 that when inverted describes the strongest possible shape for an arch. Here we encounter Euler’s constant again, this time as the function ex. Like other irrational numbers, e has some unusual properties. For instance, the function e x is the only function—other than f(x) = 0—that is equal both to its own derivative and to its own integral. You can see this clearly in the notation below.

Derivative:

Integral:

10. Bell Curve (Gaussian Distribution): f(x) = ae− x2 .

This is perhaps the function best known to the general populace, albeit one that is often misunderstood. We encountered it in chapter 3 when discussing the probabilities of craps, but it is applicable to almost any situation involving a large number of random variables, such as the Black-Scholes model used in economics for options pricing, among other applications. It is also useful for calculating the probability of a given characteristic in a large population and for determining SAT scores or academic grades (known as “grading on a curve”).

Derivative:

Integral: There is no known integral for the Bell curve. It can be calculated on a computer but not written in an explicit form.

WORKING IT OUT

Now it’s time to put all the pieces together and see how calculus really works. These are simple examples that can be done with pencil and paper, but it’s worth investing in a scientific calculator if you’re planning to delve deeper into calculus. Let the machines do the tedious task of number crunching; real