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 and the integral, exponentials and logarithms are flip sides of the same coin: Each undoes the work of the other. Start with a number, take its exponential, and then take the logarithm of the result, and you will end up with your original number.
That original number is the base; to take an exponential, you multiply the base by itself x number of times. The number of times you multiply it by itself is the power, represented in superscript: for example, 10 multiplied by itself 5 times would be written as 105. When the base is 10, you can also think of the power as denoting the number of zeros to the right of the initial 1. So an exponential function would be something like 2x, or 5x, where the exponent is the variable. A power would be something like x 2, x 5, or x 3, where the base is the variable. It’s an important distinction.
Since taking a logarithm undoes the work of the exponential, in general, the logarithm is just the number of digits in that number. Just as with exponentials, if we’re dealing with a perfect power of 10, for example, the logarithm is the number of zeros to the right of the initial 1: log(10) = 1, log(100) = 2, log(1,000) = 3, and so on. Or, to put it as generally as possible, log(10x) = x. The only catch is that you can’t take the logarithm of a negative number: no such animal exists. The logarithm inverts the exponential, but you can’t get a negative number with exponentials.
THE PLOT THICKENS
Back at the start of my foray into calculus, my physicist spouse, Sean, would leave simple problems on our home whiteboard for me to solve, like little mathy love notes. (Yes, we have a whiteboard at home. Doesn’t everyone?) The first set of problems focused on learning how to plot the points generated by specific functions onto a Cartesian grid, then connecting the dots to see the shape of the resulting curve (or “face” of the function). I quickly figured out this was much easier to do in a handy program called Grapher: You just plug in different values for the variable(s) in a given function, hit Return, and the correct curve miraculously appears. (You can do the same thing in Excel.)
It’s fun to play with Grapher, but frankly, I found it just as instructive to slowly plot out a few functions by hand. Many of us have difficulty grasping the notion of just what a function is: The textbook definitions, while technically correct, usually convey little actual meaning to nonmathy sorts like me. Literally taking a given function apart, point by point, and slowly rebuilding it again can help bridge that gap in communication.
Let’s plot the function f(x) = ax2 onto a Cartesian grid with the familiar x and y axes. Remember that f(x) is just another way of writing y for calculus purposes; so we’re working with y = ax2. The process is simple, if tedious. Assuming that a = 1, all we are doing is plugging in different values for x to get the corresponding value for y and plotting the point where they intersect onto the grid. I found it helpful to write down those initial values into columns first.
We already know this will be a parabola. I chose whole numbers, both positive and negative, for simplicity’s sake, but you can plug in any value for x along the real number line: positive, negative, fractions, and so on. (If you don’t include negative values, you only get half the parabolic curve.) Remember that the function technically comprises all possible values for x in that equation taken together—i.e., an infinite number of values. That would be tedious to plot indeed. But you can plug in enough values along the number line, plot out the corresponding points on the grid, and at some point you accumulate enough points that a definite curvy pattern emerges when you connect the dots.
I’ve described curves as representing the “faces” of functions, but those faces can have multiple expressions. Someone who is happy, sad, or angry will have the same basic features, but their faces can look quite different depending on the emotions they are experiencing. The same is true of functions. For instance, the constant a in our equation determines the size and direction of the parabola. The larger the value of a, the steeper, or thinner, the resulting parabola will be. Also, if a is positive, the parabola opens upward; if a is negative, it opens downward. Where a = 2, we get a parabola that looks like the one on page 270.
Where a = −2, we get the exact same parabolic curve, only inverted (falling below the x axis) because the sign is now negative:
Finally, we can add additional variables: f(x) = ax2 + bx + c, also known as y = x2 + bx + c. It’s fun to play with the basic equation and see firsthand how changing each value for the different variables is reflected in the shape of the resulting curve. For instance, this is what you get when you plug in the values a = 3, b = 8, and c = 10:
It’s still the same basic function; the fundamental nature of its “face” hasn’t changed, it’s just expressing different “emotions.”
TOP TEN FUNCTIONS
While it’s useful to practice graphing a few functions by hand, certain functions crop up so frequently that it’s worth committing their “faces” (curves) to memory. The top ten most common functions are listed below. They should already be somewhat familiar, since you’ve encountered all but one (the logarithm) in the text.
For good measure, I’m also including their derivatives and integrals, because it’s important information for any beginning calculus student, and why do the work of crunching those numbers a
ll over again when past generations of mathematicians have done it for you? It will also help you to see the connection between the two in practice, namely, how the derivative undoes the work of the integral, and vice versa.
1. A Constant: f (x) = c
This is the function you’d use for the velocity of a car moving at a constant speed down a straight road, for example, as discussed in chapter 2.
Derivative:
The notation to the left of the equal sign tells us we are taking a derivative of the constant c. The answer is 0 because the derivative measures a rate of change. A constant, by definition, does not change, so the rate (and hence the derivative) is 0. Integral:∫ cdx = cx
Here, the notation to the left of the equal sign tells us we are taking an integral. Remember that the integral is the flip side of the derivative. If we take a derivative of the velocity to determine the acceleration of a car moving at a constant rate, then we take an integral of the velocity to determine how far we traveled between our starting point (a) and ending point (b). The c tells us that we are dealing with a constant, and the dx tells us we are taking an integral of the derivative of that constant.
If we were taking a definite integral, we would write this differently: = (b - a) c.The a and b variables at the top and bottom of the integral sign simply define the range over which we are taking the integral. On the right side of the equal sign, the notation simply tells us that we are subtracting our starting position (a) from our ending position (b) and multiplying by the constant c to determine how far we traveled.
2. A Straight Line: f (x) = ax + b
This is the function you’d use for the velocity of a car accelerating at a constant rate, for example, also discussed in chapter 2.
Derivative:
Integral:
3. A Parabola: f(x) = ax2
This function pops up all over the place in physics, whether we’re dealing with the trajectory of a cannonball, the acceleration of a falling apple, or our motion (changing position with respect to time) on the Tower of Terror free-fall ride in chapter 4.
Derivative:
Integral:
4. Exponential Growth Curve: f(x) = 10ax
We covered the basics of exponentials earlier. For an exponential function, we fix the base number and let the power to which it is raised be the variable: In this case, the base is 10 and the power is ax. This is the function we would use to describe the almost certain annihilation of the human race by voracious zombies in chapter 6 or the rapid growth rate of the Dutch tulip trade in chapter 5.
Derivative:
Integral:
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 math is all about solving problems creatively, not rote mechanics.
Finding the Limit. We’ll start with some handy tricks for finding the limit of a given function (assuming the limit exists; sometimes there is no limit). Trust me, this will come in handy when we get to derivatives. Earlier we looked at the function f(x) = 2x + 5, representing a straight line with a slope of 2 and a y-intercept of 5. The limit of f(x) as x approaches 3 equals 11. This just means that as we plug in values for x that are closer and closer to 3, the height of the graphed function gets closer and closer to y = 11 (aka the limit).
How do we know this? Well, it becomes fairly obvious if you plug in a series of values that get closer and closer to 3. For example, x = 2.9 gives a limit of 10.8, while x = 2.95 gives a limit of 10.9, and x = 2.99999 gives a limit of 10.99998. The closer the value of x is to 3, the closer the answer is to 11. Ergo, 11 is the limit of this particular function when x = 3.
But this is a tedious and time-consuming process that merely approximates the limit; we’d prefer to determine the limit precisely. The simplest strategy is called the substitution method: You just plug in the value of whatever number is specified under the “lim” notation. For example, let’s find the limit of a parabolic function, f(x) = x2, as x approaches Plug 2 into the equation, and we get 4. So
Similarly, to
find the answer to , make x = 4, so that 42 − 4 + 2 = 14. So
You can verify this by using the graph of the function f(x) = (x2 − x + 2): another parabola. Simply plug in a few values both above and below 4, and you should see the results come closer and closer to 4 as those values trend closer and closer to 4.
Alas, it is not always that simple. Sometimes when you substitute the number specified under the “lim” notation, you get a nonsensical result, such as a 0 in the denominator, which is a mathematical taboo. In that case, you could use the factoring method to simplify things a little. Let’s say we want the answer to. If we try to plug the value −3 into the equation, we end up with This is not helpful.
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 23