Practical Calculus

(which might horrify your math teacher)

You don't need to know the exact definitions of the terms used in calculus if you want to see how they are applied to the real world, because in real life nothing is known exactly (unless it comes in discrete values, like the number of pages in a book, in which case calculus does not apply to it exactly anyway). That is, when we measure a 2 foot length, we might know that it is 2.0 ft or 2.00 ft, but there is always a limit on the precision. Later if somebody comes along with a better measuring tool and declares it to be 2.003 ft, this does not contradict the earlier measurement, it just improves the precision.

In applying the equations of physics, we should keep the above limitations in mind. For example if x = yz, even if this equation is exact (and of course we will never know if we have an equation that describes the real world exactly), we are no better off than saying that x is approximately yz, since we cannot measure exactly.

You have, no doubt, run into D meaning "change of." In calculus, d means very small D . So dx is a very small change of x. A mathematician is concerned with the exact definition of dy/dx. In practical applications, we can regard dy/dx as the ratio of two very small quantities and not worry about being exact, because there is no point to it.

The derivative: If y is a function of x, for example y = x³, then the derivative of y with respect to x is dy/dx, the slope of the graph of y vs x, with x on the horizontal axis. For this case, if you want dy/dx at x = 2, you could calculate dy as 2.01³ - 1.99³ for a dx of 0.02. Divide, and you will find dy/dx = 12. (The calculator says 12.0001. The "exact" value is 12, but in practice it doesn't matter.) This brings up the question, how small is very small. Well, that depends on how much precision is needed. Let's try dy = 2.5³ - 1.5³ divided by a dx of 1. We get dy/dx = 12.25, close enough to 12 for some purposes but not for others. Make life easier: instead of going a little above 2 to a little below 2, do a little above down to 2: y = x³ at x=2à dy/dx = (2.0001³-2³)/0.0001 = 12.000596 ≈ 12.

You might as well learn the exponent rule:
if y = x³, dy/dx = 3x², in other words the original exponent times x to the power reduced by one. In our example,
when x=2, 3x² = 12. If y = 6x^1.5, then dy/dx = (1.5)(6)x^0.5. It works for any exponent. Note that the 6 is left unchanged. Test this rule with a calculator for a few cases (as was done in the paragraph above), and you will get the hang of it. If you want the reason behind the exponent rule, (a+b)ⁿ = aⁿ + na^n-1b + n(n-1)a^n-2b²/2 …. All we need are the first two terms if b is really tiny. Now consider y = xⁿ. dy = [(x+dx)ⁿ-xⁿ]/dx
= [xⁿ + nx^n-1dx…-xⁿ]/dx = nx^n-1. If the terms left out bother you, make dx =10^-20 or something and experiment with it. In math they are more theoretical, but the outcome is the same: those terms disappear.

Two other formulas that you might as well learn: the derivative of the sine is the cosine and the derivative of the cosine is -sine. To be more precise and more general, if kx is in radians and
y = Asin(kx) then dy/dx = Akcos(kx). Similarly
if y=Acos(kx) then dy/dx=-Aksin(kx). If kx is not in radians, convert it to radians before finding the derivative.

The second derivative: This is simply the derivative of a derivative. If y = x³, dy/dx = 3x², as we have seen, and the derivative of 3x² is 6x. Rather than the awkward symbol
d(dy/dx)/dx, we label the thing d²y/dx². The slope of the y vs x graph is dy/dx, and the change of slope divided by the change of x is d²y/dx². Similarly if y is a function of t, dy/dt (the slope of the y vs t graph) is the velocity and d²y/dt² is the rate of change of v: the acceleration.

Partial derivatives: When y is a function of two or more variables, a partial derivative is the same as a derivative but with the other variables held constant. For example if y = ax³ +bxt + ct², with a, b and c constant, then the partial derivative of y with respect to x,
labeled ¶y/¶x, is 3ax² + bt, and
¶y/¶t is bx + 2ct.

Partials are useful for waves.
y = Asin(kx - w t) is a plane wave traveling in the +x direction with speed w/k. For waves on a string, sound, light, and others, we find that
¶ ²y/¶ x² is proportional to ¶ ²y/¶ t², and from this we find the speed of the wave.
If y = Asin(kx-wt),
then ¶ ²y/¶ x²=-Ak²sin(kx-w t) and
¶ ²y/¶ t² =-Aw ²sin(kx-w t), so for example when Newton's 2^nd law tells us that a string with tension F and mass/length m has
F¶ ²y/¶ x² = m ¶ ²y/¶ t² , we easily show that the wave speed is (F/m )^1/2. This is a powerful tool.

The integral: This is just a sum. ò x²dx, for example, is a bunch of x² values, each multiplied by a very small dx, then all added together. To show specifically how to do it, let's integrate the above integral from x=1 to x=2 (these are called the limits) on a calculator, using dx = 0.1. The 1 to 2 interval is split up into 10 parts, 1 to 1.1, 1.1 to 1.2,…,1.9 to 2. Using the x² value at the center of each interval, (1.05², 1.15²,…) calculate the sum of x²dx and you will find that it is 2.3325. (The exact value of _1.ò ² x²dx is 7/3 = 2.33333… so there is a very small error.) On a computer, using a spreadsheet or a programming language, you could divide it into 1000 parts and use dx = 0.001. You will then find that the integral is 2.33333…. The moral of the story is that the smaller the dx, the smaller the error. (By the way, if you tell your math teacher about dx = 0.1 or 0.001, be prepared to call 911. He or she might have a heart attack, or (s)he might inflict bodily injury on you.)

In math-speak, f(x) means function of x: for any x, there is some value of f(x). The sketch below shows a graph of y = some function of x, and when you integrate from A to B, you add up the areas of those vertical strips to get the area bounded by the function on top, the x axis on the bottom, the vertical line x=A on the left, and the vertical line x=B on the right.

To see what the integral is theoretically, let u(x) = the area under the curve starting at some point to the left of A and ending at any x; then clearly, the area that we want, the integral from A to B is u(B) - u(A). So we need to find u(x). To do this, note that as you add the area terms, your total increases by ydx each time, so du = ydx, or du/dx = y. So this is the opposite of the derivative: if y = x², for example, increase the exponent by one and divide by the new exponent to get x³/3. In general u(x) = x³/3 + C, (C= a constant), but when we calculate u(A) - u(B), the constant drops out, so forget about it. The integral of x²dx from x=1 to x=2 is 2³/3 - 1³/3 = 7/3.

Here is an important integral:
ò d(cabin)/cabin = natural log cabin.
(In other words, ò dx/x = lnx.)

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