1 2 3 4 Light and Matter Fullerton, California www.com copyright 2005 Benjamin Crowell rev. April 26, 2008 This book is licensed under the Creative Com- mons Attribution-ShareAlike license, version 1.org/licenses/by-sa/1.0/, except for those photographs and drawings of which I am not the author, as listed in the photo credits. If you agree to the license, it grants you certain privileges that you would not otherwise have, such as the right to copy the book, or download the digital version free of charge from www. At your option, you may also copy this book under the GNU Free Documentation License version 1.org/licenses/fdl.txt, with no invariant sections, no front-cover texts, and no back-cover texts.
5 1 Rates of Change 3.1 Change in discrete steps 9 Probability, 61. Two sides of the same coin, Problems.—Properties 4 Techniques of the derivative, 15. 69 Higher-order polynomials, 4.—The second derivative, 4.—Maxima and minima, 4.4 Methods of integration. Change of variable, 76.
21 Integration by parts, 78. 2 To infinity — and Problems.2 Safe use of infinitesimals 26 techniques 2.3 The product rule .1 Review of complex 2.4 The chain rule .3 Partial fractions revisited 90 The exponential, 34.7 Differentiation on a 6 Improper integrals computer .1 Integrating a function that 2.2 Limits of integration at L’Hôpital’s rule, 45. 94 Another perspective on inde- Problems.—Limits at infinity, 48. 7 Iterated integrals Problems.1 Integrals inside integrals 97 7.1 Definite and indefinite 7.4 Spherical and cylindrical integrals .2 The fundamental theorem Problems.3 Properties of the integral 59 A Detours 107 6 B Answers and solutions area, and volume, 135.— 115 Trigonometry with a right triangle, 135.—Trigonometry C Photo Credits 133 with any triangle, 135.
136 D Reference 135 Rules for differentiation, D.—Integral calculus, Algebra, 135.—Table of integrals, 136. 7 Preface thrust of the topic. These details I’ve relegated to a chapter in the Calculus isn’t a hard subject. back of the book, and the reader Algebra is hard.
I still remem- who has an interest in mathemat- ber my encounter with algebra. It ics as a career — or who enjoys a was my first taste of abstraction in nice heavy pot roast before moving mathematics, and it gave me quite on to dessert — will want to read a few black eyes and bloody noses. those details when the main text suggests the possibility of a detour. Geometry is hard.
For most peo- ple, geometry is the first time they have to do proofs using formal, ax- iomatic reasoning. I teach physics for a living. Physics is hard. There’s a reason that peo- ple believed Aristotle’s bogus ver- sion of physics for centuries: it’s because the real laws of physics are counterintuitive.
Calculus, on the other hand, is a very straightforward subject that rewards intuition, and can be eas- ily visualized. Silvanus Thompson, author of one of the most popular calculus texts ever written, opined that “considering how many fools can calculate, it is surprising that it should be thought either a diffi- cult or a tedious task for any other fool to master the same tricks.” Since I don’t teach calculus, I can’t require anyone to read this book. For that reason, I’ve written it so that you can go through it and get to the dessert course with- out having to eat too many Brus- sels sprouts and Lima beans along the way. The development of any mathematical subject involves a large number of boring details that have little to do with the main 8 1 Rates of Change 1.1 Change in discrete steps Toward the end of the eighteenth century, a German elementary school teacher decided to keep his pupils busy by assigning them a long, boring arithmetic problem.
To oversimplify a little bit (which is what textbook authors always b / A trick for finding the sum. do when they tell you about his- tory), I’ll say that the assignment ing the area of the shaded region. was to add up all the numbers Roughly half the square is shaded from one to a hundred. The chil- in, so if we want only an approxi- dren set to work on their slates, mate solution, we can simply cal- and the teacher lit his pipe, con- culate 72 /2 = 24.
fident of a long break. But al- most immediately, a boy named But, as suggested in figure b, it’s Carl Friedrich Gauss brought up not much more work to get an ex- his answer: 5,050. There are seven saw- teeth sticking out out above the di- agonal, with a total area of 7/2, so the total shaded area is (72 + 7)/2 = 28. In general, the sum of the first n numbers will be (n2 + n)/2, which explains Gauss’s re- sult: (1002 + 100)/2 = 5, 050.
Two sides of the same coin a / Adding the numbers from 1 to 7. Problems like this come up fre- quently. Imagine that each house- Figure a suggests one way of solv- hold in a certain small town sends ing this type of problem. The a total of one ton of garbage to the filled-in columns of the graph rep- dump every year.
Over time, the resent the numbers from 1 to 7, garbage accumulates in the dump, and adding them up means find- taking up more and more space. RATES OF CHANGE rate of change accumulated result 13 13n n (n2 + n)/2 The rate of change of the function x can be notated as ẋ. Given the function ẋ, we can always deter- mine the function x for any value of n by doing a running sum. Likewise, if we know x, we can de- termine ẋ by subtraction.
In the c / Carl Friedrich Gauss example where x = 13n, we can (1777-1855), a long time find ẋ = x(n) − x(n − 1) = 13n − after graduating from ele- mentary school. Or if we knew that the accumulated amount of 2 Let’s label the years as n = 1, 2, garbage was given by (n + n)/2, 3, ., and let the function1 x(n) we could calculate the town’s pop- represent the amount of garbage ulation like this: that has accumulated by the end of year n. If the population is 2 2 constant, say 13 households, then n + n (n − 1) + (n − 1) − garbage accumulates at a constant 2 2 rate, and we have x(n) = 13n. n2 + n − n2 + 2n − 1 − n + 1 = But maybe the town’s population 2 is growing.
If the population starts = n out as 1 household in year 1, and then grows to 2 in year 2, and so on, then we have the same kind of problem that the young Gauss solved. After 100 years, the accu- mulated amount of garbage will be 5,050 tons. The pile of refuse grows more and more every year; the rate of change of x is not constant. Tab- ulating the examples we’ve done so far, we have this: 1 Recall that when x is a function, the notation x(n) means the output of the d / ẋ is the slope of x.
function when the input is n. It doesn’t represent multiplication of a number x by a number n. The graphical interpretation of 1. CHANGE IN DISCRETE STEPS 11 this is shown in figure d: on a of n.
graph of x = (n2 + n)/2, the slope of the line connecting two succes- sive points is the value of the func- Some guesses tion ẋ. Even though we lack Gauss’s ge- In other words, the functions x and nius, we can recognize certain pat- ẋ are like different sides of the same terns. One pattern is that if ẋ is a coin. If you know one, you can find function that gets bigger and big- the other — with two caveats.
ger, it seems like x will be a func- tion that grows even faster than First, we’ve been assuming im- ẋ. In the example of ẋ = n and plicitly that the function x starts x = (n2 +n)/2, consider what hap- out at x(0) = 0. That might pens for a large value of n, like not be true in general. At this value of n, ẋ = 100, stance, if we’re adding water to a which is pretty big, but even with- reservoir over a certain period of out pawing around for a calculator, time, the reservoir probably didn’t we know that x is going to turn out start out completely empty.
Thus, really really big. Since n is large, if we know ẋ, we can’t find out n2 is quite a bit bigger than n, so everything about x without some roughly speaking, we can approxi- further information: the starting mate x ≈ n2 /2 = 5, 000. 100 may value of x. If someone tells you be a big number, but 5,000 is a lot ẋ = 13, you can’t conclude x = bigger.
Continuing in this way, for 13n, but only x = 13n + c, where c n = 1000 we have ẋ = 1000, but is some constant. There’s no such x ≈ 500, 000 — now x has far out- ambiguity if you’re going the op- stripped ẋ. This can be a fun game posite way, from x to ẋ. Even to play with a calculator: look at if x(0) 6= 0, we still have ẋ = which functions grow the fastest.
For instance, your calculator might have an x2 button, an ex button, Second, it may be difficult, or even and a button for x! (the factorial impossible, to find a formula for function, defined as x! = 1·2·.·x, the answer when we want to de- e. You’ll termine the running sum x given find that 502 is pretty big, but e50 a formula for the rate of change ẋ. is incomparably greater, and 50! is Gauss had a flash of insight that so big that it causes an error. led him to the result (n2 + n)/2, but in general we might only be All the x and ẋ functions we’ve able to use a computer spreadsheet seen so far have been polynomials.
to calculate a number for the run- If x is a polynomial, then of course ning sum, rather than an equation we can find a polynomial for ẋ as that would be valid for all values well, because if x is a polynomial, 12 CHAPTER 1. RATES OF CHANGE then x(n)−x(n−1) will be one too. It also looks like every polynomial we could choose for ẋ might also correspond to an x that’s a poly- nomial. And not only that, but it looks as though there’s a pattern in the power of n.
Suppose x is a polynomial, and the highest power of n it contains is a certain num- ber — the “order” of the polyno- mial. Then ẋ is a polynomial of that order minus one. Again, it’s e / Isaac Newton (1643- fairly easy to prove this going one 1727) way, passing from x to ẋ, but more difficult to prove the opposite rela- into a reservoir is smooth and con- tionship: that if ẋ is a polynomial tinuous. Or is it? Water is made of a certain order, then x must be out of molecules, after all.
It’s just a polynomial with an order that’s that water molecules are so small greater by one. that we don’t notice them as in- dividuals. Figure f shows a graph We’d imagine, then, that the run- that is discrete, but almost ap- ning sum of ẋ = n2 would be a pears continuous because the scale polynomial of order 3. If we cal- has been chosen so that the points culate x(100) = 12 + 22 +.
+ blend together visually.