CALCULUS AND PIZZA
Cliff Pickover
Copyright 1999, Cliff Pickover, www.pickover.com. All rights reserved.
Illustration by Brian Mansfield.
Motivation:
Would Calculus and Pizza sell? There are several other calculus
books on the market that aim to teach calculus in a "friendly" manner.
For example, Silvanus Thompson's Calculus Made Easy (updated by
Martin Gardner) has sold over a million copies, which suggests the potential
market for this kind of book. In fact, Calculus Made Easy book has
a better sales rank at Amazon.com then most other famous and best-selling
math and science books: James Gleick's Chaos, Michael Waldrop's
Complexity, John Paulos's Innumeracy, Edwin Abbott's Flatland,
Edward O. Wilson's Consilience, Steven J. Gould's The Mismeasure
of Man, etc.
Calculus and Pizza would have appeal outside the U.S. In Germany,
40% of all high school students take calculus -- in Japan, 90%. United
States enrollment in high school calculus stands at only 4%.
Is there a need to present calculus in an upbeat, fresh manner? Students
continually struggle with calculus, and at some U.S. schools, close to
a third of those who take calculus fail it. Student's often don't read
their big, fat, boring calculus texts. They use them only for problems
and exercises. Calculus and Pizza will be humorous and irreverent,
making it much easier to get through then other calculus books. When someone
reads it, they get sucked right in, and before they know it they've read
three chapters and learned a hell of a lot of calculus.
The ability of tomorrow's workforce to meet the demand for scientists,
engineers, technicians, and mathematicians depends on the quality and type
of education today's children receive. Recent statistics indicate that
tomorrow's work force is in grave danger of failing to meet the challenge.
In the early 1990s, U. S. students ranked 15th of 17 countries in science
scores.
The August Notices of the American Mathematics Society lists
over 500,000 first year college students taking calculus class. If we include
high-school students, there are probably around a million students taking
calculus in the U.S. alone.
CALCULUS AND PIZZA
Introduction
Hello. My name is Luigi. I own a pizza shop in New York City. If you are
reading this book, you can probably guess my two main passions: calculus
and pizza. I love pizza for its taste and its Neapolitan origin. I too
am from Naples. I love calculus because it is an intellectual triumph and,
like pizza, can be appreciated by devouring a small slice at a time.
Many students and teachers use fat calculus books. I have a fat stomach,
but I don't like my books fat. I like my books lean. Do you like your books
lean?
Chapter 1. Instantaneous Velocity and the Derivative
I throw my pizza dough high into the air. My arms are strong. My ceiling
is high. Once the pizza is near the ceiling, it falls for many seconds
before landing in my arms. I plan to add pepperonis to the pizza, but don't
worry about them. They have nothing to do with the problem. They just taste
good.
Let's look at the pizza falling from near the ceiling. Suppose the distance
that the pizza travels after t seconds is given by the formula
inches (as a function of time) = f(t) = 3t2.
After two seconds, the pizza has traveled f(2) = 3(2)2 =
12 inches. After three seconds, the pizza has traveled 27 inches. After
four seconds, the pizza has traveled 48 inches, and so forth. The pizza
is speeding up! I had better not miss catching it.
How fast is it going just as I catch it? Can we find a formula that
describes the pizza's velocity at every instant? It's a tricky question,
but that's what this book is all about. We can calculate the speed using
differential calculus
It seems almost mystical. The tricky thing about computing the velocity
at an instant is that the distance traveled by the pizza and the elapsed
time are both zero. That's what we mean by the word "instant." To find
the velocity of the pizza, we can't just divide distance traveled by the
time it took to travel. What we want to do is to compute the average velocity
over shorter and shorter time periods of length delta t. The symbol
delta is the Greek letter delta. Mathematicians like to use
it to symbolize "the change in." So, delta t is the change
in time.
Let's try to calculate the distance traveled by the pizza between time
t and (t + delta t), which is just a short
while later as indicated in the figure. We need to subtract the distances
the pizza traveled between the two times: f(t + delta
t) - f(t). Next, use the formula for distance f(t)
= 3t**2 to get 3(t + delta t)2 -
3t2. With a little shuffling, this becomes 3(t2
+ 2*delta t + delta t2 - t2).
We can simplify this formula with a little trick if you know that (a+b)2
= a2 + 2ab + b2. This gives
us the distance traveled by pizza of
3(2t * delta t + delta t2).
Are you with me so far?
If we divide this distance by delta t , which is the time needed
to travel the distance, we see that the average velocity over this period
of time is 3(2t + delta t). Now for the fun part.
As delta t gets smaller and approaches zero, we approach the instantaneous
velocity 3(2t) = 6t. This means I can tell you the speed
of my pizza at any instant in time very easily. If it hits my hands after
5 seconds falling from the ceiling, the pizza is traveling at 30 inches
per second. (And you thought it was easy to catch a falling pie?)
I hope I have made clear that from the formula for inches, f(t)=3t2,
we can obtain another formula for the speed f'(t)=6*t.
This new function f' is called the derivative of the first function
f. Finding derivatives is half the game of calculus. (Finding "integrals"
is the other half.)
Generalizing Falling Pizza Formulas
(Theory to Impress Friends and Families)
Let's be slightly more general and theoretical, so we can impress our friends
and families. For falling pizzas, we already know how to calculate the
speed function from the inches function. In general, we can determine a
speed function from an inches function as follows.
The derivative f'(t) of a function f(t)
is the function defined by the following ratio:
f(t + delta t) - f(t)
--------------------
delta t
and finding what limiting value you get as delta t approaches
zero:
f'(t) = lim(delta t -- >0) [f(t+
delta t) - f(t)] / delta t
This is the theoretical definition of the derivative. The term "lim(delta
t -- >0)" means that we want the value of this fraction as the time
interval gets close to zero. If f represents the distance traveled,
f' represents speed. If f is any function, f' gives
its rate of change. For example, if f gives the number of pizzas
eaten in my shop through time, f' gives the rate of pizza eating,
which might be 5 pizzas per minute on one of those hot Saturday nights.
Cookbook synopsis about what Luigi did to calculate instantaneous speed.
Other applications of calculus to pizza: heating and cooling, infinite
series by cutting anchovies...
Various exercises. Answers to odd numbered exercises in back.