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AP Calculus

A Sampler of Visual Proofs

in First-year CalculusAP Calculus Professional Night

June 2007

Roger B. Nelsen

Lewis & Clark College

Portland, Oregon

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8/9/2019 A Sampler of Visual Proofs in Calculus I

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Introduction

Although neither the noun proof nor the verb prove appears in the College Boards

Course Description for Calculus AB and BC [6], Im sure that theorems and their proofs

are an integral part of virtually every calculus course, AP

or otherwise. Hence I will not

attempt to convince this audience of the necessary and proper role of proof in calculussee, for example the articles Seeing is Believing by J. T. Sutcliffe [16], Why We Use

Theorem in Calculus by Lisa Townsley [17], and Some Thoughts on 2003 Calculus AB

Question 6 by Jim Hartman and Larry Riddle [7].

So, what are the characteristics of a goodproof? I agree with Yuri Manin, who wrote A

good proof is one that makes us wiser [15], and Andrew Gleason: Proofs really arentthere to convince you that something is truetheyre there to show you why it is true

[15].

A good visual proof is a picture or diagram that helps one see why a particular mathemat-

ical statement is true, and also to see how one might begin to prove it true.

Of course, there are lots of bogus visual proofs around, too. Heres one from the April2007 issue ofMath Horizons [2].

The areaA of a triangle equalsA 1. Proof:

(Sometimes I wish this were true, since it would imply that all positive integers are equal.

That would certainly make my task of grading exams much easier!)In this talk, I will discuss several topics from first-year calculus where Ive found visualproofs useful.

Better approximations to e

Although its not explicitly on the AP syllabus, virtually every calculus text (and pre-

sumably every calculus course) has the following limit expression fore:

limn

1 +1

n

n

= e , (1)

and consequently the approximation for large n,

1 + 1n

n e .

This approximation can be easily improved. But first, lets establish (1) using Napiers

inequality, for which we will give two visual proofs [12]:

0 < a < b 1

b