Course Notes for Mathematics 111: Calculus

(Copyright 2007, 2012 by Jerry Shurman. Any part of the material protected by this copyright notice may be reproduced in any form for any purpose without the permission of the copyright owner, but only the reasonable costs of reproduction may be charged. Reproduction for profit is prohibited.)

These are course notes for Mathematics 111 at Reed College. They are written for serious liberal arts students who want to understand calculus beyond memorizing formulas and procedures. The prerequisite is three years of high school mathematics, including algebra, euclidean geometry, analytic geometry, and (ideally) trigonometry. To profit from these notes, the student needn't be a math genius or possess large doses of the computational facilities that calculus courses often select for. But the student does need sufficient algebra skills, study habits, energy, and genuine interest to concentrate an investment in the material.

I have tried to put enough verbal exposition in these notes that at least portions of them will be readable outside of class. And I have tried to keep the calculations short, tidy, and lightly notated, in the hope of rendering them comprehensible stories that incur belief, rather than rituals to endure. To the extent that the notes are readable, my hope is to use classtime discussing their contents rather than conform to the model of the instructor transcribing a lecture onto the blackboard from which the students transcribe it into their notebooks in turn. The goal is that the students leave the course not having taken my word about the results, but believing truly viscerally that the results are inevitable.

Exigencies dictate that Math 111 simultaneously serve students who have taken a calculus course already and students who haven't. These notes attempt to do so in two ways,

  • by rebalancing the weight of explanation between mathematical symbols and natural language,
  • and by presenting the computations of calculus as little more than end-products of algebra that one could imagine naturally working out for oneself with some nudges in the right direction.
The presentation is meant to defamiliarize calculus for those who have seen it already, by undoing any impression of the subject as technology to use without understanding, while making calculus familiar to a wide range of readers, by which I mean comprehensible in its underlying mechanisms. Thus the notes will pose different challenges to students with prior calculus experience and to students without it. For students in the first group, the task is to consider the subject anew rather than fall back on invoking rote techniques. For students with no prior calculus, the task is to gain facility with the techniques as well as the ideas.

These notes address three subjects:

  • Integration. What is the area under a curve? More precisely, what is a procedure to calculate the area under a curve?
  • Differentiation. What is the tangent line to a curve? And again, whatever it is, how do we calculate it?
  • Approximation. What is a good polynomial approximation of a function, how do we calculate it, and what can we say about the accuracy of the approximation?
Part of the complication here is that "area under a curve" and "tangent line to a curve" are geometric notions, but we want to calculate them using analytic methods. Thus the interface between geometry and analysis needs discussion.

The basic pedagogy is to let ideas emerge from calculations. In succession, these notes define, integrate, and differentiate

  • the rational power function,
  • the logarithm function,
  • the exponential function,
  • the cosine and sine functions.
The integrals are computed without using the Fundamental Theorem of Calculus. Integrating the power function leads to the idea that an integral is not only an area, but more specifically an area that is well approximated from below and from above by suitable sums of box-areas. Although the geometrically natural idea is to integrate nonnegative-valued functions from a left endpoint to a right endpoint, the logarithm leads to the idea of integrating a function that could be negative between endpoints that need not be in order. The logarithm also illustrates the idea of defining a function as an integral and then studying its properties as such. Similarly, the exponential function illustrates the idea of defining and then studying a function as the inverse of another, and it suggests the idea of characterizing a function by a differential equation.

With the power function, the logarithm, the exponential, and the cosine and sine integrated and differentiated, we then find approximating polynomials for these functions and estimate the accuracy of the approximations.

Essentially all of the program just sketched can be carried out convincingly (if not ``fully rigorously") using only one small-but-versatile piece of technology, the finite geometric sum formula. This formula reduces many area calculations, limits of sums of many terms, to limits of quotients of two terms. In fancier language, the formula reduces integration to differentiation. This phenomenon is perhaps unsurprising since the Fundamental Theorem of Calculus says that integration and differentiation are closely related. But whereas the Fundamental Theorem is often taught as a procedure that circumvents computing integrals directly, a goal of these notes is to see differentiation emerge from integration explicitly and repeatedly. Students who learn to integrate only by using the Fundamental Theorem risk gaining no real appreciation for what integration really is, an appreciation worth having if only because the Fundamental Theorem is irrelevant to so much real-world integration.

Calculus does at some point require the technical machinery of limits. These will be treated lightly only after they are used informally. Cauchy's magnificent grammar deserves its due, but first working informally with specific examples is meant to help the reader tangibly appreciate its economy and finesse.

The last two chapters of these notes, on applications of the derivative and on the Fundamental Theorem of Calculus, are traditional. In the footsteps of so many before us, we will move ladders around corners, drain conical swimming pools, and generate blizzards of antiderivatives.

These notes are based on a set of notes by Ray Mayer. The motivation for creating a new set of notes was that when this project began, the other set of notes was not available in electronic form. That situation has now changed, and the reader of these notes is encouraged to look at Ray Mayer's notes as well.

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