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