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MATH 221 { 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. The LATEX and Python les
modern proof of the Fundamental Theorem of Calculus was written in his Lessons Given at the cole Royale Polytechnique on the Infinitesimal Calculus in 1823. Cauchy's proof finally rigorously and elegantly united the two major branches of calculus (differential and integral) into one structure.
It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor’s Manual and a student Study Guide. The complete textbook (PDF) is also available as a single file. Highlights of Calculus.
The single most important tool used to evaluate integrals is called “The Fundamental Theo-rem of Calculus”. It converts any table of derivatives into a table of integrals and vice versa. Here it is. Theorem 1 (Fundamental Theorem of Calculus). Let f(x) be a function which is defined and continuous for a ≤ x ≤ b.
The Fundamental Theorem of Calculus. May 2, 2010. The fundamental theorem of calculus has two parts: Theorem (Part I). Let f be a continuous function on [a; b] and de ne a function g: [a; b] ! R by. Z x. g(x) := f: a. Then g is di erentiable on (a; b), and for every x 2 (a; b), g0(x) = f(x):
The essential point of calculus is to see this same pattern in “continuous time.” It’s not enough to look at the total or the change every hour or every minute. The distance and speed can be changing at every instant. In that case addition and subtraction are not enough. The central idea of calculus is continuous change.
The fundamental theorem dx. mx2. of calculus states that this generalizes: Theorem (Fundamental theorem of calculus, rst version). Let f(x) be an integrable func-tion on the interval [a; b], and. y F (y) = f(x) dx. d 0 for any y between a and b. Then dyF (y) = f(y), i.e. F = f.
