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# 5 - Contents CHAPTER 4 The Chain Rule Derivatives by the...

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CHAPTER 4 4.1 4.2 4.3 4.4 CHAPTER 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 CHAPTER 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 CHAPTER 7 7.1 7.2 7.3 7.4 7.5 CHAPTER 8 8.1 8.2 8.3 8.4 8.5 Contents The Chain Rule Derivatives by the Chain Rule Implicit Differentiation and Related Rates Inverse Functions and Their Derivatives Inverses of Trigonometric Functions Integrals The Idea of the Integral 177 Antiderivatives 182 Summation vs. Integration 187 Indefinite Integrals and Substitutions 195 The Definite Integral 201 Properties of the Integral and the Average Value 206 The Fundamental Theorem and Its Consequences 213 Numerical Integration 220 Exponentials and Logarithms An Overview 228 The Exponential ex 236 Growth and Decay in Science and Economics 242 Logarithms 252 Separable Equations Including the Logistic Equation 259 Powers Instead of Exponentials 267 Hyperbolic Functions 277 Techniques of Integration Integration by Parts Trigonometric Integrals Trigonometric Substitutions Partial Fractions Improper Integrals Applications of the Integral Areas and Volumes by Slices Length of a Plane Curve Area of a Surface of Revolution Probability and Calculus Masses and Moments 8.6 Force, Work, and Energy

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C H A P T E R Integrals 5.1 The Idea of the Integral This chapter is about the idea of integration, and also about the technique of integ- ration. We explain how it is done in principle, and then how it is done in practice. Integration is a problem of adding up infinitely many things, each of which is infini- tesimally small. Doing the addition is not recommended. The whole point of calculus is to offer a better way. The problem of integration is to find a limit of sums. The key is to work backward from a limit of differences (which is the derivative). We can integrate v(x) ifit turns up as the derivative of another function f(x). The integral of v = cos x is f = sin x. The integral of v = x is f = \$x2. Basically, f(x) is an "antiderivative". The list of j ' s will grow much longer (Section 5.4 is crucial). A selection is inside the cover of this book. If we don't find a suitablef(x), numerical integration can still give an excellent answer. I could go directly to the formulas for integrals, which allow you to compute areas under the most amazing curves. (Area is the clearest example of adding up infinitely many infinitely thin rectangles, so it always comes first. It is certainly not the only problem that integral calculus can solve.) But I am really unwilling just to write down formulas, and skip over all the ideas. Newton and Leibniz had an absolutely brilliant intuition, and there is no reason why we can't share it. They started with something simple. We will do the same. SUMS A N D DIFFERENCES Integrals and derivatives can be mostly explained by working (very briefly) with sums and differences. Instead of functions, we have n ordinary numbers. The key idea is nothing more than a basic fact of algebra. In the limit as n + co, it becomes the basic fact of calculus. The step of "going to the limit" is the essential difference between algebra and calculus! It has to be taken, in order to add up infinitely many infinitesimals-but we start out this side of it.
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