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# 6 - 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 6 Exponentials and Logarithms This chapter is devoted to exponentials like 2" and 10" and above all ex. The goal is to understand them, differentiate them, integrate them, solve equations with them, and invert them (to reach the logarithm). The overwhelming importance of ex makes this a crucial chapter in pure and applied mathematics. In the traditional order of calculus books, ex waits until other applications of the . integral are complete. I would like to explain why it is placed earlier here. I believe that the equation dyldx = y has to be emphasized above techniques of integration. The laws of nature are expressed by drflerential equations, and at the center is ex. Its applications are to life sciences and physical sciences and economics and engineering (and more-wherever change is influenced by the present state). The model produces a differential equation and I want to show what calculus can do. The key is always bm+" = (bm)(b3. Section 6.1 applies that rule in three ways: 1. to understand the logarithm as the exponent; 2. to draw graphs on ordinary and semilog and log-log paper; 3. to find derivatives. The slope of b" will use bX+*" = (bx)(bh"). h 6.1 An Overview There is a good chance you have met logarithms. They turn multiplication into addition, which is a lot simpler. They are the basis for slide rules (not so important) and for graphs on log paper (very important). Logarithms are mirror images of exponentials-and those I know you have met. Start with exponentials. The numbers 10 and lo2 and lo3 are basic to the decimal system. For completeness I also include lo0, which is "ten to the zeroth power" or 1. The logarithms of those numbers are the exponents. The logarithms of 1 and 10 and 100 and 1000 are 0 and 1 and 2 and 3. These are logarithms "to base 10," because the powers are powers of 10.
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