7 - 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 242 252 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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CHAPTER 7 Techniques of Integration Chapter 5 introduced the integral as a limit of sums. The calculation of areas was started-by hand or computer. Chapter 6 opened a different door. Its new functions ex and In x led to differential equations. You might say that all along we have been solving the special differential equation dfldx = v(x). The solution is f = 1 v(x)dx. But the step to dyldx = cy was a breakthrough. The truth is that we are able to do remarkable things. Mathematics has a language, and you are learning to speak it. A short time ago the symbols dyldx and J'v(x)dx were a mystery. (My own class was not too sure about v(x) itself-the symbol for a function.) It is easy to forget how far we have come, in looking ahead to what is next. I do want to look ahead. For integrals there are two steps to take-more functions and more applications. By using mathematics we make it live. The applications are most complete when we know the integral. This short chapter will widen (very much) the range of functions we can integrate. A computer with symbolic algebra widens it more. Up to now, integration depended on recognizing derivatives. If v(x) = sec2x then f(x) = tan x. To integrate tan x we use a substitution:, I!&dx.= -1"- - - In u = - In cos x. U What we need now ,are techniques for other integrals, to change them around until we can attack them. Two examples are j x cos x dx and 5 ,/- dx, which are not immediately recognizable. With integration by parts, and a new substitution, they become simple. Those examples indicate where this chapter starts and stops. With reasonable effort (and the help of tables, which is fair) you can integrate important functions. With intense effort you could integrate even more functions. In older books that extra exertion was made-it tended to dominate the course. They had integrals like which we could work on if we had to.
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This note was uploaded on 06/13/2009 for the course TAM 455 taught by Professor Petrina during the Fall '08 term at Cornell University (Engineering School).

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7 - Contents CHAPTER 4 The Chain Rule Derivatives by the...

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