tailieunhanh - Ebook Calculus for business, economics, and the social and life sciences (10th edition): Part 2

(BQ) Part 2 book "Calculus for business, economics, and the social and life sciences" has contents: Integration, additional topics in integration, calculus of several variables. Please refer to the content | 11/17/08 4:46 PM Page 371 User-S198 201:MHDQ082:mhhof10%0:hof10ch05: CHAPTER 5 Computing area under a curve, like the area of the region spanned by the scaffolding under the roller coaster track, is an application of integration. INTEGRATION 1 Antidifferentiation: The Indefinite Integral 2 Integration by Substitution 3 The Definite Integral and the Fundamental Theorem of Calculus 4 Applying Definite Integration: Area Between Curves and Average Value 5 Additional Applications to Business and Economics 6 Additional Applications to the Life and Social Sciences Chapter Summary Important Terms, Symbols, and Formulas Checkup for Chapter 5 Review Exercises Explore! Update Think About It 371 11/17/08 4:46 PM Page 372 User-S198 201:MHDQ082:mhhof10%0:hof10ch05: 372 CHAPTER 5 Integration 5-2 SECTION Antidifferentiation: The Indefinite Integral How can a known rate of inflation be used to determine future prices? What is the velocity of an object moving along a straight line with known acceleration? How can knowing the rate at which a population is changing be used to predict future population levels? In all these situations, the derivative (rate of change) of a quantity is known and the quantity itself is required. Here is the terminology we will use in connection with obtaining a function from its derivative. Antidifferentiation ■ A function F(x) is said to be an antiderivative of f (x) if FЈ(x) ϭ f(x) for every x in the domain of f(x). The process of finding antiderivatives is called antidifferentiation or indefinite integration. NOTE Sometimes we write the equation FЈ(x) ϭ f(x) as dF ϭ f (x) dx ■ Later in this section, you will learn techniques you can use to find antiderivatives. Once you have found what you believe to be an antiderivative of a function, you can always check your answer by differentiating. You should get the original function back. Here is an example. EXAMPLE 1 Verify that F(x) ϭ x3