Chapter 6 Integration Section 1 Antiderivatives and Indefinite Integrals

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 6.1 Antiderivatives and Indefinite Integrals  The student will be able to formulate problems involving antiderivatives use the formulas and properties of antiderivatives and indefinite integrals solve applications using antiderivatives and indefinite integrals

3 Barnett/Ziegler/Byleen Business Calculus 12e The Antiderivative Many operations in mathematics have inverses. Addition & subtraction Multiplication & division Powers and roots In Calculus, we have inverse functions too! Derivative & antiderivative

4 Barnett/Ziegler/Byleen Business Calculus 12e The Antiderivative A function F is an antiderivative of a function f if F (x) = f (x).

5 Barnett/Ziegler/Byleen Business Calculus 12e Example 1

6 Barnett/Ziegler/Byleen Business Calculus 12e Example 2

7 Barnett/Ziegler/Byleen Business Calculus 12e Theorem 1: Antiderivatives Conceptual Interpretation: If F(x) and G(x) are both antiderivatives of f(x), then the graphs of F(x) and G(x) are vertical translations of each other.

8 Example 3 Barnett/Ziegler/Byleen Business Calculus 12e

9 The symbol  is called an integral sign, and the function f (x) is called the integrand. The symbol dx indicates that anti- differentiation is performed with respect to the variable x. Indefinite Integrals Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol

10 Barnett/Ziegler/Byleen Business Calculus 12e Evaluate each indefinite integral: Example 4

11 Barnett/Ziegler/Byleen Business Calculus 12e Indefinite Integral Formulas and Properties (power rule) It is important to note that property 4 states that a constant factor can be moved across an integral sign. A variable factor cannot be moved across an integral sign.

12 Barnett/Ziegler/Byleen Business Calculus 12e Examples using the Power Rule

13 Barnett/Ziegler/Byleen Business Calculus 12e Using the Power Rule   x 2/3 dx =   (x 4 + x + x 1/ x –1/2 ) dx =

14 Barnett/Ziegler/Byleen Business Calculus 12e More Examples

15 More Examples Barnett/Ziegler/Byleen Business Calculus 12e

16 Homework Barnett/Ziegler/Byleen Business Calculus 12e

17 Homework Barnett/Ziegler/Byleen Business Calculus 12e

Chapter 6 Integration Section 1 Antiderivatives and Indefinite Integrals (continued)

19 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 6.1 Antiderivatives and Indefinite Integrals  The student will be able to formulate problems involving antiderivatives use the formulas and properties of antiderivatives and indefinite integrals solve applications using antiderivatives and indefinite integrals

20 Example 1 Barnett/Ziegler/Byleen Business Calculus 12e

21 Application 1 Barnett/Ziegler/Byleen Business Calculus 12e First find f(x):Substitute (2, 6) :

22 Application 2  Find the particular antiderivative of the derivative that satisfies the given condition. Barnett/Ziegler/Byleen Business Calculus 12e

23 Review Barnett/Ziegler/Byleen Business Calculus 12e

24 Application 3 Barnett/Ziegler/Byleen Business Calculus 12e Since the fixed cost is $2,000 this means C(0)=2000

25 Application 3 (cont.)  Find the cost of producing 20 widgets. Barnett/Ziegler/Byleen Business Calculus 12e

26 Application 4 Barnett/Ziegler/Byleen Business Calculus 12e Find N(x) = number of Vogue subscribers after x months.

27 Application 4 (continued) Barnett/Ziegler/Byleen Business Calculus 12e There were 64,000 subscribers before the new magazine came out (N=64000, x=0): How long until the number of subscribers drops to 46,000? (N=46000, x=?): It will take about 15.9 months for the number of Vogue subscribers to drop to 46,000.

28 Barnett/Ziegler/Byleen Business Calculus 12e