Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 2)

2 Barnett/Ziegler/Byleen Business Calculus 12e Objectives for Section 6.5 Fundamental Theorem of Calculus ■ Solve application problems using the definite integral. ■ Calculate the average value of a function using the definite integral.

3 Review Cost & Change in Cost  Suppose the cost function for manufacturing x coffee makers by Keurig is given (in dollars) by: Barnett/Ziegler/Byleen Business Calculus 12e  When the level of production is increased from 5 to 10 coffee makers the change in cost is:

4 Continued Barnett/Ziegler/Byleen Business Calculus 12e Cost function: Marginal Cost function: So the change in cost from a production level of 5 to 10 is equal to the area under the curve of the marginal cost function from x=5 to x=10.

5 Example 1  The marginal cost function (in dollars) for a company that makes tablets is given by:  Using a definite integral, find the change in cost that results from an increase in production from 10 to 20 tablets. Barnett/Ziegler/Byleen Business Calculus 12e

6 Example 1 (continued) Barnett/Ziegler/Byleen Business Calculus 12e It will cost the company $800 more when increasing production from 10 to 20 tablets.

7 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 From past records a management service determined that the rate of increase in maintenance costs for an apartment building (in dollars per year) is given by M ’(x) = 90x 2 + 5,000, where M(x) is the total accumulated cost of maintenance for x years. Use a definite integral to calculate the total maintenance cost from the beginning of the third year to the end of the seventh year.

8 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 (continued) The total maintenance cost from the beginning of the third year to the end of the seventh year is given by: M(x) is the cost of maintenance after x years.

9 Example 3  Suppose that the temperature F (in degrees Fahrenheit) in the middle of a shallow lake from 8:00 AM (t=0) to 6:00 PM (t=10) during the month of May is given by the function and its graph: Barnett/Ziegler/Byleen Business Calculus 12e Suppose we want to calculate the average temperature from 8 AM to 6 PM.

10 Barnett/Ziegler/Byleen Business Calculus 12e Using Definite Integrals for Average Values The average value of a continuous function f over [a, b] is Note this is the area under the curve divided by the width. Hence, the result is the average height or average value of the function. Let’s use this to solve Example 3…

11 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e 8:00 AM (t=0) to 6:00 PM (t=10)

12 Example 4 Barnett/Ziegler/Byleen Business Calculus 12e

13 Barnett/Ziegler/Byleen Business Calculus 12e The total cost (in dollars) of printing x phonebooks is given by: C(x) = 20, x a)Find the average cost per phonebook if 1000 are produced. b)Find the average total cost when up to 1000 phonebooks are printed. Example 5

14 Barnett/Ziegler/Byleen Business Calculus 12e a) Find the average cost per phonebook if 1000 are produced. Solution: The average cost function is Example 5 (continued) $30 per phonebook.

15 Barnett/Ziegler/Byleen Business Calculus 12e Example 5 (continued) b) Find the average total cost when up to 1000 phonebooks are printed.

16 Homework Barnett/Ziegler/Byleen Business Calculus 12e