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Objectives for Section 13.5 Fundamental Theorem of Calculus

Objectives for Section 13.5 Fundamental Theorem of Calculus. The student will be able to evaluate definite integrals. The student will be able to calculate the average value of a function using the definite integral. . Fundamental Theorem of Calculus .

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Objectives for Section 13.5 Fundamental Theorem of Calculus

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  1. Objectives for Section 13.5 Fundamental Theorem of Calculus • The student will be able to evaluate definite integrals. • The student will be able to calculate the average value of a function using the definite integral. Barnett/Ziegler/Byleen Business Calculus 11e

  2. Fundamental Theorem of Calculus If f is a continuous function on the closed interval [a, b], and F is any antiderivative of f, then Barnett/Ziegler/Byleen Business Calculus 11e

  3. Evaluating Definite Integrals By the fundamental theorem we can evaluate easily and exactly. We simply calculate Barnett/Ziegler/Byleen Business Calculus 11e

  4. Definite Integral Properties Barnett/Ziegler/Byleen Business Calculus 11e

  5. Example 1 Make a drawing to confirm your answer. 0  x  4 - 1  y  6 Barnett/Ziegler/Byleen Business Calculus 11e

  6. Example 2 Make a drawing to confirm your answer. 0 x 4 - 1 y 4 Barnett/Ziegler/Byleen Business Calculus 11e

  7. Example 3 0 x 4 - 2 y 10 Barnett/Ziegler/Byleen Business Calculus 11e

  8. Example 4 Let u = 2x, du = 2 dx Barnett/Ziegler/Byleen Business Calculus 11e

  9. Example 5 Barnett/Ziegler/Byleen Business Calculus 11e

  10. Example 6 This is a combination of the previous three problems Barnett/Ziegler/Byleen Business Calculus 11e

  11. Example 7 Let u = x3 + 4, du = 3x2dx Barnett/Ziegler/Byleen Business Calculus 11e

  12. Example 7(revisited) On the previous slide, we made the back substitution from u back to x. Instead, we could have just evaluated the definite integral in terms of u: Barnett/Ziegler/Byleen Business Calculus 11e

  13. Numerical Integration on a Graphing Calculator Use some of the examples from previous slides: Example 5: 0 x 3 - 1 y 3 Example 7: -1 x 6 - 0.2 y 0.5 Barnett/Ziegler/Byleen Business Calculus 11e

  14. Example 8 From past records a management service determined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by M ’(x) = 90x2 + 5,000, where M(x) is the total accumulated cost of maintenance for x years. Write a definite integral that will give the total maintenance cost from the end of the second year to the end of the seventh year. Evaluate the integral. Barnett/Ziegler/Byleen Business Calculus 11e

  15. Example 8 From past records a management service determined that the rate of increase in maintenance cost for an apartment building (in dollars per year) is given by M ’(x) = 90x2 + 5,000, where M(x) is the total accumulated cost of maintenance for x years. Write a definite integral that will give the total maintenance cost from the end of the second year to the end of the seventh year. Evaluate the integral. Solution: Barnett/Ziegler/Byleen Business Calculus 11e

  16. 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. Barnett/Ziegler/Byleen Business Calculus 11e

  17. Example • Section 6.5 #70. The total cost (in dollars) of printing x dictionaries is C(x) = 20,000 + 10x • Find the average cost per unit if 1000 dictionaries are produced. • Find the average value of the cost function over the interval [0, 1000]. • Write a description of the difference between part a) and part b). Barnett/Ziegler/Byleen Business Calculus 11e

  18. Example(continued) a) Find the average cost per unit if 1000 dictionaries are produced Solution: The average cost is Barnett/Ziegler/Byleen Business Calculus 11e

  19. Example(continued) b) Find the average value of the cost function over the interval [0, 1000] Solution: Barnett/Ziegler/Byleen Business Calculus 11e

  20. Example(continued) c) Write a description of the difference between part a and part b Solution: If you just do the set-up for printing, it costs $20,000. This is the cost for printing 0 dictionaries. If you print 1,000 dictionaries, it costs $30,000. That is $30 per dictionary (part a). If you print some random number of dictionaries (between 0 and 1000), on average it costs $25,000 (part b). Those two numbers really have not much to do with one another. Barnett/Ziegler/Byleen Business Calculus 11e

  21. Summary We can evaluate a definite integral by the fundamental theorem of calculus: We can find the average value of a function f by Barnett/Ziegler/Byleen Business Calculus 11e

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