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Increasing and Decreasing Intervals Where Does the Fun End ?

Increasing and Decreasing Intervals Where Does the Fun End ?. Anne Dudley Michael Holtfrerich Joshua Whitney Glendale C ommunity College Glendale, Arizona. Increasing & Decreasing Activity 1. Determine the largest interval on which each function (on the handout) is increasing.

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Increasing and Decreasing Intervals Where Does the Fun End ?

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  1. Increasing and Decreasing IntervalsWhere Does the Fun End? Anne Dudley Michael Holtfrerich Joshua Whitney Glendale Community College Glendale, Arizona

  2. Increasing & DecreasingActivity 1 Determine the largest interval on which each function (on the handout) is increasing. Once you have determined your answers, talk to neighbors about their answers.

  3. Graph 1 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing? • (–1, 1] • (–1, 1) • [–1, 1] • Other

  4. Graph 2 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing? • Other

  5. Graph 2 DecreasingUse your clicker to indicate your answer. What is the largest interval on which the function is decreasing? • Other

  6. Discussion If you used closed intervals for your two previous answers, are you OK with the graph being both increasing and decreasing at x = 1? Will it confuse your students?

  7. Graph 3 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing? • Other

  8. Graph 4 IncreasingUse your clicker to indicate your answer.Where is the function increasing? • Other

  9. What Do Textbooks Say? Ostebee & ZornCalculus, 2ndEdition (p.56) Definition: Let I denote the interval (a, b). A function fis increasing on Iif whenever a < x1 < x2 < b. fis decreasing on Iif whenever a < x1 < x2 < b. Fact: If for all x in I, then f increases on I. If for all x in I, then f decreases on I.

  10. Graph 1 Increasing (Ostebee & Zorn )Use your clicker to indicate your answer.What is the largest interval on which the function is increasing? • Other

  11. What Do Textbooks Say? Hughes-HallettCalculus, 5th edition (p. 165) and Swokowski Calculus, 2ndedition (p.147) Let f(x) be continuous on [a, b], and differentiable on (a, b). If f’(x) > 0 on a < x < b, then f is increasing on . If f’(x)≥ 0 on a < x < b, then f is non-decreasing on .

  12. Graph 1 Increasing (Hughes-Hallett)Use your clicker to indicate your answer.What is the largest interval on which the function is increasing? • Other

  13. What Do Textbooks Say? Larson EdwardsCalculus, 5th edition (p. 219) A function f is increasing on an interval if for any two numbers x1 and x2in the interval, x1 < x2 implies f(x1) < f(x2). A function f is decreasing on an interval if for any two numbers x1 and x2in the interval, x1 < x2 implies f(x1) > f(x2).

  14. The Problem with “Any” The following function isnot increasing on [a,b]. But it does fit the precedingdefinition. For the two numbers x1< x2, f(x1) < f(x2) should imply that f(x) is increasing on [a,b].

  15. What Do Textbooks Say? Larson EdwardsCalculus, 5th edition (p. 219) (Theorem) Let f (x) be continuous on [a, b], and differentiable on (a, b). If f’(x) > 0 for all x in (a, b), then f is increasing on [a, b]. If f’(x) < 0 for all x in (a, b), then f is decreasing on [a, b]. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b]. All textbook examples and the answers in homework are open intervals.

  16. What Do Textbooks Say? Cynthia YoungPrecalculus, 1st edition (p. 128) A function f is increasing on an open interval I if for any x1 and x2 in I, where x1 < x2, then f(x1) < f(x2). A function f is decreasing on an open intervalI if for any x1 and x2 in I, where x1 < x2, then f(x1) > f(x2).

  17. Two Differing Ideas Two Point Idea of Increasing Slope of the Tangent Line Idea of Increasing(one point idea)

  18. Apply the Definitions Now determine the largest interval on which the function is increasing for the three new examples. Be prepared to clicker your choice.

  19. Graph 5 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing? • Other

  20. Graph 6 IncreasingUse your clicker to indicate your answer.What is the largest interval on which the function is increasing?

  21. Graph 7 (AKA Final Exam) Use your clicker to indicate your answer.Where is the function increasing? • Other

  22. Dudfreney Intervals We propose this definition for all textbooks at the College Algebra (Pre-Calculus) level and below: A function f is increasing on an open interval I if for all x1and x2 in I, where x1 < x2, then f(x1) < f(x2). A function f is decreasing on an open intervalI if for all x1and x2 in I, where x1 < x2, then f(x1) > f(x2).

  23. Historical Ideas Introduction to Infinitesimal Analysis, O. Veblen, 1907

  24. Historical Ideas Theory of Functions of Real Variables, J. Pierpont, 1905

  25. Historical Ideas Differential and Integral Calculus, G. Osborne, 1907

  26. Historical Ideas An Elementary Treatise on the Calculus, G. Gibson, 1901

  27. Thanks for Your Participation Anne – Anne.Dudley@gccaz.edu Michael – M.Holtfrerich@gccaz.edu Josh – Joshua.Whitney@gccaz.edu

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