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Warm-Up: December 4, 2012

Warm-Up: December 4, 2012. Find all of the x-values where f’(x)=0. Homework Questions?. Juniors…. Which class( es ) are you currently planning on taking next year? AP Calculus BC AP Statistics AP Physics. Extreme Values of Functions. Section 4.1. Absolute (Global) Extrema.

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Warm-Up: December 4, 2012

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  1. Warm-Up: December 4, 2012 • Find all of the x-values where f’(x)=0

  2. Homework Questions?

  3. Juniors… • Which class(es) are you currently planning on taking next year? • AP Calculus BC • AP Statistics • AP Physics

  4. Extreme Values of Functions Section 4.1

  5. Absolute (Global) Extrema • Let f be a function with domain D. • f(c) is the absolute maximum value on Diff for all x in D. • The highest y-value (including endpoints). • f(c) is the absolute minimum value on Diff for all x in D. • The lowest y-value (including endpoints). • Endpoints can only be included in closed interval domains.

  6. Extreme Value Theorem • If f is continuous on a closed interval [a, b], then f has both a maximum value and a minimum value on the interval.

  7. Page 184 #1-6 1) Absolute max at x=b, absolute min at x=c2 2) Absolute max at x=c, absolute min at x=b 3) Absolute max at x=c, no absolute min 4) No absolute max, no absolute min 5) Absolute max at x=c, absolute min at x=a 6) Absolute max at x=a, absolute min at x=c

  8. Local (Relative) Extrema • Let c be an interior point of the domain of f. • f(c) is a local maximum value at ciff for all x in some open interval containing c. • The top of a hill. • f(c) is a local minimum value at ciff for all x in some open interval containing c. • The bottom of a valley.

  9. More about Extrema • Local extrema are also called relative extrema. • Local extrema can also occur at endpoints. • Any absolute extremum is also a local extremum. • “Maximum” refers to an absolute maximum. • “Minimum” refers to an absolute minimum.

  10. Page 184 #7-10 Identify both local and absolute extrema. 7) Local min at (-1,0); Local max at (1,0) 8) Minima at (-2,0) and (2,0); Maximum at (0,2) 9) Maximum at (0,5) 10) Local max at (-3,0); Local min at (2,0); Maximum at (1,2); Minimum at (0,-1)

  11. Warm-Up: December 5, 2012 • Find all of the x-values where f’(x)=0 or f’(x) fails to exist.

  12. Homework Questions?

  13. Local Extreme Value Theorem • If a function f has a local maximum or a local minimum at an interior point c of its domain, and if f’ exists at c, then

  14. Critical Point • A point on the interior of the domain of a function at which f’=0 or f’ does not exist is a critical point of f.

  15. Example 1 • Use analytic methods to find the extreme values of the function on the interval and where they occur.

  16. Example 2 • Use analytic methods to find the extreme values of the function on the interval and where they occur.

  17. Assignment • Read Section 4.1 (pages 177-183) • Page 184 Exercises 1-10, 11-17 odd • Page 184 Exercises 19-29 odd • Read Section 4.2 (page 186-191)

  18. Warm-Up: December 10, 2012 • Find the extreme values of the function and where they occur. Show your work.

  19. Chapter 3 Test Questions

  20. Assignment • Read Section 4.1 (pages 177-183) • Page 184 Exercises 1-10, 11-17 odd • Page 184 Exercises 19-29 odd • Read Section 4.2 (page 186-191)

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