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Understanding Extreme Values in Calculus: Homework and Warm-Up Exercises

This resource outlines essential concepts and exercises regarding extreme values of functions, focusing on both absolute (global) and local (relative) extrema based on the Extreme Value Theorem. It includes examples of finding x-values where derivative f’(x)=0 or does not exist, critical points, and the identification of maxima and minima in given domains. Additionally, it covers relevant problems from the textbook for comprehensive practice, making it a great study aid for AP Calculus and related subjects.

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Understanding Extreme Values in Calculus: Homework and Warm-Up Exercises

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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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