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MAT 1221 Survey of Calculus

MAT 1221 Survey of Calculus. Section 3.2 Extrema and the First Derivative Test. http://myhome.spu.edu/lauw. Expectations. Check your algebra. Check your calculator works. Make sure you have all the important details for each point you test. 1 Minute….

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MAT 1221 Survey of Calculus

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  1. MAT 1221Survey of Calculus Section 3.2 Extrema and the First Derivative Test http://myhome.spu.edu/lauw

  2. Expectations • Check your algebra. • Check your calculator works. • Make sure you have all the important details for each point you test.

  3. 1 Minute… • You can learn all the important concepts in 1 minute.

  4. 1 Minute… • High/low points – most of them are at points with horizontal tangent

  5. 1 Minute… • High/low points – most of them are at points with horizontal tangent. • Highest/lowest points – at points with horizontal tangent or endpoints

  6. 1 Minute… • You can learn all the important concepts in 1 minute. • We are going to develop the theory carefully so that it works for all the functions that we are interested in. • There are a few definitions…

  7. Preview • Define • Local (relative) max./min. • Absolute max./min. • First Derivative Test • Closed Interval method

  8. Absolute Max has an absolute maximum at on if for all in ( =Domain of ) c D

  9. Absolute Min has an absolute maximum at on if for all in ( =Domain of ) c D

  10. Extreme Values • The absolute maximum and minimum values of are called the extreme values of .

  11. Example 0 y Absolute max. Absolute min. x

  12. Local (Relative) Max/Min has an local maximum at if for all in some open interval containing has an local minimum at if for all in some open interval containing

  13. Example 0 y Local max. x Local min.

  14. Note • An end point is notconsider as a local max/min.

  15. The First Derivative Test Suppose that is a critical number of a continuous function (a) If changes from positive to negative at , then has a local maximum at . (b) If changes from negative to positive at , then has a local minimum at . (c) If does not changes sign at , then has no local max. or min. at .

  16. The First Derivative Test y y =0 <0 >0 >0 <0 =0 x x a a c c b b Local max. Local min.

  17. Example 1 (Continue from 3.1) • The local min. value of f is f(3)=1 3

  18. Expectations • The first part of the problem is to find the interval of increasing/decreasing. • The second part is to find the local max./min. from the results of the first part • You are expected to answer the problem formally with a statement “The local min. value(s) of is .”

  19. How to find Absolute Max./Min.? • It is easy if the domain is a closed interval • Fact: It is difficult to do that if the domain is NOT a closed interval.

  20. The Extreme Value Theorem • If is continuous on a closed interval , then attains an absolute max value and an absolute min value at some numbers and d in . • No guarantee of absolute max/min if one of the 2 conditions are missing.

  21. How to find Absolute Max./Min.? y Absolute max. Local max. Absolute min. x Local min.

  22. The Closed Interval Method • Idea: the absolute max/min values of a continuous function on a closed interval only occurs at • the local max/min (the critical numbers) • end points of the interval

  23. The Closed Interval Method • To find the absolute max/min values of a continuous function f on a closed interval : • Find the values of at the critical numbers of in . • Find the values of f at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

  24. The Closed Interval Method • To find the absolute max/min values of a continuous function f on a closed interval : • Find the values of at the critical numbers of in . • Find the values of at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

  25. The Closed Interval Method • To find the absolute max/min values of a continuous function on a closed interval : • Find the values of at the critical numbers of in . • Find the values of at the end points. • The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of the those values from is the absolute minimum value.

  26. Example 1 • Find the absolute max/min values of

  27. Expectations • Formally answer the problem in this form:

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