2. ABSOLUTE MINIMUM AND MAXIMUMS By: Hannah Ahluwalia and Anita Vellaichamy

3. LESSON OBJECTIVES In this lesson, you will learn how to find critical numbers analytically and graphically, as well as using them to determine the absolute minimum and maximum values of a function

4. INTRODUCTION In order to continue with the lesson, you must know how to find the derivative of a function

5. EXAMPLE Find the derivative of a function through the power rule

6. INTRODUCTION PART 2 You must also know how to solve for the zeroes of a function. This is done simply by setting an equation equal to zero and solving for x.

7. EXAMPLE 2

8. CRITICAL NUMBERS Now that you know these basics, we may move into the first part of the lesson: finding critical numbers

9. CRITICAL NUMBERS CONT.

10. CRITICAL POINTS CONT. To find the critical points of a function analytically, you must take the derivative of the function. Once you have the derivative, set it equal to zero and solve for X

11. CRITICAL POINTS EX. 1 A B

12. CRITICAL POINT EX. 2 These are your critical numbers

13. NOW TRY ONE FOR YOURSELF!

14. CRITICAL POINT QUIZ

15. LESSON CONT.  Now that you know how to find critical points let’s learn how to you them! If you want to review critical points, feel free to go back at any time. Review Continue

16. LESSON CONT.  You must plug each x-value into the function. This includes critical points that you solved for, and the endpoints of the function. Continue Back

17. ABS. MAXIMUM AND MINIMUM EX.1 Function: 1. Get the derivative (you can use the power rule here): 2. Factor the derivative you found: 3. Set each equal to zero to find the critical points: This function has two critical points. ContinueBack http://tutorial.math.lamar.ed u/Classes/Cal cI/AbsExtrem a.aspx

18. EXAMPLE CONT. critical points endpoints Abs. minimum Abs. maximum ContinueBack

19. FINISHING THE EXAMPLE Back Continue

20. NOW YOU TRY ONE! Review how to find these Continue to answer https://mathway.com/exam ples/Calculus/Applications- of-Differentiation/Finding- the-Absolute-Maximum- and-Minimum-on-the- Given-Interval?id=825

21. DID YOU GET IT? If no, try again If yes, continue

22. EXAMPLE CONT. Back Continue

23. EXAMPLE CONT. If no, try again If yes, continue

24. LESSON CONT. If you are finding the abs. maximum and minimum by using your calculator, you will need to graph the function to find the critical points. Back to example problem Continue with lesson

25. CALCULATOR EX 1. Back Continue to example http://tutorial. math.lamar.edu /Classes/CalcI/ AbsExtrema.as px

26. EXAMPLE CONT. BackContinue

27. EXAMPLE CONT. Next, test the values into the equation, include critical points and endpoints. (0)=100.0 (0.604)=102.4756 (0.9661)=102.2368 (2.1755)=107.1880 (2.5369)=106.9492 ( 3.7463)=111.9004 (4)=111.7121 Abs. minimum Abs. maximum Back Finish the example

28. FINISH THE EXAMPLE Try one on your own Back

29. NOW YOU TRY ONE! Review how to find derivative Continue to answer http://archives.math.utk.edu /visual.calculus/3/max.1/3.h tml

30. PROBLEM CONT. If no, try again If yes, continue

31. PROBLEM CONT. If no, try again If yes, continue

32. PROBLEM CONT. If yes, continue If no, try again

33. GOOD JOB! If you want to review, go back, if you would like to move forward to the quiz, click continue. Review from beginning of lesson Continue to quiz

34. QUESTION 1 What is the critical points of 2.5 8.4 4.2 Review topichttp://archives.math.utk.edu/v isual.calculus/3/max.1/1.html

35. QUESTION 2 Find the Abs. maximum value of **Use calculator for this question http://archiv es.math.utk.e du/visual.calc ulus/3/max.1 /4.html Review

36. QUESTION 3 http://archives. math.utk.edu/vi sual.calculus/3/ max.1/2.html Review

37. QUESTION 4 Continue to first part of question http://archiv es.math.utk.e du/visual.cal culus/3/max.1/7.html

38. QUESTION 4 CONT. Review

39. QUESTION 4 CONT. Review

40. GREAT JOB! Now you can find both absolute extrema, but feel free to review if you need to. Back to beginning Finish

41. INCORRECT! Try this one again!

42. INCORRECT! Try this one again! Back to quiz question 1

43. CORRECT!

44. Next questionBack to question 1

45. CRITICAL NUMBERS CONT. Back to quiz question 1

46. CRITICAL NUMBERS CONT. Back to lesson

47. EXAMPLE Find the derivative of a function through the power rule Back to example

48. CRITICAL POINT EX. 2 These are your critical numbers Back to example

49. ABS. MAXIMUM AND MINIMUM EX.1 Function: 1. Get the derivative (you can use the power rule here): 2. Factor the derivative you found: 3. Set each equal to zero to find the critical points: This function has two critical points. Back

50. INCORRECT! Try this one again! Back to quiz question 2

51. CORRECT! Abs. Maximum Continue to question 3 Back to question 2

52. EXAMPLE CONT. critical points endpoints Abs. minimum Abs. maximum Back to question 2

53. EXAMPLE CONT. critical points endpoints Abs. minimum Abs. maximum Back to question 3

54. INCORRECT! Try this one again! Back to quiz question 3

55. CORRECT! Abs. minimum value Continue to question 4 Back to question 3

56. EXAMPLE CONT. critical points endpoints Abs. minimum Abs. maximum Back to question 4 part 1

57. INCORRECT! Try this one again! Back to quiz question 4 part 1

58. CORRECT! Abs. Maximum Continue to next part of question 4 Back to part 1

59. INCORRECT! Try this one again! Back to quiz question 4 part 2

60. CORRECT! Abs. Minimum Finish quiz Back to part 2

61. EXAMPLE CONT. critical points endpoints Abs. minimum Abs. maximum Back to question 4 part 2

62. FINISH!