ABSOLUTE MINIMUM AND MAXIMUMS By: Hannah Ahluwalia and Anita Vellaichamy

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

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

EXAMPLE Find the derivative of a function through the power rule

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.

EXAMPLE 2

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

CRITICAL NUMBERS CONT.

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

CRITICAL POINTS EX. 1 A B

CRITICAL POINT EX. 2 These are your critical numbers

NOW TRY ONE FOR YOURSELF!

CRITICAL POINT QUIZ

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

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

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 u/Classes/Cal cI/AbsExtrem a.aspx

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

FINISHING THE EXAMPLE Back Continue

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

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

EXAMPLE CONT. Back Continue

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

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

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

EXAMPLE CONT. BackContinue

EXAMPLE CONT. Next, test the values into the equation, include critical points and endpoints. (0)=100.0 (0.604)= (0.9661)= (2.1755)= (2.5369)= ( )= (4)= Abs. minimum Abs. maximum Back Finish the example

FINISH THE EXAMPLE Try one on your own Back

NOW YOU TRY ONE! Review how to find derivative Continue to answer /visual.calculus/3/max.1/3.h tml

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

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

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

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

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

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

QUESTION 3 math.utk.edu/vi sual.calculus/3/ max.1/2.html Review

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

QUESTION 4 CONT. Review

QUESTION 4 CONT. Review

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

INCORRECT! Try this one again!

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

CORRECT!

Next questionBack to question 1

CRITICAL NUMBERS CONT. Back to quiz question 1

CRITICAL NUMBERS CONT. Back to lesson

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

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

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

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

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

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

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

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

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

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

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

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

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

CORRECT! Abs. Minimum Finish quiz Back to part 2

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

FINISH!