Chapter 3 1 Absolute Minimum and Maximum Values

Chapter 3. 1 - Absolute Minimum and Maximum Values Emanuel Balmus and Shikhar Ratnam

Lesson Objectives Define the absolute/global maximum and minimum values of a function. Be able to use a graph or equation of f on a given interval. Define the Extreme Value Theorem and relate it any problem within the chapter.

Critical Information Absolute Max/Min: The largest/smallest y values in a function’s domain. Critical Values of a Function: f has a critical point at x-values for which f ’(x) = 0 OR f ’(x) = undefined. Extreme Value Theorem: If f is defined on a closed interval [a, b], then f has an absolute Max/Min on this interval.

What You Should Already Know… Locate an x-value for which f ‘(x) = 0 OR f ‘(x) = undefined. Must be able to take the integral and derivative of an equation. Must be able to adjust window on calculator. Must understand how to use 2 nd CALC function on a graphing calculator. Must know how to adjust window on graph for given interval

Steps to find Absolute Max/Min Find all critical points on a given interval. Evaluate the function at critical points in Step 1 as well as the endpoints on the interval. Identify the Absolute Max and Absolute Min.

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Evaluating Function at Critical Points Plug in the critical points into f (x) = f (0) = - - - -> f (2) = - - - -> f (3) = - - - -> The absolute maximum of the function f (x) is 5, because that is the largest x-value. The absolute minimum is 1, because that is the smallest x-value.

Identify Absolute Max/Min 1. ) Find the largest and smallest value 2. ) Largest answer is Abs. Max Smallest answer is Abs. Min f (0) = 5 (Largest) Absolute Max = 5 f (2) = 1 (Smallest) Absolute Min = 1 f (3) = 2

Finding the Area 4 4 4 4

How to Find the Absolute Max/Min on a graph 1) Locate all the x-values that indicate an endpoint or critical point. 2) Find the area of each section applicable to the graph. 3) Substitute those values into the equation of f (x). 4) Determine which x-values produce the highest and lowest values. x -4 -2 6 12 f(x)

How to calculate absolute max/min using a graphing calculator 1. ) Type the equation into y = 2. ) Adjust window for given domain 3. ) Finding Abs. max - go to 2 nd calculate and enter maximum 4. ) When asked for left bound move the blinking dot to the left of the observed max and click enter 5. ) When asked for right bound move the blinking dot to the right of the observed max and click enter 6. ) Repeat the same process for Absolute min Locate the absolute max/min for the function f (x) = xsinx on the interval [0, 2π]

Steps to find Absolute Max/Min Find all critical points on a given interval. Evaluate the function at critical points in Step 1 as well as the endpoints on the interval. Identify the Absolute Max and Absolute Min.

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Evaluating Function at Critical Points Plug in the critical points into f (x) = f (0) = - - - -> f (2) = - - - -> f (3) = - - - -> The absolute maximum of the function f (x) is 5, because that is the largest x-value. The absolute minimum is 1, because that is the smallest x-value.

Identify Absolute Max/Min 1. ) Find the largest and smallest value 2. ) Largest answer is Abs. Max Smallest answer is Abs. Min f (0) = 5 (Largest) Absolute Max = 5 f (2) = 1 (Smallest) Absolute Min = 1 f (3) = 2

Quiz on Lesson 1 #1 A. ) Absolute Max = 13; Absolute Min = 1 B. ) Absolute Max = 13; Absolute Min =. 075 C. ) Absolute Max = 12; Absolute Min =. 075 D. ) Absolute Max = 12; Absolute Min = 1

Steps to find Absolute Max/Min Find all critical points on a given interval. Evaluate the function at critical points in Step 1 as well as the endpoints on the interval. Identify the Absolute Max and Absolute Min.

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Identify Absolute Max/Min 1. ) Find the largest and smallest value 2. ) Largest answer is Abs. Max Smallest answer is Abs. Min f (0) = 5 (Largest) Absolute Max = 5 f (2) = 1 (Smallest) Absolute Min = 1 f (3) = 2

Quiz on Lesson 1 #2 A. ) Absolute Max = 9; Absolute Min = 3 B. ) Absolute Max = 8; Absolute Min = 3 C. ) Absolute Max = 9; Absolute Min = 2 D. ) Absolute Max = 6; Absolute Min = 2

Steps to find Absolute Max/Min Find all critical points on a given interval. Evaluate the function at critical points in Step 1 as well as the endpoints on the interval. Identify the Absolute Max and Absolute Min.

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Evaluating Function at Critical Points Plug in the critical points into f (x) = f (0) = - - - -> f (2) = - - - -> f (3) = - - - -> The absolute maximum of the function f (x) is 5, because that is the largest x-value. The absolute minimum is 1, because that is the smallest x-value.

Identify Absolute Max/Min 1. ) Find the largest and smallest value 2. ) Largest answer is Abs. Max Smallest answer is Abs. Min f (0) = 5 (Largest) Absolute Max = 5 f (2) = 1 (Smallest) Absolute Min = 1 f (3) = 2

Quiz on Lesson 1 #3 A. ) Absolute Max = 9; Absolute Min = 0 B. ) Absolute Max = 8; Absolute Min = 2 C. ) Absolute Max = 12; Absolute Min = 4 D. ) Absolute Max = 16; Absolute Min = -1

Steps to find Absolute Max/Min Find all critical points on a given interval. Evaluate the function at critical points in Step 1 as well as the endpoints on the interval. Identify the Absolute Max and Absolute Min.

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Evaluating Function at Critical Points Plug in the critical points into f (x) = f (0) = - - - -> f (2) = - - - -> f (3) = - - - -> The absolute maximum of the function f (x) is 5, because that is the largest x-value. The absolute minimum is 1, because that is the smallest x-value.

Identify Absolute Max/Min 1. ) Find the largest and smallest value 2. ) Largest answer is Abs. Max Smallest answer is Abs. Min f (0) = 5 (Largest) Absolute Max = 5 f (2) = 1 (Smallest) Absolute Min = 1 f (3) = 2

Quiz on Lesson 1 #4 A. ) Absolute Max = 9; Absolute Min = 0 B. ) Absolute Max = 4; Absolute Min = 2 C. ) Absolute Max = 8; Absolute Min = 4 D. ) Absolute Max = 9; Absolute Min = -1

Finding the Area 4 4 4 4

How to Find the Absolute Max/Min on a graph 1) Locate all the x-values that indicate an endpoint or critical point. 2) Find the area of each section applicable to the graph. 3) Substitute those values into the equation of f (x). 4) Determine which x-values produce the highest and lowest values. x -4 -2 6 12 f(x)

Finding Critical Points Find the derivative of f (x). Set the derivative equal to zero and solve for x. The critical numbers are points that were solved for x and the endpoints on the given interval Critical points {0, 2, 3}

Quiz on Lesson 2 Find the Absolut Max and Min value of g on the interval [-5, 4] A. ) Absolute Max: 6 ; Absolute Min: -3 Graph of f B. ) Absolute Max: 10 ; Absolute Min: -2 C. ) Absolute Max: 2 ; Absolute Min: -6 D. ) Absolute Max: 4 ; Absolute Min: -4

How to calculate absolute max/min using a graphing calculator 1. ) Type the equation into y = 2. ) Adjust window for given domain 3. ) Finding Abs. max - go to 2 nd calculate and enter maximum 4. ) When asked for left bound move the blinking dot to the left of the observed max and click enter 5. ) When asked for right bound move the blinking dot to the right of the observed max and click enter 6. ) Repeat the same process for Absolute min Locate the absolute max/min for the function f (x) = xsinx on the interval [0, 2π]

Quiz on Lesson 3 A. ) Absolute Max: 7. 2 ; Absolute Min: -8 B. ) Absolute Max: 6. 1 ; Absolute Min: -7. 7 C. ) Absolute Max: 5. 9 ; Absolute Min: -7. 5 D. ) Absolute Max: 6. 4 ; Absolute Min: -7. 9