Calculus I (MAT 145) Dr. Day Monday March 25, 2019 Chapter 4: Using All Your Derivative Knowledge! Absolute and Relative Extremes What is a “critical number?” Increasing and Decreasing Behavior of Functions Connecting f and f’ Concavity of Functions: A function’s curvature Connecting f and f” Graphing a Function: Putting it All together! Max-Mins Problems: Determine Solutions for Contextual Situations Other Applications Finally . . . What if We Reverse the Derivative Process? Monday, March 25, 2019 MAT 145

Critical Numbers Where could relative extrema occur? Critical numbers are the locations where local extrema could occur. Critical points are the points (x- and y-values) that describe both the locations and function values at those points. Determine critical numbers for 1. 2. 3. Monday, March 25, 2019 MAT 145

Absolute Extrema Where and what are the absolute and local extrema? Monday, March 25, 2019 MAT 145

Absolute Extrema Where and what are the absolute and local extrema? Monday, March 25, 2019 MAT 145

What does f’ tell us about f? If f’(c)= 0, there is a horizontal tangent to the curve at x=c. This may mean there is a local max or min at x=c. If f’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) at x=c. If f(x) is continuous at x=c, there may be a local max or min at x=c. Monday, March 25, 2019 MAT 145

First derivative test Monday, March 25, 2019 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Monday, March 25, 2019 MAT 145

Concavity of a Function Concavity Animations More Concavity Animations Monday, March 25, 2019 MAT 145

What does f’’ tell us about f? If f’’(c)> 0, then the original curve f(x) is concave up at x=c. If f’’(c)< 0, then the original curve f(x) is concave down at x=c. If f’’(c)= 0, then f(x) is neither concave up nor concave down at x=c. And there could be an inflection point on f(x) at x=c. If f’’(c) is undefined, there could be a discontinuity, a vertical tangent, or a cusp (sharp point) in f’(x) at x=c. There may be a change of concavity in f(x) at x=c. Monday, March 25, 2019 MAT 145

Inflection Point Monday, March 25, 2019 MAT 145

Info about f from f ’ Here’s a graph of g’(x). Determine all intervals over which g is increasing and over which g is decreasing. Identify and justify where all local extremes occur. Monday, March 25, 2019 MAT 145

Info about f from f ’’ Here’s a graph of h”(x). Determine all intervals over which h is concave up and over which h is concave down. Identify and justify where all points of inflection occur. Monday, March 25, 2019 MAT 145

Pulling it all together For f(x) shown below, use calculus to determine and justify: All x-axis intervals for which f is increasing All x-axis intervals for which f is decreasing The location and value of every local & absolute extreme All x-axis intervals for which f is concave up All x-axis intervals for which f is concave down The location of every point of inflection. Monday, March 25, 2019 MAT 145

Monday, March 25, 2019 MAT 145

Identify Extrema From a Graph Graph each function. Identify all global and local extremes. For each of those, write a sentence based on this template: At x = ?, there is a (local/global) (max/min) of y = ? Monday, March 25, 2019 MAT 145

Identify Extrema From a Graph At x = 1 there is a global max of 5. At x = 4 there is a global min of 3. At x = 0 there is a global max of 4. There is no global min. At x = -1 there is a global min of 1/e. There is no global max. Monday, March 25, 2019 MAT 145

Determine the Critical Numbers For each function, determine every critical number. Unless otherwise restricted, assume that each function’s domain includes all possible values for which that function is defined. Monday, March 25, 2019 MAT 145

Determine the Critical Numbers For each function, determine every critical number. Unless otherwise restricted, assume that each function’s domain includes all possible values for which that function is defined. Monday, March 25, 2019 MAT 145

Identify Extrema Using Critical Numbers For each function, determine every critical number, and then use those critical numbers to determine all absolute extreme values. Note the domain restrictions. For each extreme value, write a sentence based on this template: At x = _?_, there is an absolute (max/min) of _?_. Monday, March 25, 2019 MAT 145

Identify Extrema Using Critical Numbers At x = √8 there is a global max of 8. At x = −1 there is a global min of −√15. At x = −2 there is a global max of 92. At x = 3 there is a global min of −158. At x = 1 there is a global max of 1. At x = 0 there is a global min of 0. Monday, March 25, 2019 MAT 145

Absolute and Relative Extremes Ways to Find Extrema Local Extremes: Examine behavior at critical points. Absolute Extremes: Examine behavior at critical points and at endpoints. Example Determine critical numbers, absolute extrema, and relative extrema for the unrestricted function (all possible domain values) and then for the restricted domain [−3,1]. Monday, March 25, 2019 MAT 145