Calculus I (MAT 145) Dr. Day Monday March 25, 2019

1. 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? MAT 145

2. Critical Numbers • Where could relative extrema occur? Critical numbers are the locations where local extremacould 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. MAT 145

3. Absolute Extrema • Where and what are the absolute and local extrema? MAT 145

4. Absolute Extrema • Where and what are the absolute and local extrema? MAT 145

5. 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 maybe a local max or min at x=c. MAT 145

6. First derivative test MAT 145

7. Concavity of a Function Concavity Animations More Concavity Animations MAT 145

8. Concavity of a Function Concavity Animations More Concavity Animations MAT 145

9. 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. MAT 145

10. Inflection Point MAT 145

11. 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. MAT 145

12. 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. MAT 145

13. 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. MAT 145

14. MAT 145

15. Identify ExtremaFrom a Graph Graph each function. Identify all global and local extremes. For each of those, write a sentence based on this template: Atx = ?, there is a (local/global) (max/min) ofy =? MAT 145

16. Identify ExtremaFrom a Graph Atx = 0 there is a global max of 4. There is no global min. Atx = 1 there is a global max of 5. Atx = 4 there is a global min of 3. Atx = -1 there is a global min of1/e. There is no global max. MAT 145

17. 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. MAT 145

18. 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. MAT 145

19. Identify ExtremaUsing 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: Atx = _?_, there is an absolute (max/min) of_?_. MAT 145

20. Identify ExtremaUsing Critical Numbers Atx = √8 there is a global max of 8. Atx = −1 there is a global min of −√15. Atx = −2 there is a global max of 92. Atx = 3 there is a global min of −158. Atx = 1 there is a global max of 1. Atx = 0 there is a global min of 0. MAT 145

21. 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]. MAT 145