1. Calculus I (MAT 145) Dr. Day Wednesday March 20, 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
Friday, March 22, 2019
MAT 145

2. 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
Friday, March 22, 2019
MAT 145

3. Fermat’s Theorem
Friday, March 22, 2019
MAT 145

4. Absolute Extrema Where and what are the absolute and local extrema?
Friday, March 22, 2019
MAT 145

5. Absolute Extrema Where and what are the absolute and local extrema?
Friday, March 22, 2019
MAT 145

6. 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.
Friday, March 22, 2019
MAT 145

7. First derivative test
Friday, March 22, 2019
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8. Absolute and Relative Maximums and Minimums
Use the graph of f ’(x) to describe a strategy for identifying the global and local extrema of f, knowing f ’(x). m a t h
Friday, March 22, 2019
MAT 145

9. Concavity of a Function
Concavity Animations
More Concavity Animations
Friday, March 22, 2019
MAT 145

10. Concavity of a Function
Concavity Animations
More Concavity Animations
Friday, March 22, 2019
MAT 145

11. 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.
Friday, March 22, 2019
MAT 145

12. Inflection Point
Friday, March 22, 2019
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13. First Derivative Test, Concavity, Second Derivative Test (Sec. 4.3)
Determining Increasing or Decreasing Nature of a Function
If f’(x) > 0, then f is _?_.
If f’(x) < 0, then f is _?_.
Using the First Derivative to Determine Whether an Extreme Value Exists: The First Derivative Test (and first derivative sign charts)
If f’ changes from positive to negative at x=c, then f has a _?_ _?_ at c.
If f’ changes from negative to positive at x=c, then f has a _?_ _?_ at c.
If f’ does not change sign at x=c, then f has neither a local max or min at c.
Concavity of f
If f’’(x) > 0 for all x in some interval I, then the graph is concave up on I.
If f’’(x) < 0 for all x in some interval I, then the graph is concave down on I.
Second derivative Test
If f’(c) = 0 and f’’(c) > 0, then f has a local min at c.
If f’(c) = 0 and f’’(c) < 0, then f has a local max at c.
Friday, March 22, 2019
MAT 145

14. 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.
Friday, March 22, 2019
MAT 145

15. 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.
Friday, March 22, 2019
MAT 145

16. 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.
Friday, March 22, 2019
MAT 145

17. Friday, March 22, 2019
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18. 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 = ?
Friday, March 22, 2019
MAT 145

19. 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.
Friday, March 22, 2019
MAT 145

20. 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.
Friday, March 22, 2019
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21. 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.
Friday, March 22, 2019
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22. 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 _?_.
Friday, March 22, 2019
MAT 145

23. 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.
Friday, March 22, 2019
MAT 145

24. 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].
Friday, March 22, 2019
MAT 145