Warm Up

1. Warm Up Page 214 Quick Review #1 as a class #2 individually Correct Homework

2. Lesson 4.3 Connecting f ’ and f ” with the graph of f What you’ll learn First Derivative Test for Local Extrema Concavity Points of Inflection Second Derivative Test for Local Extrema Learning about Functions from Derivatives Why? Differential calculus is a powerful problem-solving tool precisely because of its usefulness for analyzing functions.

3. a) Find f’(x) and critical pts where f’(x) = 0. Divide graph into intervals Use critical points and find the sign of f’ on each interval. Use 1st derivative test to identify the behavior of y. Check Graph Identify extrema b) Example 1:For each of the following functions, use the First Derivative Test to find the local extreme values. Identify any absolute extrema.

4. Concavity The graph of a differentiable function y = f(x) is • Concave up on an open interval I if y” is increasing on I • Concave down on an open interval I if y’ is decreasing on I. If a function has a 2nd derivative we can conclude that y’ increases when y”> 0, and y’ decreases if y” < 0. http://justmathtutoring.com/ Use 2nd derivative test to find concavity, max, min

5. Homework Page 214 Quick Review 3-10 Page 215 Exercises 1-15 odds

6. 4.3 Continued Warm Up Page 211 Example 6 Put data into your calculator and verify logistics equation Graph equation over points using window [12, 108] by [0, 730,000]

7. Example 6 Population Growth in Alaska(using a calculator to find derivatives, analyze data) Table 4.2 shows the population of Alaska in each 10-year census between 1920 and 2000. • Find the logistic regression for the data • Use the regression equation to predict the Alaskan population in the 2020 census. • Find the inflection point of the regression equation. What significance does the inflection point have in terms of population growth in Alaska? • What does the regression equation indicate about the population of Alaska in the long run?

8. Theorem 4 (review) First Derivative Test for Local ExtremaP205 / 206 for graphs The following test applies to a continuous function f(x). At a critical point c: • If f’ changes sign from positive to negative at c, then f has a local maximum value at c. • If f’ changes sign from negative to positive at c, then f has a local minimum value at c. • If f’ does not change sign at c, then f has no local extreme value at c.

9. 1st Derivative Test Continued… At a Left Endpoint a: If f ’ < 0, for x near a, then f has a local maximum at a. If f ’ > 0, for x near a, then f has a local minimum at a. At a Right Endpoint b: If f ’ < 0, for x near b, then f has a local minimum at b. If f ’ > 0, for x near b, then f has a local maximum at b.

10. Concavity Test The graph of a twice-differentiable function y = f(x) is a) Concave up on any interval where y” > 0. b) Concave down on any interval where y” < 0. Example 2: Use the Concavity Test to determine the concavity of the given functions on the given intervals.

11. Point of Inflection (reminder) A point where the graph of a function has a tangent line (derivative exists & function is continuous) and where the concavity changes is a Point of Inflection. f” = 0

12. Example 3 Finding Points of Inflection Find all points of inflection of the graph of HOW? Find the 2nd derivative – recalling product and chain rules. Find critical points of 2nd derivative, set y” = 0 and solve for x (these are the x values of your points of inflection) Evaluate f(critical points) to find the y values of your points of inflection. (use original function!)

13. Example 4 Reading the Graph of the Derivative Caution: It is tempting to oversimplify a point of inflection as a point where the second derivative is zero, but that can be wrong for 2 reasons. • The 2nd derivative can be zero at a non-inflection point. Consider f(x) = x4 and figure 4.27 on p 209 • The 2nd derivative need not be zero at an inflection point. Consider f(x) = cube root of x. At x = 0 there is a vertical tangent and both f’ and f” fail to exist although there is a change in concavity. • The safest way to test for a point of inflection is to confirm algebraically that there was a sign change in the 2nd derivative!

14. Given the graph of the derivative function on p209, figure 4.25, answer the following questions about f, justifying each answer with information obtained from the graph of f’. • On what intervals is f increasing? (f’ +) • On what intervals is the graph of f concave up? (f” +) • At which x-coordinates does f have local extrema? (f’ = 0) • What are the x-coordinates of all inflection points of the graph of f? (f” = 0) • Use info from 1-4 to sketch a possible graph of f on the interval [-4,4].

15. Example 5 Studying Motion along a Line A particle is moving along the x-axis with position function , t ≥ 0. Find the velocity and acceleration – take first and 2nd derivative Describe the motion of the particle – use derivatives f ’ positive: increasing, right / f ’ negative: decreasing, left f ” positive: concave up, accelerating f ” negative: concave down, decelerating

16. Theorem 5 Second Derivative Test for Local Extrema • If f’(c) = 0 and f”(c) < 0, then f has a local maximum at x = c. 2. If f’(c) = 0 and f”(c) > 0, then f has a local minimum at x = c. Good news: Only need to know f’ and f” at c itself so easy to apply. Bad news: Test fails if f”(c) = 0 or f”(c) fails to exist. When that happens we revert to the 1st Derivative test.

17. Example 7: Using the 2nd Derivative Test Find the local extreme values of . Find f’ Find f” Find the critical points (f’(x) = 0) and test them with f”. Give your conclusions in sentences.

18. Example 8 Using f’ and f” to Graph f Let • Identify where the extrema of f occur. • Find the intervals on which f is increasing and the intervals on which f is decreasing. • Find where the graph of f is concave up and where it is concave down. • Sketch a possible graph for f.

19. Learning about Functions from Derivatives We are able to recover almost everything we need to know about a differentiable function y = f(x) by examining y’ and y”. The only info we can’t get from the derivative is where to place the graph on the coordinate plane, we need one point from f(x) to position the graph. Study the graphs on p213 to further develop an eye for finding f(x) from its derivative. Also, interactive derivatives at http://www.slu.edu/classes/maymk/GeoGebra/Derivatives.html

20. Example 9: Analyzing a Discontinuous Derivative A function f is continuous on the interval [-4,4]. The discontinuous function f ’, with domain [-4,0) U (0,2) U (2,4], is shown in figure 4.33 on page 214. • Find the x-coordinates of all local extrema ( where f ’ changes sign) and points of inflection (where f ” changes sign so f ’ changes direction) of f. b) Sketch a possible graph of f.

21. Homework Summary AP Test Practice P121 thru 124 (due Friday) Homework Page 216 Exercises 18-48 (3n, nЄI) Study for quiz on 4.1 thru 4.3