Building Understanding for the Connection between f, f’, and f”

1. Building Understanding for the Connection between f, f’, and f” Jim Rahn www.jamesrahn.com James.rahn@verizon.net

2. Equity & Access The College Board and the Advanced Placement Program encourage teachers, AP Coordinators, and school administrators to make equitable access a guiding principle for their AP programs. The College Board is committed to the principle that all students deserve an opportunity to participate in rigorous and academically challenging courses and programs. All students who are willing to accept the challenge of a rigorous academic curriculum should be considered for admission to AP courses. The Board encourages the elimination of barriers that restrict access to AP courses for students from ethnic, racial, and socioeconomic groups that have been traditionally underrepresented in the AP Program. Schools should make every effort to ensure that their AP classes reflect the diversity of their student Population.

3. Set your viewing window to: [-2, 2, 1, -20, 20, 10, 1] • Enter the equations • y1 = -2x2and • y2 = nderiv(y1,x,x) or . • Create their graphs. Set y2 to be a bold line.

4. When is the graph of y1 increasing? • When is the graph of y1 decreasing? • When does the graph of y1 reach a maximum? • When is the graph of the derivative of y1 (y2) positive? • When is the graph of the derivative of y1 (y2) negative? • When does the graph of the derivative of y1 (y2) change from negative to positive and equal zero?

5. How is it possible to describe the behavior of y1 by looking at its derivative?

6. Enter the equations • Set your viewing window to:[-2,2,1,-2,2,1,1]. • Create a graph of y1, y2, and y3. Set y2 to be a bold line. Make a sketch of each.

7. When is the graph of y1 increasing? • When is the graph of y1 decreasing? • When does the graph of y1 reach a maximum? • When does the graph of y1 reach a minimum? • When is the graph of the derivative of y1 (y2) positive? • When is the graph of the derivative of y1 (y2) negative? • When does the graph of the derivative of y1 (y2) change from negative to positive and equal zero? • When does the graph of the derivative of y1 (y2) change from positive to negative and equal zero?

8. From the graph of a derivative of a function how can you tell if the original function is increasing? • From the graph of a derivative of a function how can you tell if the original function is decreasing? • From the graph of a derivative of a function, how can you tell the location of a local maximum on the original function? • From the graph of a derivative of a function, how can you tell the location of a local minimum on the original function? • The graph of the derivative of the function is positive. • The graph of the derivative of the function is negative. • The graph of the derivative of the function has a zero and is changing from being positive to being negative. • The graph of the derivative of the function has a zero and is changing from being negative to being positive.

9. When is the graph of y1 concave up? • When is the graph of y1 concave down? • When does the graph of y1 change concavity or have a point of inflection? • When is the graph of y3 positive? • When is the slope of y2 positive? • When is the graph of y3 negative? • When is the slope of y2 negative? • When does the graph of y3 equal change sign from positive to negative or negative to positive? • When does the slope of y2 change sign?

10. How can you tell from the graph of the second derivative of a function, where the original function is concave up? • How can you tell from the graph of the second derivative of a function, where the original function is concave down? • How can you tell from the graph of the second derivative of a function, where the original function has a point of inflection? • When the second derivative graph is positive. • When the slope of the first derivative is positive. • When the graph of the second derivative is negative. • When the slope of the first derivative is negative. • When the graph of the second derivative changes sign. • When the slope of the first derivative changes sign.

11. Let’s Apply What We Have Learned • Set your window to [-4.7,4.7, 1, -12, 12, 2,1]. • Let represent the derivative of a function y = f(x). • Create a graph of y1 or the derivative of y=f(x). • Make a sketch of y=f(x) below. Sketch of the graph of f Graph of Y1 or the derivative of f

12. Describe how you knew where y=f(x) was going to be increasing, decreasing, and any location of relative maximums or minimums.

13. What would the derivative of y1 look like? • What would this say about the concavity of y=f(x)? Explain why.

14. Set your window to [-1, 1.5, 1, -2, 4, 1, 1]. • Suppose represents the derivative of a function y = f(x). • Let be the second derivative of y = f(x).

15. Graph y1 (the derivative of y=f(x)). • Graph of y2 (the second derivative of y=f(x)) in the same window. • Make a sketch of each in the space below.

16. Make some observations about the behavior of the function f by studying the graph of the derivative of f or y1 and the graph of the second derivative of f or y2. • Use the information you collected about y1 and y2 to create a graph of y=f(x) in the space below.

17. 2010 AB 5

18. If this is a graph of f’(x), create a possible graph of f. Given the graph of f’(x), create a graph of f”.

19. These graphs were based upon

20. Connecting Graphs of f, f ', and f " www.jamesrahn.com Calculus AP Understanding the Derivative