1. The Second Derivative

2. The graph above is the derivative f ‘(x) of a function y=f(x)
The graph above is the derivative f ‘(x) of a function y=f(x). What information about f(x) can you obtain from its derivative? Be as detail as possible.

3. GIVEN THE GRAPH OF f(x) CHOOSE
In the next slides you will be presented (on the left) with the graph of a function, and on the right with some choices for its derivative function. Choose its derivative function and
Give reason for your choice
For each of the graphs you did not choose give one reason why it was not chosen

6. GIVEN THE GRAPH OF f’(x) CHOOSE
In the next slides you will be presented (on the left) with the graph of the derivative function, and on the right some choices for the graph of the function (an anti-derivative function). Choose the graph corresponding to the function.
Give reason for your choice
For each of the graphs you did not choose give one reason why it was not chosen

9. GIVEN THE GRAPH OF f’(x) CONSTRUCT A
Click on the link below to work on this activity.
The graph in red is the derivative function of a function f(x). I will do the first one with your help. You then practice on one. Finally, we will see what group is the best. Total time: 10 minutes
Reconstructing a function from f ’

10. Second Derivative

11. Function and Its First Derivative

12. First Derivative and Second Derivative
The derivative function of the derivative function is called the second derivative function

14. Function and Its Second Derivative

15. Function, First, and Second Derivatives
Local Max of f ’(x)
Inflection Point of f(x)
Change Concave up to concave down
f “ changes from positive to negative

16. Function, First, and Second Derivatives
f “ changes from negative to positive
Inflection Point of f(x)
Change Concave down to concave up
Local min of f ’(x)

17. PRACTICE Select the correct answer in each case.
On an interval where a function is concave up, the first derivative is
positive negative
Increasing decreasing
any of the above is possible
On an interval where a function is concave up, the second derivative is
Positive negative
any of the above is possible

18. Identifying f, f’, f’’
You will be presented with three graphs. They represent f, f ‘, f” . Determine which one is which. Give reasons for your choice

21. Higher Order Derivatives

22. Practice For each of the functions below determine: Domain
Where the function is increasing/decreasing
Where the function is concave up/down
Any critical points
Any inflection points