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The Second Derivative

The Second Derivative. 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. . GIVEN THE GRAPH OF f(x) CHOOSE THE GRAPH OF f ‘(x).

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The Second Derivative

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  1. The Second Derivative

  2. 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 THE GRAPH OF f ‘(x) 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 functionand • Give reason for your choice • For each of the graphs you did not choose give one reason why it was not chosen

  4. GIVEN THE GRAPH OF f’(x) CHOOSE THE GRAPH OF f (x) 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 (ananti-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

  5. GIVEN THE GRAPH OF f’(x) CONSTRUCT A GRAPH OF f (x) 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 ’

  6. Second Derivative

  7. Function and Its First Derivative

  8. First Derivative and Second Derivative The derivative function of the derivative function is called the secondderivative function

  9. Function and Its Second Derivative

  10. 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

  11. 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)

  12. 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 • Increasing -- decreasing • any of the above is possible

  13. 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

  14. Higher Order Derivatives

  15. 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

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