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.

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

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 (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

GIVEN THE GRAPH OF f’(x) CHOOSE THE GRAPH OF f (x)

GIVEN THE GRAPH OF f’(x) CONSTRUCT A

GRAPH OF f (x)

Reconstructing a function from f ’

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

Second Derivative

Function and Its First Derivative

First Derivative and Second Derivative

The derivative function of the derivative function is called the second derivative function

Function and Its Second Derivative

Function, First, and Second Derivatives

Inflection Point of f(x)

Change Concave up to concave

down

Local Max of f ’(x)

f “ changes from positive to negative

Function, First, and Second Derivatives

Inflection Point of f(x)

Change Concave down to concave up

Local min of f ’(x)

f “ changes from negative to positive

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

PRACTICE

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

Higher Order Derivatives

PracticeFor 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