The Second Derivative
Feb 15, 2016
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