Example 2: Graph the 2nd Derivative, f″(x), Given the Graph of f′(x).

Sketch the graph of the 2 nd Derivative, f ( x ) , determine the concavity of the original graph, f ( x ) , and identify any inflection points based on the given derivative graph.

Step 1: Identify and then boldly label the graph you are working with.

In this example you have been given the derivative graph, the f ( x ) graph.

 

Step 2: Determine if you are being asked to go move down a level or move up a level.

In this example you are being asked to move down a level. You are starting with the derivative graph, f ( x ) , and being asked to move down a level, find the 2 nd Derivativegraph, f ( x ) .

Step 3 (If you are moving down ): Mark your given graph with plus (+), minus (-), and zero (0) based on the 1 st Derivative Triangle. The 1 st Derivative of f ( x ) is the 2 nd Derivative, f ( x ) .

 

Step 4 (If you are moving down ): Use your plus (+), minus (-), and zero (0) that you marked along with their estimated slope to determine the y-values on the graph you are moving down to.

 

The first points that you will always want to graph will be the zeros that you have marked. You know where a zero value will always be graphed, it will always be graphed on the x-axis . After that you want to work your way left to right keeping in mind that the steeper the graph the larger the number, and increasing means a positive slope, and decreasing means a negative slope. I find it helpful to use approximated values to keep track of it all. A steep positive slope I would use m=15 , or a flat negative slope I use m=-1 in my head or writing on the actual graph. If you are given a piecewise graph with constants and linear equations , you can actually use real values if needed. The derivative of a constant is always zero, and the derivative of a linear equation ( y= m x+b ) is always the m , the slope.

 

( x , y ) = ( x , slope   of the   Derivative Graph )

( x , y ) = ( x , m )

Final Result: In this example you would draw a final version of your derivative graph based on the sketch from Step 4 . Remember to always make sure your final answer passes the vertical line test to ensure it is a function. The sketch of the 2 nd Derivative, f ( x ) , given this derivative graph, f ( x ) ,would look like:

 

Concavity:

Where the 2 nd Derivative graph has positive y-values , the original graph, f ( x ) , is concave up.

The x-interval would be ( , 2 ) ( 2 , ) . Notice that you would not include the end points, and we use the union symbol to connect the two x-intervals .

Where the 2 nd Derivative graph has negative y-values , the original graph, f ( x ) , is concave down.

The x-interval would be ( 2 , 2 ) .

 

Inflection Point:

Inflection points occur when i) the 2 nd Derivative equals zero and ii) the concavity changes .

Here you have two inflection points.

First at x=-2 and again at x=2 . At those two x-values the 2 nd Derivative graph equals zero, the graph switches from positive to negative or negative to positive.

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