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Lesson 11-filled

# Lesson 11-filled - Math 1314 Lesson 11 Applications of the...

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Math 1314 Lesson 11 Applications of the Second Derivative Concavity Earlier in the course, we saw that the second derivative is the rate of change of the first derivative. The second derivative can tell us if the rate of change of the function is increasing or decreasing. In business, for example, the first derivative might tell us that our profits are increasing, but the second derivative will tell us if the pace of the increase is increasing or decreasing. Example 1 : From these graphs, you can see that the shape of the curve change differs depending on whether the slopes of tangent lines are increasing or decreasing. This is the idea of concavity . Definition : Let the function f be differentiable on an interval ( a , b ). Then f is concave upward on ( a , b ) if f is increasing on ( a , b ) and f is concave downward on ( a , b ) if f is decreasing on ( a , b ). Determining Where a Function is Concave Upward and Where it is Concave Downward By Analyzing the Sign of the Second Derivative Algebraically We can also determine concavity algebraically. The procedure for doing this should look pretty familiar: 1. Find the second derivative of the function. 2. Determine all values of x for which 0 ) ( = x f or is undefined.

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Lesson 11-filled - Math 1314 Lesson 11 Applications of the...

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