Section 2.5 The Second Derivative. The following two graphs represent the velocity a car is traveling in 3 seconds –Describe what is going on in each.

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Presentation transcript:

Section 2.5 The Second Derivative

The following two graphs represent the velocity a car is traveling in 3 seconds –Describe what is going on in each case\ –Who went farther? t t v v

Complete the work sheet What is the relationship between the concavity of the graph of f and the 2 nd derivative?

Complete the work sheet What is the relationship between the concavity of the graph of f and the 2 nd derivative?

If f’’ > 0 over an interval, than f’ is increasing and the graph of f is concave up on that interval If f’’ < 0 over an interval, than f’ is decreasing and the graph of f is concave down on that interval If f’’ = 0, then we have cases –The graph of f may still be concave up or concave down –If the concavity switches, we have an inflection point

The following table gives the number of passenger cars, C = f(t), in millions, in the US in the year t Is f’ positive or negative over the given periods? –What does it tell us about the situation? Is f’’ positive or negative over the given periods? –What does it tell us about the situation? t (year) C (cars, in millions)

Notations for 2 nd Derivative We are taking the derivative of the first derivative Note: