Section 2.5The 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 2nd derivative?

• Complete the work sheet• What is the relationship between the concavity of

the graph of f and the 2nd 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) 1940 1950 1960 1970 1980

C (cars, in millions) 27.5 40.3 61.7 89.3 121.6

Notations for 2nd Derivative

• We are taking the derivative of the first derivative

• Note:

2

2If ( ) then ''( )

d yy f x f x

dx

2

2

d dy d y

dx dx dx

2

2''( )

x a

d yf a

dx