AP CALCULUS BC Section 3.4: Concavity and the Second Derivative Test, pg. 190

DO NOW:

1. Given the graph of the derivative function of f(x), state: a) Where f(x) is increasing:

__________________________

b) Where f(x) is decreasing:

__________________________

c) The x-coordinate for all extrema and corresponding classification:

_____________________________________________________

2. Given the graph of the derivative function of f(x), state: d) Where f(x) is increasing:

__________________________

e) Where f(x) is decreasing:

__________________________

f) The x-coordinate for all extrema, and corresponding classification:

_____________________________________________________

CONCAVITY

SAMPLE PROBLEMS

For each of the following functions analyze a) Intervals of increasing, decreasing b) Points of extrema c) Points of inflection d) Concavity e) Graph

1. 2

6( )

3f x

x=

+

2.

2

2

1( )

4

xg x

x

+=

−

3. 4 3( ) 4m x x x= −

4. 5 3( ) 3 5c x x x= − +

5. Let’s go back to the DO NOW activity and discuss the concavity of the original function f(x).

CLASSWORK – HOMEWORK

1. You are given the graph of the derivative of f(x), sketch a possible graph for f(x).

2. Sketch a possible graph of f(x) given the following information.

3. Given f(x) determine: Domain, Range, Asymptote(s), Hole(s) if any, intercepts, critical points, local and absolute extrema, point(s) of inflection, intervals of inc. and dec., intervals of concavity. SKETCH. Label all asymptotes and points of interest.

a)

2

2

9( )

16

xj x

x

−=

−

b) 2( ) 2 8g x x= −