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.
f ( x) =
6
x2 + 3
2.
x2 + 1
g ( x) = 2
x −4
3.
m( x ) = x 4 − 4 x 3
4.
c( x) = −3 x5 + 5 x3
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)
x2 − 9
j ( x) = 2
x − 16
b)
g ( x) = 2 x 2 − 8