Warm-up
1.
Find all values of c on the interval that satisfy
the mean value theorem.
f ( x)  x( x 2  x  2) on [-1,1]
2. Find where increasing and decreasing.
f ( x)  x 2  x  2
Table of Contents
• 26. Section 4.4 The Shape of a Graph
The Shape of a graph
• Essential Question – What is the 2nd
derivative test and what does it tell
you about a function?
Concavity
• Concave up – would catch
water
• Concave down – water
would roll off curve
• Concave up – curve lies
above tangents
• Concave down – curve lies
below tangents
Concavity test
• Concave up where y’ is increasing
(y” > 0)
• Concave down where y’ is decreasing
(y” < 0)
Example
• Where is this concave up and where
concave down?
y  x on [3,10]
2
y '  2x
y"  2
2 is always positive,
so always concave up
Example
• Where is this concave up and where
concave down?
y  3  sin x on (0,2 )
y '  cos x
y "   sin x
sin is + on (0, ) so y" is negative
concave down
sin is - on ( , 2 ) so y" is positive
concave up
Points of Inflection
• Points where concavity changes
• Y”=0 or is undefined at points of
inflection
• A graph crosses its tangent at point
of inflection
Example
• Find all points of inflection of x  4 x
4
y '  4 x3  12 x 2
Plug in values in each interval to f”
y ''  12 x 2  24 x
12 x( x  2)  0
0
Points of inflection
2
x  0 and x  2
3
Example
• Use the graph of f to estimate where
f’ and f” are 0, positive and negative
f '  0 x  1.5, 0, 0.5
f '  0 (1.5, 0) (0.5, )
f '  0 (,1.5) (0, 0.5)
f ''  0 x  1, 0.25
f ''  0 (, 1) (0.25, )
f ''  0 (1, 0.25)
Looking at a graph
• On intervals f is
• f’ is pos
increasing
• f’ is increasing, f’’ is pos
• On intervals f is
concave up
• At local extremes of f • f’ =0
• Inflection points of f
• f’’ = 0
Particle movement
• A particle is moving along x-axis
x(t )  2t 3  14t 2  22t  5
• Find velocity and acceleration and
describe motion
v(t )  6t 2  28t  22
t  1, t  11/ 3
Going right until t=1, then left until t=3.7, then right
a (t )  12t  28
t 7/3
Slowing down before t=2.3, then speeding up
Second derivative test
for local extrema
• If f’(c)=0 and f’’(c)<0, then f has a
local max at x=c
• If f’(c)=0 and f’’(c)>0, then f has a
local min at x=c.
Example
x3  12 x  5
y '  3x2  12
y "  6x
3( x 2  4)  0
y "(2)  12
x  2
local min
y "(2)  12 local max
Assignment
• Pg 243 #1-13 odd, 22, 23, 29, 37, 43,
53-58 all