4'3 How Derivatives Affect the Shape of a Graph

1
4.3 How Derivatives Affect the Shape of a Graph
2
Increasing and Decreasing
How can we determine the intervals of x (from
left to right) for which f is increasing or
decreasing?
m lt 0
m gt 0
Decreasing
Increasing
Increasing
m gt 0
b
a
(a) If f '(x) gt 0 on an interval, then f is
increasing on that interval.
(b) If f '(x) lt 0 on an interval, then f is
decreasing on that interval.
3
The First Derivative Test
Can you use the slopes of tangent lines to help
you identify local maximums and minimums?
Maximum
m lt 0
m gt 0
m gt 0
b
a
Minimum
4
The First Derivative Test
Suppose that c is a critical number of a
continuous function f.
(a) If f '(x) changes from positive to negative
at c, then f has a local maximum at c.
(b) If f '(x) changes from negative to positive
at c, then f has a local minimum at c.
(c) If f '(x) does not changes sign at c, then f
has no local maximum or minimum at c.
5
Examples
For each of the following, (a) Find the intervals
of x for which f is increasing or decreasing. (b)
Find the local maximum and minimum values of f.
6
Concavity
What does concavity mean to you? Which parts of
the graph are concave up and which are concave
down? How can you tell?
Concave Upward
P
?
c
b
a
Concave Downward
If the graph of f lies above all its tangent
lines on an interval I, then it is called concave
upward on I. If the graph of f lies below all of
its tangent lines on I, it is called concave
downward on I.
7
Concavity
Sketch a graph of the first and second
derivative. Can these be used to determine
concavity?
P
?
c
a
b
8
Concavity
(a) If f ?(x) gt 0 for all x in I, then the graph
of f is concave upward on I.
(b) If f ?(x) lt 0 for all x in I, then the graph
of f is concave downward
Definition The point P on the curve y f (x) is
called an inflection point if f is continuous
there and the curve changes concavity.
9
Concavity
Can you use the first derivative and concavity to
determine if a value x c is a maximum or a
minimum?
P
?
a
b
10
The Second Derivative Test
Suppose f ? is continuous near c.
(a) If f '(c) 0 and f ?(c) gt 0, then f has a
local minimum at c.
(b) If f '(c) 0 and f ?(c) lt 0, then f has a
local maximum at c.
11
Examples
For each of the following, (a) Determine all
critical values of f. (b) Determine the intervals
of x for which the function is concave up or
concave down. (c) Find any points of inflection.
(d) Find the local minimum and maximum values.
12
Examples
(a) Determine the vertical and horizontal
asymptotes. (b) Find the intervals of increase
and decrease. (c) Find the local maximum and
minimum values. (d) Find the intervals of
concavity and the inflection points. (e) Sketch a
graph.