Transcript 3 3 Inc Dec 1st Derv Test

3-3: Increasing/Decreasing Functions and the 1st
Derivative Test
Objectives:
Assignment:
1. To determine the
intervals on which a
function is increasing or
decreasing
• P. 186-189: 1-15 odd, 9598
2. To apply the First
Derivative Test to find
relative extrema
• P. 186-189: 21-33 eoo,
41-49 eoo, 55-64, 72, 76,
79, 80, 87, 99
Warm-Up
Given the graph of 𝑓′(𝑥),
what must be true about
the graph of 𝑓(𝑥)?
6
f'(x)
4
2
5
2
4
6
Objective 1
You will be able to
determine the
intervals on which
a function is
increasing or
decreasing
Exercise 1
6
4
2
5
Let 𝑓(𝑥) be defined
by the graphed
function.
Intuitively, what does
it mean for 𝑓(𝑥) to
be increasing on
(0, 3)?
Decreasing on (5, 8)?
Exercise 2
6
4
2
5
Find 𝑓(1) and 𝑓(2).
Since 𝑓(𝑥) is
increasing on (0,3)
and
𝑥1 = 1 < 𝑥2 = 2,
what must be true
about 𝑓(1) and
𝑓(2)?
Exercise 3
6
4
2
5
Find 𝑓(6) and 𝑓(7).
Since 𝑓(𝑥) is
decreasing on (0,3)
and
𝑥1 = 1 < 𝑥2 = 2,
what must be true
about 𝑓(6) and
𝑓(7)?
Increasing Functions
From left to right, the graph goes up
6
4
2
5
A function 𝑓 is
increasing on an
interval if for any two
numbers 𝑥1 and 𝑥2 in the
interval, 𝑥1 < 𝑥2 implies
𝑓 𝑥1 < 𝑓 𝑥2 .
Decreasing Functions
Constant
6
4
2
A function 𝑓 is
decreasing on an
interval if for any two
numbers 𝑥1 and 𝑥2 in the
interval, 𝑥1 < 𝑥2 implies
𝑓 𝑥1 > 𝑓 𝑥2 .
5
From left to right, the graph goes down
Investigation
In this Investigation,
we’ll be exploring
the relationship
between the
graph of a
function and its
derivative.
Testing for Increasing/Decreasing
Let 𝑓 be a function that is
continuous on [𝑎, 𝑏] and
differentiable on (𝑎, 𝑏).
Constant
6
𝑓′ 𝑥 = 0
If 𝑓′ (𝑥) > 0 for all 𝑥 in (𝑎, 𝑏),
1. Increasing
then 𝑓 is increasing on [𝑎, 𝑏].
4
If 𝑓 ′ 𝑥 = 0 for all 𝑥 in (𝑎, 𝑏),
2. Constant
then 𝑓 is constant on [𝑎, 𝑏].
2
𝑓′ 𝑥 > 0
𝑓′ 𝑥 < 0
5
If 𝑓 ′ 𝑥 < 0 for all 𝑥 in (𝑎, 𝑏),
3. Decreasing
then 𝑓 is decreasing on [𝑎, 𝑏].
Exercise 4
Find the open intervals on which
3 2
3
𝑓(𝑥) = 𝑥 − 𝑥 is increasing or
2
decreasing.
If
,
then is decreasing on
the test interval.
If
,
then is constant on
the test interval.
If
,
then is increasing on
the test interval.
Determine the sign of
one test value in
each interval.
Locate the critical
numbers of in
,
and use them to
determine test
intervals.
Increasing/Decreasing Algorithm
Let 𝑓 be continuous on (𝑎, 𝑏). To find the
open intervals on which 𝑓 is increasing or
decreasing:
Exercise 5
Find the open intervals on which 𝑓(𝑥) = 𝑥 3 is
increasing or decreasing.
Exercise 6
A function is strictly monotonic on an
interval if it is either increasing on the entire
interval or decreasing on the entire interval.
Find two functions that are strictly monotonic
over some interval, one increasing and the
other decreasing.
Objective 2
You will be able to apply the First
Derivative Test to find relative
extrema
Exercise 7
Given the graph of 𝑓′(𝑥),
what must be true about
the graph of 𝑓(𝑥)?
Exercise 8
Given the graph of 𝑓′(𝑥),
what must be true about
the graph of 𝑓(𝑥)?
Exercise 9
Given the graph of 𝑓′(𝑥),
what must be true about
the graph of 𝑓(𝑥)?
First Derivative Test
Let 𝑐 be a critical number of a function 𝑓 that is
continuous on an open interval 𝐼 containing 𝑐. If
𝑓 is differentiable on the interval, except possibly
at 𝑐, then 𝑓(𝑐) can be classified as follows.
If 𝑓′(𝑥) changes from
negative to positive at 𝑐,
then 𝑓 has a relative
minimum at 𝑐, 𝑓 𝑐 .
First Derivative Test
Let 𝑐 be a critical number of a function 𝑓 that is
continuous on an open interval 𝐼 containing 𝑐. If
𝑓 is differentiable on the interval, except possibly
at 𝑐, then 𝑓(𝑐) can be classified as follows.
If 𝑓′(𝑥) changes from
positive to negative at 𝑐,
then 𝑓 has a relative
maximum at 𝑐, 𝑓 𝑐 .
First Derivative Test
Let 𝑐 be a critical number of a function 𝑓 that is
continuous on an open interval 𝐼 containing 𝑐. If
𝑓 is differentiable on the interval, except possibly
at 𝑐, then 𝑓(𝑐) can be classified as follows.
If 𝑓′(𝑥) positive on both
sides of 𝑐 or negative on
both sides of 𝑐, then
𝑓(𝑐) is neither a relative
minimum nor maximum.
First Derivative Test
Let 𝑐 be a critical number of a function 𝑓 that is
continuous on an open interval 𝐼 containing 𝑐. If
𝑓 is differentiable on the interval, except possibly
at 𝑐, then 𝑓(𝑐) can be classified as follows.
If 𝑓′(𝑥) positive on both
sides of 𝑐 or negative on
both sides of 𝑐, then
𝑓(𝑐) is neither a relative
minimum nor maximum.
Exercise 10
Find the relative extrema of the function
1
𝑓(𝑥) = 𝑥 − sin 𝑥 in the interval 0,2𝜋 .
2
Exercise 11
Find the relative extrema of
𝑓 𝑥 = 𝑥 2 − 4 2/3 .
Exercise 12
Find the relative extrema of 𝑓 𝑥 =
𝑥 4 +1
.
2
𝑥
3-3: Increasing/Decreasing Functions and the 1st
Derivative Test
Objectives:
Assignment:
1. To determine the
intervals on which a
function is increasing or
decreasing
• P. 186-189: 1-15 odd, 9598
2. To apply the First
Derivative Test to find
relative extrema
• P. 186-189: 21-33 eoo,
41-49 eoo, 55-64, 72, 76,
79, 80, 87, 99