Download Finding increasing and decreasing intervals

Problem of the Day
An equation of the line tangent to the
graph of y = x + cos x at the point (0,
1) is
a) y = 2x + 1
d) y = x - 1
b) y = x + 1
e) y = 0
c) y = x
Problem of the Day
An equation of the line tangent to the
graph of y = x + cos x at the point (0,
1) is
a) y = 2x + 1
d) y = x - 1
b) y = x + 1
e) y = 0
c) y = x
Finding increasing and decreasing
intervals
1. Increasing function - graph moves up as x
moves
to the right.
2. Decreasing function - graph moves down as x
moves to the right.
What do you know about the slope of an
increasing function?
a decreasing function?
a constant function?
Finding increasing and decreasing
intervals function - graph moves up as x
1. Increasing
moves
to the right.
2. Decreasing function - graph moves down as x
moves to the right.
What do you know about the slope of an
increasing function?
a decreasing function?
a constant function?
In Calculus what gives you the slope of the
curve?
Finding increasing and decreasing
intervals
Theorem 3.5 - Derivative tests
If
f '(x) > 0
f '(x) < 0
f '(x) = 0
f is increasing
f is decreasing
f is constant
Additional derivative test
information
f '(x) goes from - to +
+
(relative minimum)
+
-
f '(x) goes from + to (relative maximum)
+
f '(x) goes from + to +
+
(neither*)
f '(x) goes from - to (neither*)
(*strictly monotone - neither on the
interval)
-
Example
for
find the increasing/decreasing intervals
4
3
f(x) = x - 2x
Example
for
find the increasing/decreasing intervals
4
3
f(x) = x - 2x
1) need derivative
Example
for
find the increasing/decreasing intervals
4
3
f(x) = x - 2x
3
2
1) need derivative
f '(x) = 4x - 6x
2) find critical numbers
Example find the increasing/decreasing intervals
for
3
2
4
3
1) need derivative
f '(x) = 4x - 6x
f(x) = x - 2x
2) find critical numbers
3
2
0 = 4x - 6x
2
0 = 2x (2x - 3)
2
2x = 0
or 2x - 3 = 0
x = 0 or 3/2
Example find the increasing/decreasing intervals
for
3
2
4
3
1) need derivative
f '(x) = 4x - 6x
f(x) = x - 2x
2) find critical numbers x = 0 or 3/2
3) describe interval
behavior
Example find the increasing/decreasing intervals
for
3
2
4
3
1) need derivative
f '(x) = 4x - 6x
f(x) = x - 2x
2) find critical numbers x = 0 or 3/2
3) describe interval
behavior
if x < 0
if x > 0 and < 3/2
if x > 3/2
3
2
f '(-1) = 4(-1) - 6(-1) = -10
3
2
f '(½) = 4(½) - 6(½) = -1
3
2
f '(2) = 4(2) - 6(2) = 8
Example
for
find the increasing/decreasing intervals
4
3
3
2
f(x)
=
x
2x
1) need derivative
f '(x) = 4x - 6x
2) find critical numbers x = 0 or 3/2
3) describe interval
behavior
f '(x) - 0
x
0
0
+
3/2
3
2
3
2
f '(-1) = 4(-1) - 6(-1) = -10
f '(½) = 4(½) - 6(½) = -1
3
2
f '(2) = 4(2) - 6(2) = 8
Example find the increasing/decreasing intervals
for
4
3
3) describe interval
f(x) = x - 2x
- 0 +
behavior f '(x) - 0
3/2
x
0
so what does the graph look
like?
Example find the increasing/decreasing intervals
for
4
3
3) describe interval
f(x) = x - 2x
- 0 +
behavior f '(x) - 0
3/2
x
0
so what does the graph look
like?
Example
for
find the increasing/decreasing intervals
f(x) =
x
domain
x+1
Note: -1 is not in
Example
for
find the increasing/decreasing intervals
f(x) =
domain
1)
need derivative
x
x -1+ 1
Note: -1 is not in
f(x) = x(x + 1)
f
-2
-1
'(x) = x (-1)(x + 1) + 1(x + 1)
-2
= (x + 1) (-x + x + 1)
=
1
2
(x + 1)
Example
for
find the increasing/decreasing intervals
f(x) =
domain
1)
need derivative
x
Note: -1 is not in
f x'(x)
+ 1=
2) find critical numbers
0=
1
2
(x + 1)
1
2
(x + 1)
only critical number is -1 which is not in
domain
Example
for
find the increasing/decreasing intervals
f(x) =
1)
need derivative
domain
f '(x) =
1
(x + 1)2
x
Note: -1 is not in
2) find critical numbers
x+1
0=
1
(x + 1)2
only critical number is -1 which is not in
domain
3) describe interval
behavior
f '(x) + undef +
x
-1
f '(-2) = 1
f '(0) = 1
Example
for
find the increasing/decreasing intervals
f(x) =
1)
need derivative
domain
f '(x) =
1
(x + 1)2
x
Note: -1 is not in
2) find critical numbers
x+1
0=
1
(x + 1)2
only critical number is -1 which is not in
domain
3) describe interval
behavior
f '(x) + undef +
x
-1
What does it look like?
f '(-2) = 1
f '(0) = 1
Example
for
find the increasing/decreasing intervals
f(x) = x + 3
domain
x2
Note: 0 is not in
Example
for
find the increasing/decreasing intervals
f(x) = x + 3
-1
domain
Note: 0 is not in
-2
f(x) = x + 3x
x -3
f
-2
'(x) = -x + 3(-2)x
-3
= -x (x + 6)
2
f '(x)
x
-
0
-6
critical numbers are 0 and f '(-8) = -.003
+ undef6 f '(1) = -7
0
f '(-1) = 5
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