13 graphs of log and exp functions
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Graphs of Log and Exponential Functions
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
In this section we examine the graphs of functions related to log and exponential functions.
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
Graphs of Log and Exponential Functions
exx
In this section we examine the graphs of functions related to log and exponential functions.
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
Graphs of Log and Exponential Functions
exx
We gather the following information:
In this section we examine the graphs of functions related to log and exponential functions.
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept* The behavior as x ±∞ and as x v-asymptotes
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept
* The critical points where f ' = 0 or f ' is not defined* The behavior as x ±∞ and as x v-asymptotes
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept
* The critical points where f ' = 0 or f ' is not defined* The inflection points
* The behavior as x ±∞ and as x v-asymptotes
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept
* The critical points where f ' = 0 or f ' is not defined* The inflection points
Domain: All real numbers
* The behavior as x ±∞ and as x v-asymptotes
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
In this section we examine the graphs of functions related to log and exponential functions.
Graphs of Log and Exponential Functions
exx
We gather the following information:
* The domain, find the vertical asymptotes, if any.* The x intercept and y intercept
* The critical points where f ' = 0 or f ' is not defined* The inflection points
Domain: All real numbers
* The behavior as x ±∞ and as x v-asymptotes
x intercept and y intercept: (0, 0)
Example: Sketch the graph y = . Find the criticalpoints and the inflection points, if any.
Graphs of Log and Exponential Functions
The behavior as x ±∞:
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞0+
Lim
Graphs of Log and Exponential Functions
exx
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
Graphs of Log and Exponential FunctionsDomain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
y
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
=
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
=
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 =
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1,= and y=1/e.
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
For the inflection point:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. = and y=1/e.
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. =
f " = -e-x(1 – x) – e-x
and y=1/e.For the inflection point:
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. =
f " = -e-x(1 – x) – e-x = -2e-x + xe-x
and y=1/e.For the inflection point:
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. =
f " = -e-x(1 – x) – e-x = -2e-x + xe-x
= e-x(2 – x)
and y=1/e.For the inflection point:
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. =
f " = -e-x(1 – x) – e-x = -2e-x + xe-x
= e-x(2 – x) = 0 x = 2 and y = 2/e2.
and y=1/e.For the inflection point:
Lim
Graphs of Log and Exponential Functions
exx
The critical points where f ' = 0 or f ' is not defined:
The behavior as x ±∞:
x∞=
L'Hopital's Rule
Lim ex1
x∞= 0+
Lim exx
x-∞= -∞
0+
f ' = e2xex – xex
e2xex(1 – x)
= e-x(1 – x) = 0 when x = 1. =
Put all this information together:
f " = -e-x(1 – x) – e-x = -2e-x + xe-x
= e-x(2 – x) = 0 x = 2 and y = 2/e2.
and y=1/e.For the inflection point:
Graphs of Log and Exponential FunctionsDomain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
(1, 1/e) critical point
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
(1, 1/e) critical point
(2, 2/e2 ) inflection point
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
(1, 1/e) critical point
(2, 2/e2 ) inflection point
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
(2, 2/e2 ) inflection point
x
y
Graphs of Log and Exponential Functions
x-int. and y-int.: (0, 0)
y 0+ as x ∞
y -∞ as x -∞
(1, 1/e) critical point
(2, 2/e2 ) inflection point
Domain: All real numbersx intercept and y intercept: (0, 0)as x +∞, y 0+ as x –∞, y –∞,
(1, 1/e) critical point
(2, 2/e2 ) inflection point
x
y
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
-x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
-x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept:
-x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1),
-x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞:
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e
x∞
-x2 -x2
x- ∞
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points:
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: f ' = -2xe = 0 -x2
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
f ' = -2xe = 0 x = 0 and y = 1
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
The inflection points:
f ' = -2xe = 0 x = 0 and y = 1
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
The inflection points: f " = -2e + 4x2e -x2 -x2
f ' = -2xe = 0 x = 0 and y = 1
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
The inflection points: f " = -2e + 4x2e = -2e (1 – 2x2)-x2 -x2 -x2
f ' = -2xe = 0 x = 0 and y = 1
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: f ' = -2xe = 0 x = 0 and y = 1 -x2
The inflection points: f " = -2e + 4x2e = -2e (1 – 2x2)-x2 -x2 -x2
1 – 2x2 = 0
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
The inflection points: f " = -2e + 4x2e = -2e (1 – 2x2)-x2 -x2 -x2
1 – 2x2 = 0 x = ±1/2
f ' = -2xe = 0 x = 0 and y = 1
Example: Sketch the graph y = e .
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: All real numbers
x intercept and y intercept: y-int: (0, 1), there is no x-int. (e = 0)
-x2
-x2
The behavior as x ±∞: Lim e = lim e = 0+
x∞
-x2 -x2
x- ∞
The critical points: -x2
The inflection points: f " = -2e + 4x2e = -2e (1 – 2x2)-x2 -x2 -x2
1 – 2x2 = 0 x = ±1/2 and y = 1/e
f ' = -2xe = 0 x = 0 and y = 1
Graphs of Log and Exponential FunctionsDomain: All real numbersy-int: (0, 1), there is no x-int.
(±1/2,1/ e ) are inflection points
as x ± ∞, y 0+ (0, 1) critical point
x
y
Graphs of Log and Exponential FunctionsDomain: All real numbersy-int: (0, 1), there is no x-int.
(±1/2,1/ e ) are inflection points
as x ± ∞, y 0+ (0, 1) critical point
y-int: (0, 1), critical pt.
x
y
Graphs of Log and Exponential Functions
y 0+ as x ∞y 0+ as x -∞
Domain: All real numbersy-int: (0, 1), there is no x-int.
(±1/2,1/ e ) are inflection points
as x ± ∞, y 0+ (0, 1) critical point
y-int: (0, 1), critical pt.
x
y
Graphs of Log and Exponential Functions
y 0+ as x ∞y 0+ as x -∞
(1/2, e-1/2 ) inf. pt(-1/2, e-1/2 ) inf. pt
y-int: (0, 1), critical pt.
Domain: All real numbersy-int: (0, 1), there is no x-int.
(±1/2,1/ e ) are inflection points
as x ± ∞, y 0+ (0, 1) critical point
x
y
Graphs of Log and Exponential Functions
y 0+ as x ∞y 0+ as x -∞
(1/2, e-1/2 ) inf. pt(-1/2, e-1/2 ) inf. pt
y-int: (0, 1), critical pt.
Domain: All real numbersy-int: (0, 1), there is no x-int.
(±1/2,1/ e ) are inflection points
as x ± ∞, y 0+ (0, 1) critical point
x
y
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int.
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞:
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x
x∞
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x
x∞ x∞
L'Hopital's Rule
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x = 0
x∞ x∞
L'Hopital's Rule
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x = 0
x∞ x∞
L'Hopital's Rule
The behavior as x 0+:
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x = 0
x∞ x∞
L'Hopital's Rule
The behavior as x 0+: Lim Ln(x)/x x0+
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x = 0
x∞ x∞
L'Hopital's Rule
The behavior as x 0+: Lim Ln(x)/xx0+
-∞
0+
Example: Sketch the graph y = Ln(x)/x.
Find the critical points and the inflection points, if any.
Graphs of Log and Exponential Functions
Domain: x > 0
x intercept and y intercept: No y-int. For x-int: Ln(x) = 0 x = 1, so (1, 0) is the x-int.
The behavior as x ∞: Lim Ln(x)/x = lim 1/x = 0
x∞ x∞
L'Hopital's Rule
The behavior as x 0+: Lim Ln(x)/x = -∞x0+
-∞
0+
The critical pt:
Graphs of Log and Exponential Functions
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e, , and y=1/e.
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3)
and y=1/e.
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
and y=1/e.
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
So -3 + 2Ln(x) = 0
and y=1/e.
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
and y=1/e.
and y=(3/2)/e3/2 .
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
x-int: (1, 0)
and y=1/e.
and y=(3/2)/e3/2 .
x
y
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
y 0+ as x ∞
x-int: (1, 0)
y - ∞ as x 0+
and y=1/e.
and y=(3/2)/e3/2 .So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
x
y
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
y 0+ as x ∞
(e, 1/e) critical pt.
x-int: (1, 0)
y - ∞ as x 0+
and y=1/e.
and y=(3/2)/e3/2 .So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
x
y
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
y 0+ as x ∞
(e, 1/e) critical pt.
x-int: (1, 0)
y - ∞ as x 0+
(3/2, 3e-3/2/2 ) inf. pt
and y=1/e.
and y=(3/2)/e3/2 .So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
x
y
The critical pt:
Graphs of Log and Exponential Functions
f ' = 1/x2 – Ln(x)/x2 = 0 1 – Ln(x) = 0 or x = e,
The inflection pt:
f " = -2/x3 – (1/x3 – 2Ln(x)/x3) = -3/x3 + 2Ln(x)/x3 = 0
y 0+ as x ∞
(e, 1/e) critical pt.
x-int: (1, 0)
y - ∞ as x 0+
(3/2, 3e-3/2/2 ) inf. pt
and y=1/e.
and y=(3/2)/e3/2 .
Graph y = x2e-2x
Graph y = x2Ln(x)
Your Turn:
So -3 + 2Ln(x) = 0 Ln(x) = 3/2 or x = e3/2,
x
y
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