Definition
Curvature
Curve Sketching
Second Derivative Test
Higher Derivatives
Bernd Schröder
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
x
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
a
Bernd Schröder
Higher Derivatives
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
y
6
v
v
a
Bernd Schröder
Higher Derivatives
b
x
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
2. Velocities can change, just like positions can. The rate of
change of the velocity is the acceleration.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Do We Need Higher Derivatives?
1. Just because we know a function is, say, decreasing, does
not mean we really know the function’s behavior.
2. Velocities can change, just like positions can. The rate of
change of the velocity is the acceleration.
3. Higher order derivatives can be used to approximate
functions with (Taylor) polynomials.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. Let f be a function whose derivative f 0 is
differentiable, too.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. Let f be a function whose derivative f 0 is
differentiable, too.
Then f is called twice differentiable and we
00
0 0
denote f := f .
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. Let f be a function whose derivative f 0 is
differentiable, too.
Then f is called twice differentiable and we
00
0 0
denote f := f . In general, if f is n-times differentiable with
nth derivative f (n) , then f is called (n + 1)-times differentiable
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. Let f be a function whose derivative f 0 is
differentiable, too.
Then f is called twice differentiable and we
00
0 0
denote f := f . In general, if f is n-times differentiable with
nth derivative f (n) , thenf is called
(n + 1)-times differentiable
if and only if f (n+1) := f (n)
Bernd Schröder
Higher Derivatives
0
exists.
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. Let f be a function whose derivative f 0 is
differentiable, too.
Then f is called twice differentiable and we
00
0 0
denote f := f . In general, if f is n-times differentiable with
nth derivative f (n) , thenf is called
(n + 1)-times differentiable
if and only if f (n+1) := f (n)
0
exists.
Another notation for the nth derivative is
Bernd Schröder
Higher Derivatives
dn f
(x) := f (n) (x).
dxn
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
finish line
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
finish line
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
finish line
u
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
finish line
u
car 1
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Which Car Do You Think Will Win?
s
6
finish line
u
car 1
car 2
t
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b]
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b].
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b].
It is called concave down on [a, b]
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b].
It is called concave down on [a, b] if and only if f 00 (x) < 0 for
all x in [a, b].
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b].
It is called concave down on [a, b] if and only if f 00 (x) < 0 for
all x in [a, b].
A point x where the function changes from concave down to
concave up or vice versa is called an inflection point.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Definition. A twice differentiable function f is called concave
up on the interval [a, b] if and only if f 00 (x) > 0 for all x in [a, b].
It is called concave down on [a, b] if and only if f 00 (x) < 0 for
all x in [a, b].
A point x where the function changes from concave down to
concave up or vice versa is called an inflection point. An
inflection point necessarily occurs where f 00 is zero or
undefined.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
Bernd Schröder
Higher Derivatives
decreasing
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
concave
down
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
concave
down
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
concave
down
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
concave
down
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
All Combinations of Growth Behavior and
Concavity are Possible
increasing
decreasing
concave
up
concave
down
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞
x→∞
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
Bernd Schröder
Higher Derivatives
x→−∞
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f0
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
−1
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
−1
Bernd Schröder
Higher Derivatives
1
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f0
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 0− − −
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 0− − −
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 0− − −
−−−
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 0− − −
−−−
+++
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f decr.
f 0− − −
−−−
+++
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f decr.
f 0− − −
incr.
+++
−−−
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f decr.
f 0− − −
incr.
+++
decr.
−−−
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f 00
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
0
f decr.
f 0− − −
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
−−−
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
−−−
+++
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
conc. up
+++
−−−
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
conc. up
+++
conc. down
−−−
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
Second Derivative Test
6
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
Second Derivative Test
6
-
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
Second Derivative Test
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
x
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
x
1
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
x
2
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
2
conc. down
−−−
x
3
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
2
conc. down
−−−
x
3
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
2
conc. down
−−−
x
3
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
2
conc. down
−−−
x
3
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
f
f 00
1
conc. up
+++
2
conc. down
−−−
x
3
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
1
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
2
1
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
1
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
1
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
6
3
2
b
b
1
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
x
3
b
−1
f
f 00
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
2
b
−1
f
f 00
x
3
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
y
Sketch the Graph of f
From the Information
Below
6
3
2
b
b
b
1
b
f 0 > 0 on (−1, 1) and on (3, ∞)
f 0 < 0 on (−∞, −1) and on
(1, 3)
f 00 > 0 on (−∞, 0) and on
(2, ∞)
f 00 < 0 on (0, 2)
f (−1) = 1, f (0) = 32 , f (1) = 2,
f (2) = 21 , f (3) = −1
lim f (x) = ∞, lim f (x) = ∞
x→∞
x→−∞
-
−1
1
b
−1
f
f 00
x
3
2
conc. up
+++
conc. down
−−−
conc. up
+++
f decr.
f 0− − −
0
2
incr.
+++
decr.
−−−
incr.
+++
−1
Bernd Schröder
Higher Derivatives
1
3
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test. Let f be a function that
is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is
continuous in this interval and f 0 (a) = 0.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test. Let f be a function that
is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is
continuous in this interval and f 0 (a) = 0.
1. If f 00 (a) < 0, then f has a local maximum at a.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test. Let f be a function that
is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is
continuous in this interval and f 0 (a) = 0.
1. If f 00 (a) < 0, then f has a local maximum at a.
2. If f 00 (a) > 0, then f has a local minimum at a.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test. Let f be a function that
is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is
continuous in this interval and f 0 (a) = 0.
1. If f 00 (a) < 0, then f has a local maximum at a.
2. If f 00 (a) > 0, then f has a local minimum at a.
For single variable functions, the second derivative test can
often be replaced with an analysis of growth behavior.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Theorem. The second derivative test. Let f be a function that
is twice differentiable in an interval (a − ∆, a + ∆), so that f 00 is
continuous in this interval and f 0 (a) = 0.
1. If f 00 (a) < 0, then f has a local maximum at a.
2. If f 00 (a) > 0, then f has a local minimum at a.
For single variable functions, the second derivative test can
often be replaced with an analysis of growth behavior. But for
multivariable functions, the higher-dimensional analogue of this
theorem is just about the only way to identify extrema.
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f 00
Bernd Schröder
Higher Derivatives
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f 00
Bernd Schröder
Higher Derivatives
f 00 < 0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
a
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
f 0 (a) = 0
a
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
···
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f0
···
f0 > 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
implies
6
f0
···
f0 > 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f increasing
implies
6
f0
···
f0 > 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f increasing
implies
6
f0
···
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f increasing
implies
6
f0
···
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f increasing
f0
···
f decreasing
implies
6
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f
f increasing
f0
···
f decreasing
implies
6
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f
f increasing
f0
···
f decreasing
implies
6
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f
a
f increasing
f0
···
f decreasing
implies
6
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science
Definition
Curvature
Curve Sketching
Second Derivative Test
Why Does the Second Derivative Test Work?
f
···
···
a
f increasing
f0
···
f decreasing
implies
6
implies
6
f0 > 0
f0 < 0
PP f 0 (a) = 0
PP
PP
a
PP
···
f 0 decreasing
f 00
Bernd Schröder
Higher Derivatives
implies
6
f 00
<0
logo1
Louisiana Tech University, College of Engineering and Science