Lesson 21: Curve Sketching (Section 041 slides)

..
Section 4.4
Curve Sketching
V63.0121.041, Calculus I
New York University
November 17, 2010
Announcements
Quiz 4 this week in recitation on 3.3, 3.4, 3.5, 3.7
There is class on November 24
. . . . . .

. . . . . .
Announcements
Quiz 4 this week in
recitation on 3.3, 3.4, 3.5,
3.7
There is class on
November 24
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 2 / 55

. . . . . .
Objectives
given a function, graph it
completely, indicating
zeroes (if easy)
asymptotes if applicable
critical points
local/global max/min
inflection points

. . . . . .
Why?
Graphing functions is like
dissection

. . . . . .
Why?
dissection … or diagramming
sentences

. . . . . .
Why?
dissection … or diagramming
sentences
You can really know a lot about
a function when you know all of
its anatomy.

. . . . . .
The Increasing/Decreasing Test
Theorem (The Increasing/Decreasing Test)
If f′
> 0 on (a, b), then f is increasing on (a, b). If f′
< 0 on (a, b), then f
is decreasing on (a, b).
Example
Here f(x) = x3
+ x2
, and f′
(x) = 3x2
+ 2x.
..
f(x)
.
f′
(x)

. . . . . .
Testing for Concavity
Theorem (Concavity Test)
If f′′
(x) > 0 for all x in (a, b), then the graph of f is concave upward on
(a, b) If f′′
(x) < 0 for all x in (a, b), then the graph of f is concave
downward on (a, b).
Example
Here f(x) = x3
+ x2
, f′
(x) = 3x2
+ 2x, and f′′
(x) = 6x + 2.
..
f(x)
.
f′
(x)
.
f′′
(x)

. . . . . .
Graphing Checklist
To graph a function f, follow this plan:
0. Find when f is positive, negative, zero,
not defined.
1. Find f′
and form its sign chart. Conclude
information about increasing/decreasing
and local max/min.
2. Find f′′
and form its sign chart. Conclude
concave up/concave down and inflection.
3. Put together a big chart to assemble
monotonicity and concavity data
4. Graph!

. . . . . .
Outline
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic

. . . . . .
Graphing a cubic
Example
Graph f(x) = 2x3
− 3x2
− 12x.

. . . . . .
Graphing a cubic
Example
Graph f(x) = 2x3
− 3x2
− 12x.
(Step 0) First, let’s find the zeros. We can at least factor out one power
of x:
f(x) = x(2x2
− 3x − 12)
so f(0) = 0. The other factor is a quadratic, so we the other two roots
are
x =
3 ±
√
32
− 4(2)(−12)
4
=
3 ±
√
105
4
It’s OK to skip this step for now since the roots are so complicated.

. . . . . .
Step 1: Monotonicity
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
.

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.
x + 1
..
−1
.
f′
(x)
.
f(x)
..
2
..
−1

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
f′
(x)
.
f(x)
..
2
..
−1

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+
.
↗

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+
.
↗
.
↘

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+
.
↗
.
↘
.
↗

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+
.
↗
.
↘
.
↗
.
max

. . . . . .
f(x) = 2x3
− 3x2
− 12x
=⇒ f′
(x) = 6x2
− 6x − 12 = 6(x + 1)(x − 2)
.. x − 2..
2
.− . −. +.
x + 1
..
−1
.
+
.
+
.
−
.
f′
(x)
.
f(x)
..
2
..
−1
.
+
.
−
.
+
.
↗
.
↘
.
↗
.
max
.
min

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
Another sign chart: .

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
Another sign chart: ..
f′′
(x)
.
f(x)
..
1/2

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
f′′
(x)
.
f(x)
..
1/2
.
−−

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
f′′
(x)
.
f(x)
..
1/2
.
−−
.
++

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
f′′
(x)
.
f(x)
..
1/2
.
−−
.
++
.
⌢

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
f′′
(x)
.
f(x)
..
1/2
.
−−
.
++
.
⌢
.
⌣

. . . . . .
Step 2: Concavity
f′
(x) = 6x2
− 6x − 12
=⇒ f′′
(x) = 12x − 6 = 6(2x − 1)
f′′
(x)
.
f(x)
..
1/2
.
−−
.
++
.
⌢
.
⌣
.
IP

. . . . . .
Step 3: One sign chart to rule them all
Remember, f(x) = 2x3
− 3x2
− 12x.
.

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP

. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV

. . . . . .
..
I
.
II
.
III
.
IV
.
decreasing,
concave
down

. . . . . .
..
I
.
II
.
III
.
IV
.
decreasing,
concave
down
.
increasing,
concave
down

. . . . . .
..
I
.
II
.
III
.
IV
.
decreasing,
concave
down
.
increasing,
concave
down
.
decreasing,
concave up

. . . . . .
..
I
.
II
.
III
.
IV
.
decreasing,
concave
down
.
increasing,
concave
down
.
decreasing,
concave up
.
increasing,
concave up

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP
.

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP
..

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP
...

. . . . . .
− 3x2
− 12x.
..
f′
(x)
.
monotonicity
..
−1
..
2
. +.
↗
.− .
↘
. −.
↘
.+ .
↗
.
f′′
(x)
.
concavity
..
1/2
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP
....

. . . . . .
Step 4: Graph
..
f(x) = 2x3
− 3x2
− 12x
. x.
f(x)
.
f(x)
.
shape of f
..
−1
.
7
.
max
..
2
.
−20
.
min
..
1/2
.
−61/2
.
IP
......
(
3−
√
105
4 , 0
)
..
(−1, 7)
..
(0, 0)
..
(1/2, −61/2)
..
(2, −20)
..
(
3+
√
105
4 , 0
)

. . . . . .
Graphing a quartic
Example
Graph f(x) = x4
− 4x3
+ 10

. . . . . .
Graphing a quartic
Example
Graph f(x) = x4
− 4x3
+ 10
(Step 0) We know f(0) = 10 and lim
x→±∞
f(x) = +∞. Not too many other
points on the graph are evident.

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
We make its sign chart.
.

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−
.
+

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−
.
+
.
↘

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−
.
+
.
↘
.
↘

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−
.
+
.
↘
.
↘
.
↗

. . . . . .
f(x) = x4
− 4x3
+ 10
=⇒ f′
(x) = 4x3
− 12x2
= 4x2
(x − 3)
.. 4x2..
0
.0.+ . +. +.
(x − 3)
..
3
.
0
.
−
.
−
.
+
.
f′
(x)
.
f(x)
..
3
.
0
..
0
.
0
.
−
.
−
.
+
.
↘
.
↘
.
↗
.
min

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
Here is its sign chart:
.

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.−

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++
.
⌣

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++
.
⌣
.
⌢

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++
.
⌣
.
⌢
.
⌣

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++
.
⌣
.
⌢
.
⌣
.
IP

. . . . . .
Step 2: Concavity
f′
(x) = 4x3
− 12x2
=⇒ f′′
(x) = 12x2
− 24x = 12x(x − 2)
.. 12x..
0
.0.− . +. +.
x − 2
..
2
.
0
.
−
.
−
.
+
.
f′′
(x)
.
f(x)
..
0
.
0
..
2
.
0
.
++
.
−−
.
++
.
⌣
.
⌢
.
⌣
.
IP
.
IP

. . . . . .
Step 3: Grand Unified Sign Chart
Remember, f(x) = x4
− 4x3
+ 10.
..
f′
(x)
.
monotonicity
..
3
.
0
..
0
.
0
.
−
.
↘
.
−
.
↘
.
−
.
↘
.
+
.
↗
.
f′′
(x)
.
concavity
..
0
.
0
..
2
.
0
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min

. . . . . .
Remember, f(x) = x4
− 4x3
+ 10.
..
f′
(x)
.
monotonicity
..
3
.
0
..
0
.
0
.
−
.
↘
.
−
.
↘
.
−
.
↘
.
+
.
↗
.
f′′
(x)
.
concavity
..
0
.
0
..
2
.
0
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min
.

. . . . . .
Remember, f(x) = x4
− 4x3
+ 10.
..
f′
(x)
.
monotonicity
..
3
.
0
..
0
.
0
.
−
.
↘
.
−
.
↘
.
−
.
↘
.
+
.
↗
.
f′′
(x)
.
concavity
..
0
.
0
..
2
.
0
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min
..

. . . . . .
Remember, f(x) = x4
− 4x3
+ 10.
..
f′
(x)
.
monotonicity
..
3
.
0
..
0
.
0
.
−
.
↘
.
−
.
↘
.
−
.
↘
.
+
.
↗
.
f′′
(x)
.
concavity
..
0
.
0
..
2
.
0
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min
...

. . . . . .
Remember, f(x) = x4
− 4x3
+ 10.
..
f′
(x)
.
monotonicity
..
3
.
0
..
0
.
0
.
−
.
↘
.
−
.
↘
.
−
.
↘
.
+
.
↗
.
f′′
(x)
.
concavity
..
0
.
0
..
2
.
0
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min
....

. . . . . .
Step 4: Graph
..
f(x) = x4
− 4x3
+ 10
. x.
y
.
f(x)
.
shape
..
0
.
10
.
IP
..
2
.
−6
.
IP
..
3
.
−17
.
min
......
(0, 10)
..
(2, −6)
..
(3, −17)

. . . . . .
Outline
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Logarithmic

. . . . . .
Graphing a function with a cusp
Example
Graph f(x) = x +
√
|x|

. . . . . .
Graphing a function with a cusp
Example
Graph f(x) = x +
√
|x|
This function looks strange because of the absolute value. But
whenever we become nervous, we can just take cases.

. . . . . .
Step 0: Finding Zeroes
f(x) = x +
√
|x|
First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if
x is positive.

. . . . . .
f(x) = x +
√
|x|
x is positive.
Are there negative numbers which are zeroes for f?

. . . . . .
f(x) = x +
√
|x|
x is positive.
Are there negative numbers which are zeroes for f?
x +
√
−x = 0
√
−x = −x
−x = x2
x2
+ x = 0
The only solutions are x = 0 and x = −1.

. . . . . .
Step 0: Asymptotic behavior
f(x) = x +
√
|x|
lim
x→∞
f(x) = ∞, because both terms tend to ∞.

. . . . . .
f(x) = x +
√
|x|
lim
x→∞
lim
x→−∞
f(x) is indeterminate of the form −∞ + ∞. It’s the same as
lim
y→+∞
(−y +
√
y)

. . . . . .
f(x) = x +
√
|x|
lim
x→∞
lim
x→−∞
f(x) is indeterminate of the form −∞ + ∞. It’s the same as
lim
y→+∞
(−y +
√
y)
lim
y→+∞
(−y +
√
y) = lim
y→∞
(
√
y − y) ·
√
y + y
√
y + y
= lim
y→∞
y − y2
√
y + y
= −∞

. . . . . .
Step 1: The derivative
Remember, f(x) = x +
√
|x|.
To find f′
, first assume x > 0. Then
f′
(x) =
d
dx
(
x +
√
x
)
= 1 +
1
2
√
x

. . . . . .
√
|x|.
To find f′
f′
(x) =
d
dx
(
x +
√
x
)
= 1 +
1
2
√
x
Notice
f′
(x) > 0 when x > 0 (so no critical points here)

. . . . . .
√
|x|.
To find f′
f′
(x) =
d
dx
(
x +
√
x
)
= 1 +
1
2
√
x
Notice
f′
lim
x→0+
f′
(x) = ∞ (so 0 is a critical point)

. . . . . .
√
|x|.
To find f′
f′
(x) =
d
dx
(
x +
√
x
)
= 1 +
1
2
√
x
Notice
f′
lim
x→0+
f′
(x) = ∞ (so 0 is a critical point)
lim
x→∞
f′
(x) = 1 (so the graph is asymptotic to a line of slope 1)

. . . . . .
√
|x|.
If x is negative, we have
f′
(x) =
d
dx
(
x +
√
−x
)
= 1 −
1
2
√
−x
Notice
lim
x→0−
f′
(x) = −∞ (other side of the critical point)

. . . . . .
√
|x|.
f′
(x) =
d
dx
(
x +
√
−x
)
= 1 −
1
2
√
−x
Notice
lim
x→0−
f′
lim
x→−∞
f′
(x) = 1 (asymptotic to a line of slope 1)

. . . . . .
√
|x|.
f′
(x) =
d
dx
(
x +
√
−x
)
= 1 −
1
2
√
−x
Notice
lim
x→0−
f′
lim
x→−∞
f′
(x) = 1 (asymptotic to a line of slope 1)
f′
(x) = 0 when
1 −
1
2
√
−x
= 0 =⇒
√
−x =
1
2
=⇒ −x =
1
4
=⇒ x = −
1
4

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
We can’t make a multi-factor sign chart because of the absolute value,
but we can test points in between critical points.
..
f′
(x)
.
f(x)

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .−

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +.
↗

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +.
↗
.
↘

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +.
↗
.
↘
.
↗

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +.
↗
.
↘
.
↗
.
max

. . . . . .
f′
(x) =



1 +
1
2
√
x
if x > 0
1 −
1
2
√
−x
if x < 0
..
f′
(x)
.
f(x)
..
−1
4
.0 ..
0
.∞.+ .− . +.
↗
.
↘
.
↗
.
max
.
min

. . . . . .
Step 2: Concavity
If x > 0, then
f′′
(x) =
d
dx
(
1 +
1
2
x−1/2
)
= −
1
4
x−3/2
This is negative whenever x > 0.

. . . . . .
Step 2: Concavity
If x > 0, then
f′′
(x) =
d
dx
(
1 +
1
2
x−1/2
)
= −
1
4
x−3/2
If x < 0, then
f′′
(x) =
d
dx
(
1 −
1
2
(−x)−1/2
)
= −
1
4
(−x)−3/2
which is also always negative for negative x.

. . . . . .
Step 2: Concavity
If x > 0, then
f′′
(x) =
d
dx
(
1 +
1
2
x−1/2
)
= −
1
4
x−3/2
If x < 0, then
f′′
(x) =
d
dx
(
1 −
1
2
(−x)−1/2
)
= −
1
4
(−x)−3/2
In other words, f′′
(x) = −
1
4
|x|−3/2
.

. . . . . .
Step 2: Concavity
If x > 0, then
f′′
(x) =
d
dx
(
1 +
1
2
x−1/2
)
= −
1
4
x−3/2
If x < 0, then
f′′
(x) =
d
dx
(
1 −
1
2
(−x)−1/2
)
= −
1
4
(−x)−3/2
In other words, f′′
(x) = −
1
4
|x|−3/2
.
Here is the sign chart:
..
f′′
(x)
.
f(x)
..
0
.−∞.−− .
⌢
... −−.
⌢

. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x +
√
|x|
..
f′
(x)
.
monotonicity
..
−1
4
.0 ..
0
.∞.+1 .
↗
.+ .
↗
.− .
↘
. +.
↗
. +1.
↗
.
f′′
(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−1
4
.
1
4
.
max
..
0
.
0
.
min
.
−∞
.
+∞

. . . . . .
Step 3: Synthesis
f(x) = x +
√
|x|
..
f′
(x)
.
monotonicity
..
−1
4
.0 ..
0
.∞.+1 .
↗
.+ .
↗
.− .
↘
. +.
↗
. +1.
↗
.
f′′
(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−1
4
.
1
4
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.

. . . . . .
Step 3: Synthesis
f(x) = x +
√
|x|
..
f′
(x)
.
monotonicity
..
−1
4
.0 ..
0
.∞.+1 .
↗
.+ .
↗
.− .
↘
. +.
↗
. +1.
↗
.
f′′
(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−1
4
.
1
4
.
max
..
0
.
0
.
min
.
−∞
.
+∞
..

. . . . . .
Step 3: Synthesis
f(x) = x +
√
|x|
..
f′
(x)
.
monotonicity
..
−1
4
.0 ..
0
.∞.+1 .
↗
.+ .
↗
.− .
↘
. +.
↗
. +1.
↗
.
f′′
(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−1
4
.
1
4
.
max
..
0
.
0
.
min
.
−∞
.
+∞
...

. . . . . .
Step 3: Synthesis
f(x) = x +
√
|x|
..
f′
(x)
.
monotonicity
..
−1
4
.0 ..
0
.∞.+1 .
↗
.+ .
↗
.− .
↘
. +.
↗
. +1.
↗
.
f′′
(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−1
4
.
1
4
.
max
..
0
.
0
.
min
.
−∞
.
+∞
....

. . . . . .
Graph
f(x) = x +
√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−1
4
.
1
4
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
..... x.
f(x)

. . . . . .
Graph
f(x) = x +
√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−1
4
.
1
4
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
..... x.
f(x)
..
(−1, 0)

. . . . . .
Graph
f(x) = x +
√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−1
4
.
1
4
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
..... x.
f(x)
..
(−1, 0)
..
(−1
4 , 1
4 )

. . . . . .
Graph
f(x) = x +
√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−1
4
.
1
4
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
..... x.
f(x)
..
(−1, 0)
..
(−1
4 , 1
4 )
..
(0, 0)

. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2

. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2
Before taking derivatives, we notice that f is odd, that f(0) = 0, and
lim
x→∞
f(x) = 0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
is always positive so it doesn’t figure into the sign of
f′
(x). So our sign chart looks like this:
.. 1 −
√
2x.. √
1/2
. 0.
1 +
√
2x
..
−
√
1/2
.
0
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .
1 +
√
2x
..
−
√
1/2
.
0
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+.
1 +
√
2x
..
−
√
1/2
.
0
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
+
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
↘
.
+
.
−
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
↘
.
+
.
↗
.
−
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
↘
.
+
.
↗
.
−
.
↘
..
−
√
1/2
.
0
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
↘
.
+
.
↗
.
−
.
↘
..
−
√
1/2
.
0
.
min
..
√
1/2
.
0

. . . . . .
If f(x) = xe−x2
, then
f′
(x) = 1 · e−x2
+ xe−x2
(−2x) =
(
1 − 2x2
)
e−x2
=
(
1 −
√
2x
) (
1 +
√
2x
)
e−x2
The factor e−x2
f′
.. 1 −
√
2x.. √
1/2
. 0.+ .+. −.
1 +
√
2x
..
−
√
1/2
.
0
.
−
.
+
.
+
.
f′
(x)
.
f(x)
.
−
.
↘
.
+
.
↗
.
−
.
↘
..
−
√
1/2
.
0
.
min
..
√
1/2
.
0
.
max

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.
√
2x −
√
3
..
√
3/2
.
0
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .
√
2x −
√
3
..
√
3/2
.
0
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− .
√
2x −
√
3
..
√
3/2
.
0
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +.
√
2x −
√
3
..
√
3/2
.
0
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
++
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
++
.
−−
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
−−
.
++
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
++
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
..
−
√
3/2
.
0
.
IP
..
0
.
0
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
..
−
√
3/2
.
0
.
IP
..
0
.
0
.
IP
..
√
3/2
.
0

. . . . . .
Step 2: Concavity
If f′
(x) = (1 − 2x2
)e−x2
, we know
f′′
(x) = (−4x)e−x2
+ (1 − 2x2
)e−x2
(−2x) =
(
4x3
− 6x
)
e−x2
= 2x(2x2
− 3)e−x2
.. 2x..
0
.0.− .− . +. +.
√
2x −
√
3
..
√
3/2
.
0
.
−
.
−
.
−
.
+
.
√
2x +
√
3
..
−
√
3/2
.
0
.
−
.
+
.
+
.
+
.
f′′
(x)
.
f(x)
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
..
−
√
3/2
.
0
.
IP
..
0
.
0
.
IP
..
√
3/2
.
0
.
IP

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
.

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
..

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
...

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
....

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
.....

. . . . . .
Step 3: Synthesis
f(x) = xe−x2
..
f′
(x)
.
monotonicity
..
−
√
1/2
.0 .. √
1/2
. 0.− .
↘
.− .
↘
.+ .
↗
. +.
↗
. −.
↘
. −.
↘
.
f′′
(x)
.
concavity
..
−
√
3/2
.
0
..
0
.
0
..
√
3/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
++
.
⌣
.
−−
.
⌢
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
..
√
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.
0
.
IP
..
√
3/2
.
√
3
2e3
.
IP
......

. . . . . .
Step 4: Graph
..
x
.
f(x)
.
f(x) = xe−x2
..
(
−
√
1/2, − 1√
2e
)
..
(√
1/2, 1√
2e
)
..
(
−
√
3/2, −
√
3
2e3
)
..
(0, 0)
..
(
√
3/2,
√
3
2e3
)
.
f(x)
.
shape
..
−
√
1/2
.
− 1√
2e
.
min
.. √
1/2
.
1√
2e
.
max
..
−
√
3/2
.
−
√
3
2e3
.
IP
..
0
.0.
IP
.. √
3/2
.
√
3
2e3
.
IP
......

. . . . . .
Example with Vertical Asymptotes
Example
Graph f(x) =
1
x
+
1
x2

. . . . . .
Step 0
Find when f is positive, negative, zero, not defined.

. . . . . .
Step 0
Find when f is positive, negative, zero, not defined. We need to factor f:
f(x) =
1
x
+
1
x2
=
x + 1
x2
.
This means f is 0 at −1 and has trouble at 0. In fact,
lim
x→0
x + 1
x2
= ∞,
so x = 0 is a vertical asymptote of the graph.

. . . . . .
Step 0
Find when f is positive, negative, zero, not defined. We need to factor f:
f(x) =
1
x
+
1
x2
=
x + 1
x2
.
This means f is 0 at −1 and has trouble at 0. In fact,
lim
x→0
x + 1
x2
= ∞,
so x = 0 is a vertical asymptote of the graph. We can make a sign
chart as follows:
.. x + 1..0 .
−1
.− . +.
x2
..
0
.
0
.
+
.
+
.
f(x)
..
∞
.
0
..
0
.
−1
.
−
.
+
.
+

. . . . . .
Step 0, continued
For horizontal asymptotes, notice that
lim
x→∞
x + 1
x2
= 0,
so y = 0 is a horizontal asymptote of the graph. The same is true at
−∞.

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
The critical points are x = −2 and x = 0. We have the following sign
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
min

. . . . . .
We have
f′
(x) = −
1
x2
−
2
x3
= −
x + 2
x3
.
chart:
.. −(x + 2)..0 .
−2
.+ . −.
x3
..
0
.
0
.
−
.
+
.
f′
(x)
.
f(x)
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
min
.
VA

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
The critical points of f′
are −3 and 0. Sign chart:
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
IP

. . . . . .
Step 2: Concavity
We have
f′′
(x) =
2
x3
+
6
x4
=
2(x + 3)
x4
.
.. (x + 3)..0 .
−3
.− . +.
x4
..
0
.
0
.
+
.
+
.
f′′
(x)
.
f(x)
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
IP
.
VA

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
.

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
.

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
.

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0
.

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0
..
VA

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0
..
VA
.

. . . . . .
Step 3: Synthesis
..
f′
.
monotonicity
..
∞
.
0
..
0
.
−2
.
−
.
+
.
−
.
↘
.
↗
.
↘
.
f′′
.
concavity
..
∞
.
0
..
0
.
−3
.
−−
.
++
.
++
.
⌢
.
⌣
.
⌣
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0
..
VA
..
HA

. . . . . .
Step 4: Graph
.. x.
y
..
(−3, −2/9)
..
(−2, −1/4)
.
f
.
shape of f
..
∞
.
0
..
0
.
−1
..
−2
.
−1/4
..
−3
.
−2/9
.
−∞
.
0
.
∞
.
0
.
−
.
+
.
+
.
HA
..
IP
..
min
..
0
..
VA
..
HA

. . . . . .
Problem
Graph f(x) = cos x − x

. . . . . .
Step 0: intercepts and asymptotes
f(0) = 1 and f(−π/2) = −π/2. So by the Intermediate Value
Theorem there is a zero in between. We don’t know it’s precise
value, though.
Since −1 ≤ cos x ≤ 1 for all x, we have
−1 − x ≤ cos x − x ≤ 1 − x
for all x. This means that lim
x→∞
f(x) = −∞ and lim
x→−∞
f(x) = ∞.

. . . . . .
If f(x) = cos x − x, then f′
(x) = − sin x − 1 = (−1)(sin x + 1).
f′
(x) = 0 if x = 3π/2 + 2πk, where k is any integer
f′
(x) is periodic with period 2π
Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x + 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x + 1) ≤ 0
for all x. This means f′
(x) is negative at all other points.
..
f′
(x)
.
f(x)
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0

. . . . . .
(x) = − sin x − 1 = (−1)(sin x + 1).
f′
f′
0 ≤ sin x + 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x + 1) ≤ 0
..
f′
(x)
.
f(x)
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −

. . . . . .
(x) = − sin x − 1 = (−1)(sin x + 1).
f′
f′
0 ≤ sin x + 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x + 1) ≤ 0
..
f′
(x)
.
f(x)
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −. −

. . . . . .
(x) = − sin x − 1 = (−1)(sin x + 1).
f′
f′
0 ≤ sin x + 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x + 1) ≤ 0
..
f′
(x)
.
f(x)
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −

. . . . . .
(x) = − sin x − 1 = (−1)(sin x + 1).
f′
f′
0 ≤ sin x + 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x + 1) ≤ 0
..
f′
(x)
.
f(x)
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
This is 0 when x = π/2 + πk, where k is any integer.
This is periodic with period 2π
..
f′′
(x)
.
f(x)
..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−. ++..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−. ++. −−..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−. ++. −−. ++..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++. −−. ++..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−. ++..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 ..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 .
IP
..
π/2
. 0..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 .
IP
..
π/2
. 0.
IP
..
3π/2
. 0..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 .
IP
..
π/2
. 0.
IP
..
3π/2
. 0.
IP
..
5π/2
. 0..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 .
IP
..
π/2
. 0.
IP
..
3π/2
. 0.
IP
..
5π/2
. 0.
IP
..
7π/2
. 0

. . . . . .
Step 2: Concavity
If f′
(x) = − sin x − 1, then f′′
(x) = − cos x.
..
f′′
(x)
.
f(x)
.−−.
⌢
. ++.
⌣
. −−.
⌢
. ++.
⌣
..
−π/2
.0 .
IP
..
π/2
. 0.
IP
..
3π/2
. 0.
IP
..
5π/2
. 0.
IP
..
7π/2
. 0.
IP

. . . . . .
Step 3: Synthesis
..
f′
(x)
.
mono
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘
.
f′′
(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP

. . . . . .
Step 3: Synthesis
..
f′
(x)
.
mono
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘
.
f′′
(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.

. . . . . .
Step 3: Synthesis
..
f′
(x)
.
mono
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘
.
f′′
(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
..

. . . . . .
Step 3: Synthesis
..
f′
(x)
.
mono
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘
.
f′′
(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
...

. . . . . .
Step 3: Synthesis
..
f′
(x)
.
mono
..
−π/2
.0 ..
3π/2
. 0..
7π/2
. 0. −.
↘
. −.
↘
.
f′′
(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
....

Lesson 21: Curve Sketching (Section 041 slides)

Lesson 21: Curve Sketching (Section 041 slides)

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