Lesson 21: Curve Sketching (Section 021 handout)

Section 4.4
Curve Sketching
V63.0121.021, Calculus I
New York University
November 18, 2010
Announcements
There is class on November 23. The homework is due on
November 24. Turn in homework to my mailbox or bring to class on
November 23.
Announcements
There is class on
November 23. The
homework is due on
November 24. Turn in
homework to my mailbox or
bring to class on
November 23.
V63.0121.021, Calculus I (NYU) Section 4.4 Curve Sketching November 18, 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
Notes
Notes
Notes
1
Section 4.4 : Curve SketchingV63.0121.021, Calculus I November 18, 2010

Why?
Graphing functions is like
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)
Notes
Notes
Notes
2

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.
(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.
Notes
Notes
Notes
3

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:
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:
f (x)
f (x)1/2
−− ++
IP
Step 3: One sign chart to rule them all
Remember, f (x) = 2x3
− 3x2
− 12x.
f (x)
monotonicity−1 2
+− −+
f (x)
concavity1/2
−− −− ++ ++
f (x)
shape of f−1
7
max
2
−20
min
1/2
−61/2
IP
Notes
Notes
Notes
4

Combinations of monotonicity and concavity
III
III IV
decreasing,
concave
down
increasing,
concave
down
decreasing,
concave up
increasing,
concave up
Step 3: One sign chart to rule them all
Remember, f (x) = 2x3
− 3x2
− 12x.
f (x)
monotonicity−1 2
+− −+
f (x)
concavity1/2
−− −− ++ ++
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
Notes
Notes
Notes
5

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)
We make its sign chart.
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)
Here is its sign chart:
12x
0
0− + +
x − 2
2
0− − +
f (x)
f (x)0
0
2
0++ −− ++
IP IP
Notes
Notes
Notes
6

Step 3: Grand Uniﬁed Sign Chart
Remember, f (x) = x4
− 4x3
+ 10.
f (x)
monotonicity3
0
0
0− − − +
f (x)
concavity0
0
2
0++ −− ++ ++
f (x)
shape0
10
IP
2
−6
IP
3
−17
min
Step 4: Graph
f (x) = x4
− 4x3
+ 10
x
y
f (x)
shape0
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
Notes
Notes
Notes
7

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.
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 ∞.
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
= −∞
Notes
Notes
Notes
8

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
Notice
f (x) > 0 when x > 0 (so no critical points here)
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)
Step 1: The derivative
Remember, f (x) = x + |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)
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)−1
4
0
0
∞+ − +
max min
Notes
Notes
Notes
9

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.
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.
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)
concavity0
−∞−− −− −−−∞ −∞
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)
(−1, 0)
(−1
4, 1
4)
(0, 0)
Notes
Notes
Notes
10

Example with Horizontal Asymptotes
Example
Graph f (x) = xe−x2
Step 2: Concavity
Notes
Notes
Notes
11

Step 3: Synthesis
Step 4: Graph
x
f (x)
Example with Vertical Asymptotes
Example
Graph f (x) =
1
x
+
1
x2
Notes
Notes
Notes
12

Step 0
Step 2: Concavity
Notes
Notes
Notes
13

Step 3: Synthesis
Step 4: Graph
x
y
Problem
Graph f (x) = cos x − x
Notes
Notes
Notes
14

Step 0: intercepts and asymptotes
Step 2: Concavity
Notes
Notes
Notes
15

Step 3: Synthesis
Step 4: Graph
f (x) = cos x − x
x
y
Logarithmic
Problem
Graph f (x) = x ln x2
Notes
Notes
Notes
16

Step 0: Intercepts and Asymptotes
Step 2: Concavity
Notes
Notes
Notes
17

Step 3: Synthesis
Step 4: Graph
x
y
Summary
Graphing is a procedure that gets easier with practice.
Remember to follow the checklist.
Notes
Notes
Notes
18

Lesson 21: Curve Sketching (Section 021 handout)

More Related Content

What's hot

Similar to Lesson 21: Curve Sketching (Section 021 handout)

More from Matthew Leingang

Lesson 21: Curve Sketching (Section 021 handout)