Lesson 21: Curve Sketching (Section 021 handout)

V63.0121.021, Calculus I Section 4.4 : Curve Sketching November 18, 2010

Notes
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
Notes

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
Notes

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


1


Why?
Notes

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
Notes
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) = x 3 + x 2 , and f (x) = 3x 2 + 2x.

f (x)
f (x)


Testing for Concavity
Notes
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) = x 3 + x 2 , f (x) = 3x 2 + 2x, and f (x) = 6x + 2.
f (x) f (x)
f (x)


2


Graphing Checklist
Notes

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
Notes

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
Notes

Example
Graph f (x) = 2x 3 − 3x 2 − 12x.

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


3


Step 1: Monotonicity
Notes

f (x) = 2x 3 − 3x 2 − 12x
=⇒ f (x) = 6x 2 − 6x − 12 = 6(x + 1)(x − 2)

We can form a sign chart from this:

− − +
x −2
2
− + +
x +1
−1
+ − + f (x)
−1 2 f (x)
max min


Step 2: Concavity
Notes

f (x) = 6x 2 − 6x − 12
=⇒ f (x) = 12x − 6 = 6(2x − 1)

Another sign chart:

−− ++ f (x)
1/2 f (x)
IP


Step 3: One sign chart to rule them all
Notes

Remember, f (x) = 2x 3 − 3x 2 − 12x.

+ − − + f (x)
−1 2 monotonicity
−− −− ++ ++ f (x)
1/2 concavity
7 −61/2 −20 f (x)
−1 1/2 2 shape of f
max IP min


4


Combinations of monotonicity and concavity
Notes
increasing, decreasing,
concave concave
down down

II I

III IV

decreasing, increasing,
concave up concave up


Step 3: One sign chart to rule them all
Notes

Remember, f (x) = 2x 3 − 3x 2 − 12x.

+ − − + f (x)
−1 2 monotonicity
−− −− ++ ++ f (x)
1/2 concavity
7 −61/2 −20 f (x)
max IP min


Step 4: Graph
f (x) Notes

f (x) = 2x 3 − 3x 2 − 12x

√
(−1, 7)
3− 105
4 ,0 (0, 0)
√
x
(1/2, −61/2) 3+ 105
4 ,0

(2, −20)

7 −61/2 −20 f (x)
max IP min

5


Graphing a quartic
Notes

Example
Graph f (x) = x 4 − 4x 3 + 10

(Step 0) We know f (0) = 10 and lim f (x) = +∞. Not too many other
x→±∞
points on the graph are evident.


Notes

f (x) = x 4 − 4x 3 + 10
=⇒ f (x) = 4x 3 − 12x 2 = 4x 2 (x − 3)

We make its sign chart.

+ 0 + +
4x 2
0
− − 0 +
(x − 3)
3
− 0 − 0 + f (x)
0 3 f (x)
min


Step 2: Concavity
Notes

f (x) = 4x 3 − 12x 2
=⇒ f (x) = 12x 2 − 24x = 12x(x − 2)

Here is its sign chart:

− 0 + +
12x
0
− − 0 +
x −2
2
++ 0 −− 0 ++ f (x)
0 2 f (x)
IP IP


6


Step 3: Grand Uniﬁed Sign Chart
Notes

Remember, f (x) = x 4 − 4x 3 + 10.

− 0 − − 0 + f (x)
0 3 monotonicity
++ 0 −− 0 ++ ++ f (x)
0 2 concavity
10 −6 −17 f (x)
0 2 3 shape
IP IP min


Step 4: Graph
y Notes

f (x) = x 4 − 4x 3 + 10

(0, 10)
x
(2, −6)
(3, −17)

10 −6 −17 f (x)
0 2 3 shape
IP IP min

Outline
Notes

Simple examples
A cubic function
A quartic function

More Examples
Points of nondiﬀerentiability
Horizontal asymptotes
Vertical asymptotes
Logarithmic


7


Graphing a function with a cusp
Notes

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
Notes

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 = x 2
2
x +x =0

The only solutions are x = 0 and x = −1.


Step 0: Asymptotic behavior
Notes

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


8


Step 1: The derivative
Notes

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

Notice
f (x) > 0 when x > 0 (so no critical points here)
lim f (x) = ∞ (so 0 is a critical point)
x→0+
lim f (x) = 1 (so the graph is asymptotic to a line of slope 1)
x→∞


Step 1: The derivative
Notes
Remember, f (x) = x + |x|.
If x is negative, we have
d √ 1
f (x) = x + −x = 1 − √
dx 2 −x
Notice
lim f (x) = −∞ (other side of the critical point)
x→0−
lim f (x) = 1 (asymptotic to a line of slope 1)
x→−∞
f (x) = 0 when

1 √ 1 1 1
1− √ = 0 =⇒ −x = =⇒ −x = =⇒ x = −
2 −x 2 4 4


Notes

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

+ 0 − ∞ + f (x)
−1 0 f (x)
4
max min


9


Step 2: Concavity
Notes
If x > 0, then
d 1 1
f (x) = 1 + x −1/2 = − x −3/2
dx 2 4
This is negative whenever x > 0.
If x < 0, then
d 1 1
f (x) = 1 − (−x)−1/2 = − (−x)−3/2
dx 2 4
which is also always negative for negative x.
1
In other words, f (x) = − |x|−3/2 .
4
Here is the sign chart:

−− −∞ −− f (x)
0 f (x)


Step 3: Synthesis
Notes

Now we can put these things together.

f (x) = x + |x|

+1 + 0 − ∞ + +1 (x)
f
−1 0 monotonicity
4
−∞ −− −−−∞ −− −∞ (x)
f
0 concavity
1
−∞ 0 4 0 +∞(x)
f
−1 −1 0 shape
4
zero max min


Graph
Notes

f (x) = x + |x|

f (x)

(− 1 , 1 )
4 4
(−1, 0)
x
(0, 0)

1
−∞ 0 4 0 +∞ f (x)
−1 −1 0 shape
4
zero max min


10


Example with Horizontal Asymptotes
Notes

Example
2
Graph f (x) = xe −x


Notes


Step 2: Concavity
Notes


11


Step 3: Synthesis
Notes


Step 4: Graph
Notes
f (x)

x


Example with Vertical Asymptotes
Notes

Example
1 1
Graph f (x) = + 2
x x


12


Step 0
Notes


Notes


Step 2: Concavity
Notes


13


Step 3: Synthesis
Notes


Step 4: Graph
Notes

y

x


Notes

Problem
Graph f (x) = cos x − x


14


Step 0: intercepts and asymptotes
Notes


Notes


Step 2: Concavity
Notes


15


Step 3: Synthesis
Notes


Step 4: Graph
Notes
f (x) = cos x − x
y

x


Logarithmic
Notes

Problem
Graph f (x) = x ln x 2


16


Step 0: Intercepts and Asymptotes
Notes


Notes


Step 2: Concavity
Notes


17


Step 3: Synthesis
Notes


Step 4: Graph
Notes

y

x


Summary
Notes

Graphing is a procedure that gets easier with practice.
Remember to follow the checklist.


18

Lesson 21: Curve Sketching (Section 021 handout)

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