lesson 19: curve sketching

. . . . . .

Section 4.4Curve Sketching

V63.0121.006/016, Calculus I

New York University

April 1, 2010

. . . . . .

Second-chance Midterm: Tomorrow in Recitation

I 12 free-response questions, no multiple choiceI Covers all sections so far, through todayI Your score on this exam will replace your midterm score

V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 2 / 47

. . . . . .


. . . . . .

Quiz 3 tomorrow in recitation

I Section 2.6: implicit differentiationI Section 2.8: linear approximation and differentialsI Section 3.1: exponential functionsI Section 3.2: logarithmsI Section 3.3: derivatives of logarithmic and exponential functionsI Section 3.4: exponential growth and decayI Section 3.5: inverse trigonometric functions


. . . . . .

Outline

The Procedure

Simple examplesA cubic functionA quartic function

More ExamplesPoints of nondifferentiabilityHorizontal asymptotesVertical asymptotesTrigonometric and polynomial togetherLogarithmic


. . . . . .

Objective

Given a function, graph itcompletely, indicating

I zeroesI asymptotes if applicableI critical pointsI local/global max/minI inflection points


. . . . . .

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 fis 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 concavedownward 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/decreasingand local max/min.

2. Find f′′ and form its sign chart. Concludeconcave up/concave down and inflection.

3. Put together a big chart to assemblemonotonicity and concavity data

4. Graph!


. . . . . .

Outline

The Procedure




. . . . . .

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 powerof x:

f(x) = x(2x2 − 3x− 12)

so f(0) = 0. The other factor is a quadratic, so we the other two rootsare

x =3±

√32 − 4(2)(−12)

4=

3±√105

4It’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:

.

.x− 2..2

.− .− .+

.x+ 1..−1

.+.+.−

.f′(x)

.f(x)..2

..−1

.+ .− .+.↗ .↘ .↗.max .min


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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: .

.f′′(x)

.f(x).

.1/2.−− .++.⌢ .⌣

.IP


. . . . . .

Step 2: Concavity

f′(x) = 6x2 − 6x− 12=⇒ f′′(x) = 12x− 6 = 6(2x− 1)


.f′′(x)

.f(x).

.1/2

.−− .++.⌢ .⌣

.IP


. . . . . .

Step 2: Concavity

f′(x) = 6x2 − 6x− 12=⇒ f′′(x) = 12x− 6 = 6(2x− 1)


.f′′(x)

.f(x).

.1/2.−−

.++.⌢ .⌣

.IP


. . . . . .

Step 2: Concavity

f′(x) = 6x2 − 6x− 12=⇒ f′′(x) = 12x− 6 = 6(2x− 1)


.f′′(x)

.f(x).

.1/2.−− .++

.⌢ .⌣.IP


. . . . . .

Step 2: Concavity

f′(x) = 6x2 − 6x− 12=⇒ f′′(x) = 12x− 6 = 6(2x− 1)


.f′′(x)

.f(x).

.1/2.−− .++.⌢

.⌣.IP


. . . . . .

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.

.

.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

." . . . "


. . . . . .


Remember, f(x) = 2x3 − 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

.decreasing,concavedown

.increasing,concavedown

.decreasing,concave up

.increasing,concave up


. . . . . .


Remember, f(x) = 2x3 − 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."

. . . "


. . . . . .


Remember, f(x) = 2x3 − 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." .

. . "


. . . . . .


Remember, f(x) = 2x3 − 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." . .

. "


. . . . . .


Remember, f(x) = 2x3 − 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

(Step 0) We know f(0) = 10 and limx→±∞

f(x) = +∞. Not too many otherpoints 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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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


. . . . . .


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)

Here is its sign chart:

.

.12x..0.0

.− .+ .+

.x− 2..2.0.− .− .+

.f′′(x)

.f(x)..0.0 .

.2

.0.++ .−− .++.⌣ .⌢ .⌣

.IP .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 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 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 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 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 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 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 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 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 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 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 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 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 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 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

The Procedure




. . . . . .

Example

Graph f(x) = x+√

|x|

This function looks strange because of the absolute value. Butwhenever we become nervous, we can just take cases.


. . . . . .

Step 0: Finding Zeroes

f(x) = x+√

|x|I First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if

x is positive.

I 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: Finding Zeroes

f(x) = x+√

|x|I First, look at f by itself. We can tell that f(0) = 0 and that f(x) > 0 if

x is positive.I 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|I lim

x→∞f(x) = ∞, because both terms tend to ∞.

I limx→−∞

f(x) is indeterminate of the form −∞+∞. It’s the same aslim

y→+∞(−y+

√y)

limy→+∞

(−y+√y) = lim

y→∞(√y− y) ·

√y+ y√y+ y

= limy→∞

y− y2√y+ y

= −∞


. . . . . .

Step 1: The derivative

Remember, f(x) = x+√

|x|.To find f′, first assume x > 0. Then

f′(x) =ddx

(x+

√x)= 1+

12√x

NoticeI f′(x) > 0 when x > 0 (so no critical points here)I lim

x→0+f′(x) = ∞ (so 0 is a critical point)

I limx→∞

f′(x) = 1 (so the graph is asymptotic to a line of slope 1)


. . . . . .




f′(x) =ddx

(x+

√x)= 1+

12√x

NoticeI f′(x) > 0 when x > 0 (so no critical points here)

I limx→0+

f′(x) = ∞ (so 0 is a critical point)

I limx→∞



. . . . . .




f′(x) =ddx

(x+

√x)= 1+

12√x

NoticeI f′(x) > 0 when x > 0 (so no critical points here)I lim

x→0+f′(x) = ∞ (so 0 is a critical point)

I limx→∞



. . . . . .



|x|.If x is negative, we have

f′(x) =ddx

(x+

√−x

)= 1− 1

2√−x

NoticeI lim

x→0−f′(x) = −∞ (other side of the critical point)

I limx→−∞

f′(x) = 1 (asymptotic to a line of slope 1)

I f′(x) = 0 when

1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−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).

.−14

.0 ..0

.∓∞

.+ .− .+.↗ .↘ .↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+

.− .+.↗ .↘ .↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .−

.+.↗ .↘ .↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗ .↘ .↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+.↗

.↘ .↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+.↗ .↘

.↗. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+.↗ .↘ .↗

. max. min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+.↗ .↘ .↗. max

.min


. . . . . .


f′(x) =

1+

12√x

if x > 0

1− 12√−x

if x < 0


. .f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+.↗ .↘ .↗. max. min


. . . . . .

Step 2: Concavity

I If x > 0, then

f′′(x) =ddx

(1+

12x−1/2

)= −1

4x−3/2

This is negative whenever x > 0.

I If x < 0, then

f′′(x) =ddx

(1− 1

2(−x)−1/2

)= −1

4(−x)−3/2

which is also always negative for negative x.

I In other words, f′′(x) = −14|x|−3/2.

Here is the sign chart:

. .f′′(x)

.f(x)..0

.−∞.−−.⌢

.

..−−.⌢


. . . . . .

Step 2: Concavity

I If x > 0, then

f′′(x) =ddx

(1+

12x−1/2

)= −1

4x−3/2

This is negative whenever x > 0.I If x < 0, then

f′′(x) =ddx

(1− 1

2(−x)−1/2

)= −1

4(−x)−3/2

which is also always negative for negative x.

I In other words, f′′(x) = −14|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.

.−14

.0 ..0

.∓∞.+1.↗

.+.↗

.−.↘

.+.↗

.+1.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.−∞.⌢

.−∞.⌢.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

.min

.−∞ .+∞

." ." . ."


. . . . . .

Step 3: Synthesis


f(x) = x+√

|x|

. .f′(x)

.monotonicity.

.−14

.0 ..0

.∓∞.+1.↗

.+.↗

.−.↘

.+.↗

.+1.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.−∞.⌢

.−∞.⌢.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

.min

.−∞ .+∞."

." . ."


. . . . . .

Step 3: Synthesis


f(x) = x+√

|x|

. .f′(x)

.monotonicity.

.−14

.0 ..0

.∓∞.+1.↗

.+.↗

.−.↘

.+.↗

.+1.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.−∞.⌢

.−∞.⌢.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

.min

.−∞ .+∞." ."

. ."


. . . . . .

Step 3: Synthesis


f(x) = x+√

|x|

. .f′(x)

.monotonicity.

.−14

.0 ..0

.∓∞.+1.↗

.+.↗

.−.↘

.+.↗

.+1.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.−∞.⌢

.−∞.⌢.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

.min

.−∞ .+∞." ." .

."


. . . . . .

Step 3: Synthesis


f(x) = x+√

|x|

. .f′(x)

.monotonicity.

.−14

.0 ..0

.∓∞.+1.↗

.+.↗

.−.↘

.+.↗

.+1.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.−∞.⌢

.−∞.⌢.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

.min

.−∞ .+∞." ." . ."


. . . . . .

Graph

f(x) = x+√

|x|

.

.f(x)

.shape.

.−1.0

. zero

.−∞ .+∞..−1

4

.14

. max

.−∞ .+∞..0.0

.min

.−∞ .+∞." ." . ."

.x

.f(x)

..(−1,0) .

.(−14 ,

14)

..(0,0)


. . . . . .

Example

Graph f(x) = xe−x2

Before taking derivatives, we notice that f is odd, that f(0) = 0, andlim

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 off′(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

.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

.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

.−√

32e3

.IP

..0.0

.IP

..√

3/2

.√

32e3

.IP

. . . " ." . .


. . . . . .

Step 4: Graph

.

.x

.f(x)

.f(x) = xe−x2

..(−√

1/2,− 1√2e

)

..(√

1/2, 1√2e

)

.

.(−√

3/2,−√

32e3

)..(0,0)

..(√

3/2,√

32e3

)

.f(x)

.shape.

.−√

1/2

.− 1√2e

.min

..√

1/2

. 1√2e

. max

..−√

3/2

.−√

32e3

.IP

..0.0

.IP

..√

3/2

.√

32e3

.IP

. . . " ." . .


. . . . . .

Example

Graph f(x) =1x+

1x2


. . . . . .

Step 0

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

We need to factor f:

f(x) =1x+

1x2

=x+ 1x2

.

This means f is 0 at −1 and has trouble at 0. In fact,

limx→0

x+ 1x2

= ∞,

so x = 0 is a vertical asymptote of the graph. We can make a signchart as follows:

. .x+ 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+


. . . . . .

Step 0

Find when f is positive, negative, zero, not defined. We need to factor f:

f(x) =1x+

1x2

=x+ 1x2

.


limx→0

x+ 1x2

= ∞,

so x = 0 is a vertical asymptote of the graph.

We can make a signchart as follows:

. .x+ 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+


. . . . . .

Step 0

Find when f is positive, negative, zero, not defined. We need to factor f:

f(x) =1x+

1x2

=x+ 1x2

.


limx→0

x+ 1x2

= ∞,

so x = 0 is a vertical asymptote of the graph. We can make a signchart as follows:

. .x+ 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+


. . . . . .

Step 0, continued

For horizontal asymptotes, notice that

limx→∞

x+ 1x2

= 0,

so y = 0 is a horizontal asymptote of the graph. The same is true at−∞.


. . . . . .


We havef′(x) = − 1

x2− 2

x3= −x+ 2

x3.

The critical points are x = −2 and x = 0. We have the following signchart:

. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−

.↘ .↗ .↘.min .VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘

.↗ .↘.min .VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗

.↘.min .VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min

.VA


. . . . . .



x2− 2

x3= −x+ 2

x3.


. .−(x+ 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=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

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−−

.++ .++.⌢ .⌣ .⌣

.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++

.++.⌢ .⌣ .⌣

.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++

.⌢ .⌣ .⌣.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢

.⌣ .⌣.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣

.⌣.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=2(x+ 3)

x4.


. .(x+ 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP

.VA


. . . . . .

Step 2: Concavity

We havef′′(x) =

2x3

+6x4

=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

.− .+ .+

.HA . .IP . .min . " .0 . " .VA . .HA


. . . . . .

Step 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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 3: Synthesis

.

.f′


..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


. . . . . .

ProblemGraph f(x) = cos x− x

. .x

.y


. . . . . .

ProblemGraph f(x) = x ln x2

. .x

.y


. . . . . .

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/decreasingand local max/min.

2. Find f′′ and form its sign chart. Concludeconcave up/concave down and inflection.

3. Put together a big chart to assemblemonotonicity and concavity data

4. Graph!


lesson 19: curve sketching

Education