lesson 21: curve sketching
DESCRIPTION
TRANSCRIPT
Section 4.4Curve Sketching
V63.0121.002.2010Su, Calculus I
New York University
June 10, 2010
AnnouncementsI Homework 4 due Tuesday
. . . . . .
. . . . . .
Announcements
I Homework 4 due Tuesday
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 2 / 45
. . . . . .
Objectives
I given a function, graph itcompletely, indicating
I zeroes (if easy)I asymptotes if applicableI critical pointsI local/global max/minI inflection points
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 3 / 45
. . . . . .
Why?
Graphing functions is likedissection
… or diagrammingsentencesYou can really know a lot abouta function when you know all ofits anatomy.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
. . . . . .
Why?
Graphing functions is likedissection … or diagrammingsentences
You can really know a lot abouta function when you know all ofits anatomy.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
. . . . . .
Why?
Graphing functions is likedissection … or diagrammingsentencesYou can really know a lot abouta function when you know all ofits anatomy.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 5 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 6 / 45
. . . . . .
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!
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 7 / 45
. . . . . .
Outline
Simple examplesA cubic functionA quartic function
More ExamplesPoints of nondifferentiabilityHorizontal asymptotesVertical asymptotesTrigonometric and polynomial togetherLogarithmic
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 8 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
. . . . . .
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
." . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
. . . . . .
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
." . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
. . . . . .
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
." . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
. . . . . .
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
." . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
. . . . . .
Combinations of monotonicity and concavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
. . . . . .
Combinations of monotonicity and concavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
. . . . . .
Combinations of monotonicity and concavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
. . . . . .
Combinations of monotonicity and concavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
. . . . . .
Combinations of monotonicity and concavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
. . . . . .
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."
. . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
. . . . . .
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." .
. . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
. . . . . .
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." . .
. "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
. . . . . .
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." . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
. . . . . .
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
. . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
. . . . . .
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.
. . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
. . . . . .
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. .
. . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
. . . . . .
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. . .
. "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
. . . . . .
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. . . . "
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 19 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 20 / 45
. . . . . .
Outline
Simple examplesA cubic functionA quartic function
More ExamplesPoints of nondifferentiabilityHorizontal asymptotesVertical asymptotesTrigonometric and polynomial togetherLogarithmic
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 21 / 45
. . . . . .
Graphing a function with a cusp
Example
Graph f(x) = x+√
|x|
This function looks strange because of the absolute value. Butwhenever we become nervous, we can just take cases.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 22 / 45
. . . . . .
Graphing a function with a cusp
Example
Graph f(x) = x+√
|x|
This function looks strange because of the absolute value. Butwhenever we become nervous, we can just take cases.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 22 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
. . . . . .
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 23 / 45
. . . . . .
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
= −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
. . . . . .
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
= −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
. . . . . .
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
= −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 24 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
. . . . . .
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 limx→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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 25 / 45
. . . . . .
Step 1: The derivative
Remember, f(x) = x+√
|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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
. . . . . .
Step 1: The derivative
Remember, f(x) = x+√
|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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
. . . . . .
Step 1: The derivative
Remember, f(x) = x+√
|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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 26 / 45
. . . . . .
Step 1: Monotonicity
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)
..−1
4
.0 ..0
.∓∞.+ .− .+.↗ .↘ .↗. max. min
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 27 / 45
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 28 / 45
. . . . . .
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
.−∞ .+∞
." ." . ."
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
. . . . . .
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
.−∞ .+∞."
." . ."
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
. . . . . .
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
.−∞ .+∞." ."
. ."
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
. . . . . .
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
.−∞ .+∞." ." .
."
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
. . . . . .
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
.−∞ .+∞." ." . ."
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 29 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
Graph
f(x) = x+√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
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)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 30 / 45
. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2
Before taking derivatives, we notice that f is odd, that f(0) = 0, andlim
x→∞f(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 31 / 45
. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2
Before taking derivatives, we notice that f is odd, that f(0) = 0, andlim
x→∞f(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 31 / 45
. . . . . .
Step 1: Monotonicity
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
.−.↘
.+.↗
.−.↘
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 32 / 45
. . . . . .
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
.−−.⌢
.++.⌣
.−−.⌢
.++.⌣
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 33 / 45
. . . . . .
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
. . . " ." . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 34 / 45
. . . . . .
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
. . . " ." . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 35 / 45
. . . . . .
Example with Vertical Asymptotes
Example
Graph f(x) =1x+
1x2
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 36 / 45
. . . . . .
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
.− .+ .+
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
. . . . . .
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
.− .+ .+
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
. . . . . .
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
.− .+ .+
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 37 / 45
. . . . . .
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−∞.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 38 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
Step 1: Monotonicity
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 39 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 40 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 41 / 45
. . . . . .
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
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 42 / 45
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 43 / 45
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 43 / 45
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 44 / 45
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 44 / 45
. . . . . .
Summary
I Graphing is a procedure that gets easier with practice.I Remember to follow the checklist.I Graphing is like dissection—or is it vivisection?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 45 / 45