lesson 21: curve sketching (section 041 slides)
DESCRIPTION
We can put all of our graph-description techniques into a single picture. (The problem I did on the sketchpad is now prettified.)TRANSCRIPT
![Page 1: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/1.jpg)
..
Section 4.4Curve Sketching
V63.0121.041, Calculus I
New York University
November 17, 2010
AnnouncementsI Quiz 4 this week in recitation on 3.3, 3.4, 3.5, 3.7I There is class on November 24
. . . . . .
![Page 2: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/2.jpg)
. . . . . .
Announcements
I Quiz 4 this week inrecitation on 3.3, 3.4, 3.5,3.7
I There is class onNovember 24
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 2 / 55
![Page 3: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/3.jpg)
. . . . . .
Objectives
I given a function, graph itcompletely, indicating
I zeroes (if easy)I asymptotes if applicableI critical pointsI local/global max/minI inflection points
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 3 / 55
![Page 4: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/4.jpg)
. . . . . .
Why?
Graphing functions is likedissection
… or diagrammingsentencesYou can really know a lot abouta function when you know all ofits anatomy.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 4 / 55
![Page 5: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/5.jpg)
. . . . . .
Why?
Graphing functions is likedissection … or diagrammingsentences
You can really know a lot abouta function when you know all ofits anatomy.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 4 / 55
![Page 6: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/6.jpg)
. . . . . .
Why?
Graphing functions is likedissection … or diagrammingsentencesYou can really know a lot abouta function when you know all ofits anatomy.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 4 / 55
![Page 7: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/7.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 5 / 55
![Page 8: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/8.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 6 / 55
![Page 9: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/9.jpg)
. . . . . .
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!
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 7 / 55
![Page 10: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/10.jpg)
. . . . . .
Outline
Simple examplesA cubic functionA quartic function
More ExamplesPoints of nondifferentiabilityHorizontal asymptotesVertical asymptotesTrigonometric and polynomial togetherLogarithmic
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 8 / 55
![Page 11: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/11.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 9 / 55
![Page 12: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/12.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 9 / 55
![Page 13: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/13.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 14: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/14.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 15: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/15.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 16: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/16.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 17: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/17.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 18: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/18.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 19: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/19.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 20: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/20.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 21: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/21.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 22: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/22.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 23: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/23.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 24: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/24.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 10 / 55
![Page 25: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/25.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 26: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/26.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 27: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/27.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 28: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/28.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 29: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/29.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 30: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/30.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 31: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/31.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 11 / 55
![Page 32: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/32.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 12 / 55
![Page 33: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/33.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 12 / 55
![Page 34: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/34.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 12 / 55
![Page 35: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/35.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 12 / 55
![Page 36: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/36.jpg)
. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV
.
decreasing,concavedown
.
increasing,concavedown
.
decreasing,concave up
.
increasing,concave up
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 13 / 55
![Page 37: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/37.jpg)
. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV
.
decreasing,concavedown
.
increasing,concavedown
.
decreasing,concave up
.
increasing,concave up
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 13 / 55
![Page 38: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/38.jpg)
. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV
.
decreasing,concavedown
.
increasing,concavedown
.
decreasing,concave up
.
increasing,concave up
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 13 / 55
![Page 39: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/39.jpg)
. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV
.
decreasing,concavedown
.
increasing,concavedown
.
decreasing,concave up
.
increasing,concave up
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 13 / 55
![Page 40: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/40.jpg)
. . . . . .
Combinations of monotonicity and concavity
..
I
.
II
.
III
.
IV
.
decreasing,concavedown
.
increasing,concavedown
.
decreasing,concave up
.
increasing,concave up
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 13 / 55
![Page 41: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/41.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 14 / 55
![Page 42: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/42.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 14 / 55
![Page 43: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/43.jpg)
. . . . . .
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
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 14 / 55
![Page 44: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/44.jpg)
. . . . . .
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
.
"
.
.
."
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 14 / 55
![Page 45: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/45.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 15 / 55
![Page 46: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/46.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 15 / 55
![Page 47: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/47.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 15 / 55
![Page 48: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/48.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 15 / 55
![Page 49: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/49.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 15 / 55
![Page 50: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/50.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 16 / 55
![Page 51: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/51.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 16 / 55
![Page 52: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/52.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 53: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/53.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 54: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/54.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 55: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/55.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 56: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/56.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 57: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/57.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 58: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/58.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 59: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/59.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 60: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/60.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 61: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/61.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 62: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/62.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 63: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/63.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 64: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/64.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 65: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/65.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 66: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/66.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 67: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/67.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 68: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/68.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 69: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/69.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 17 / 55
![Page 70: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/70.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 71: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/71.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 72: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/72.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 73: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/73.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 74: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/74.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 75: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/75.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 76: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/76.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 77: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/77.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 78: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/78.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 79: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/79.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 80: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/80.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 81: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/81.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 82: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/82.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 83: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/83.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 84: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/84.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 85: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/85.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 86: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/86.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 87: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/87.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 88: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/88.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 18 / 55
![Page 89: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/89.jpg)
. . . . . .
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
.
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 19 / 55
![Page 90: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/90.jpg)
. . . . . .
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
.
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 19 / 55
![Page 91: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/91.jpg)
. . . . . .
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
.
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 19 / 55
![Page 92: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/92.jpg)
. . . . . .
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
.
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 19 / 55
![Page 93: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/93.jpg)
. . . . . .
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
.
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 19 / 55
![Page 94: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/94.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 20 / 55
![Page 95: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/95.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 20 / 55
![Page 96: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/96.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 20 / 55
![Page 97: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/97.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 20 / 55
![Page 98: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/98.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 20 / 55
![Page 99: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/99.jpg)
. . . . . .
Outline
Simple examplesA cubic functionA quartic function
More ExamplesPoints of nondifferentiabilityHorizontal asymptotesVertical asymptotesTrigonometric and polynomial togetherLogarithmic
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 21 / 55
![Page 100: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/100.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 22 / 55
![Page 101: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/101.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 22 / 55
![Page 102: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/102.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 23 / 55
![Page 103: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/103.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 23 / 55
![Page 104: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/104.jpg)
. . . . . .
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.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 23 / 55
![Page 105: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/105.jpg)
. . . . . .
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
= −∞
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 24 / 55
![Page 106: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/106.jpg)
. . . . . .
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
= −∞
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 24 / 55
![Page 107: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/107.jpg)
. . . . . .
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
= −∞
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 24 / 55
![Page 108: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/108.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 25 / 55
![Page 109: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/109.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 25 / 55
![Page 110: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/110.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 25 / 55
![Page 111: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/111.jpg)
. . . . . .
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)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 25 / 55
![Page 112: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/112.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 26 / 55
![Page 113: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/113.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 26 / 55
![Page 114: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/114.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 26 / 55
![Page 115: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/115.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 116: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/116.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 117: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/117.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 118: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/118.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 119: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/119.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 120: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/120.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 121: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/121.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 122: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/122.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 123: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/123.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 124: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/124.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 125: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/125.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 27 / 55
![Page 126: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/126.jpg)
. . . . . .
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.−∞.−− .
⌢... −−.
⌢
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 28 / 55
![Page 127: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/127.jpg)
. . . . . .
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.−∞.−− .
⌢... −−.
⌢
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 28 / 55
![Page 128: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/128.jpg)
. . . . . .
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.−∞.−− .
⌢... −−.
⌢
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 28 / 55
![Page 129: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/129.jpg)
. . . . . .
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.−∞.−− .
⌢... −−.
⌢
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 28 / 55
![Page 130: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/130.jpg)
. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x+√
|x|
.. f′(x).monotonicity
..−1
4
.0 ..0.∓∞.+1 .
↗.+ .
↗.− .
↘. +.
↗. +1.
↗.
f′′(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−14
.
14
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 29 / 55
![Page 131: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/131.jpg)
. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x+√
|x|
.. f′(x).monotonicity
..−1
4
.0 ..0.∓∞.+1 .
↗.+ .
↗.− .
↘. +.
↗. +1.
↗.
f′′(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−14
.
14
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 29 / 55
![Page 132: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/132.jpg)
. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x+√
|x|
.. f′(x).monotonicity
..−1
4
.0 ..0.∓∞.+1 .
↗.+ .
↗.− .
↘. +.
↗. +1.
↗.
f′′(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−14
.
14
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 29 / 55
![Page 133: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/133.jpg)
. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x+√
|x|
.. f′(x).monotonicity
..−1
4
.0 ..0.∓∞.+1 .
↗.+ .
↗.− .
↘. +.
↗. +1.
↗.
f′′(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−14
.
14
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 29 / 55
![Page 134: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/134.jpg)
. . . . . .
Step 3: Synthesis
Now we can put these things together.
f(x) = x+√
|x|
.. f′(x).monotonicity
..−1
4
.0 ..0.∓∞.+1 .
↗.+ .
↗.− .
↘. +.
↗. +1.
↗.
f′′(x)
.
concavity
..
0
.
−∞
.
−−
.
⌢
.
−−
.
⌢
.
−−
.
⌢
.
−∞
.
⌢
.
−∞
.
⌢
.
f(x)
.
shape
..
−1
.
0
.
zero
..
−14
.
14
.
max
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 29 / 55
![Page 135: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/135.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 136: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/136.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 137: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/137.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 138: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/138.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 139: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/139.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 140: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/140.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 141: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/141.jpg)
. . . . . .
Graph
f(x) = x+√
|x|
..
f(x)
.
shape
..
−1
.
0
.
zero
.
−∞
.
+∞
..
−14
.
14
.
max
.
−∞
.
+∞
..
0
.
0
.
min
.
−∞
.
+∞
.
"
.
"
.
.
"
. x.
f(x)
..(−1,0)
..(−1
4 ,14)
..(0,0)
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 30 / 55
![Page 142: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/142.jpg)
. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2
Before taking derivatives, we notice that f is odd, that f(0) = 0, andlimx→∞
f(x) = 0
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 31 / 55
![Page 143: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/143.jpg)
. . . . . .
Example with Horizontal Asymptotes
Example
Graph f(x) = xe−x2
Before taking derivatives, we notice that f is odd, that f(0) = 0, andlimx→∞
f(x) = 0
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 31 / 55
![Page 144: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/144.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 145: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/145.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 146: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/146.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 147: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/147.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 148: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/148.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 149: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/149.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 150: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/150.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 151: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/151.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 152: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/152.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 153: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/153.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 154: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/154.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 155: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/155.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 156: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/156.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 157: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/157.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 158: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/158.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 32 / 55
![Page 159: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/159.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 160: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/160.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 161: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/161.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 162: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/162.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 163: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/163.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 164: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/164.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 165: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/165.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 166: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/166.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 167: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/167.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 168: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/168.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 169: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/169.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 170: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/170.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 171: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/171.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 172: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/172.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 173: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/173.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 174: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/174.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 175: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/175.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 176: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/176.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 177: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/177.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 178: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/178.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 179: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/179.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 180: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/180.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 181: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/181.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 182: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/182.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 33 / 55
![Page 183: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/183.jpg)
. . . . . .
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
.
.
.
"
.
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 184: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/184.jpg)
. . . . . .
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
.
.
.
"
.
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 185: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/185.jpg)
. . . . . .
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
.
.
.
"
.
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 186: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/186.jpg)
. . . . . .
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
.
.
.
"
.
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 187: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/187.jpg)
. . . . . .
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
.
.
.
".
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 188: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/188.jpg)
. . . . . .
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
.
.
.
".
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 189: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/189.jpg)
. . . . . .
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
.
.
.
".
"
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 34 / 55
![Page 190: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/190.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 191: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/191.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 192: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/192.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 193: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/193.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 194: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/194.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 195: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/195.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 196: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/196.jpg)
. . . . . .
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
. . ." . ". .
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 35 / 55
![Page 197: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/197.jpg)
. . . . . .
Example with Vertical Asymptotes
Example
Graph f(x) =1x+
1x2
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 36 / 55
![Page 198: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/198.jpg)
. . . . . .
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
.
−
.
+
.
+
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 37 / 55
![Page 199: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/199.jpg)
. . . . . .
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
.
−
.
+
.
+
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 37 / 55
![Page 200: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/200.jpg)
. . . . . .
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
.
−
.
+
.
+
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 37 / 55
![Page 201: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/201.jpg)
. . . . . .
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−∞.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 38 / 55
![Page 202: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/202.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 203: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/203.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 204: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/204.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 205: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/205.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 206: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/206.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 207: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/207.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 208: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/208.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 39 / 55
![Page 209: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/209.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 210: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/210.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 211: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/211.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 212: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/212.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 213: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/213.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 214: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/214.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 215: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/215.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 216: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/216.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 217: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/217.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 40 / 55
![Page 218: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/218.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
![Page 219: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/219.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
![Page 220: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/220.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
![Page 221: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/221.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
![Page 222: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/222.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
![Page 223: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/223.jpg)
. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 41 / 55
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. . . . . .
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
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 42 / 55
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. . . . . .
Trigonometric and polynomial together
ProblemGraph f(x) = cos x− x
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 43 / 55
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. . . . . .
Step 0: intercepts and asymptotes
I f(0) = 1 and f(−π/2) = −π/2. So by the Intermediate ValueTheorem there is a zero in between. We don’t know it’s precisevalue, though.
I Since −1 ≤ cos x ≤ 1 for all x, we have
−1− x ≤ cos x− x ≤ 1− x
for all x. This means that limx→∞
f(x) = −∞ and limx→−∞
f(x) = ∞.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 44 / 55
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. . . . . .
Step 1: Monotonicity
If f(x) = cos x− x, then f′(x) = − sin x− 1 = (−1)(sin x+ 1).I f′(x) = 0 if x = 3π/2+ 2πk, where k is any integerI f′(x) is periodic with period 2πI Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x+ 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x+ 1) ≤ 0
for all x. This means f′(x) is negative at all other points.
.. f′(x).f(x)
..−π/2
.0 ..3π/2
. 0..7π/2
. 0
. −.↘
. −.↘
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 45 / 55
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. . . . . .
Step 1: Monotonicity
If f(x) = cos x− x, then f′(x) = − sin x− 1 = (−1)(sin x+ 1).I f′(x) = 0 if x = 3π/2+ 2πk, where k is any integerI f′(x) is periodic with period 2πI Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x+ 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x+ 1) ≤ 0
for all x. This means f′(x) is negative at all other points.
.. f′(x).f(x)
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −
.↘
. −.↘
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 45 / 55
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. . . . . .
Step 1: Monotonicity
If f(x) = cos x− x, then f′(x) = − sin x− 1 = (−1)(sin x+ 1).I f′(x) = 0 if x = 3π/2+ 2πk, where k is any integerI f′(x) is periodic with period 2πI Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x+ 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x+ 1) ≤ 0
for all x. This means f′(x) is negative at all other points.
.. f′(x).f(x)
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −
.↘
. −
.↘
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 45 / 55
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. . . . . .
Step 1: Monotonicity
If f(x) = cos x− x, then f′(x) = − sin x− 1 = (−1)(sin x+ 1).I f′(x) = 0 if x = 3π/2+ 2πk, where k is any integerI f′(x) is periodic with period 2πI Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x+ 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x+ 1) ≤ 0
for all x. This means f′(x) is negative at all other points.
.. f′(x).f(x)
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −
.↘
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 45 / 55
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. . . . . .
Step 1: Monotonicity
If f(x) = cos x− x, then f′(x) = − sin x− 1 = (−1)(sin x+ 1).I f′(x) = 0 if x = 3π/2+ 2πk, where k is any integerI f′(x) is periodic with period 2πI Since −1 ≤ sin x ≤ 1 for all x, we have
0 ≤ sin x+ 1 ≤ 2 =⇒ −2 ≤ (−1)(sin x+ 1) ≤ 0
for all x. This means f′(x) is negative at all other points.
.. f′(x).f(x)
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 45 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−
.⌢. ++.
⌣. −−.
⌢. ++.
⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−
.⌢
. ++
.⌣
. −−.⌢
. ++.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 241: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/241.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−
.⌢
. ++
.⌣
. −−
.⌢
. ++.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 242: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/242.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−
.⌢
. ++
.⌣
. −−
.⌢
. ++
.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 243: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/243.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++
.⌣
. −−
.⌢
. ++
.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 244: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/244.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−
.⌢
. ++
.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++
.⌣
..−π/2
.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0
.
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
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. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0 .
IP
..π/2
. 0
.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 248: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/248.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0 .
IP
..π/2
. 0.
IP
..3π/2
. 0
.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 249: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/249.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0 .
IP
..π/2
. 0.
IP
..3π/2
. 0.
IP
..5π/2
. 0
.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 250: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/250.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0 .
IP
..π/2
. 0.
IP
..3π/2
. 0.
IP
..5π/2
. 0.
IP
..7π/2
. 0
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 251: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/251.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = − sin x− 1, then f′′(x) = − cos x.I This is 0 when x = π/2+ πk, where k is any integer.I This is periodic with period 2π
.. f′′(x).f(x)
.−−.⌢. ++.
⌣. −−.
⌢. ++.
⌣..
−π/2.0 .
IP
..π/2
. 0.
IP
..3π/2
. 0.
IP
..5π/2
. 0.
IP
..7π/2
. 0.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 46 / 55
![Page 252: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/252.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
.
f′′(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 47 / 55
![Page 253: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/253.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
.
f′′(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 47 / 55
![Page 254: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/254.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
.
f′′(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 47 / 55
![Page 255: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/255.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
.
f′′(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 47 / 55
![Page 256: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/256.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..−π/2
.0 ..3π/2
. 0..7π/2
. 0. −.↘
. −.↘
.
f′′(x)
.
conc
..
−π/2
.
0
..
π/2
.
0
..
3π/2
.
0
..
5π/2
.
0
..
7π/2
.
0
.
−−
.
⌢
.
++
.
⌣
.
−−
.
⌢
.
++
.
⌣
.
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 47 / 55
![Page 257: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/257.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 258: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/258.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 259: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/259.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 260: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/260.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 261: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/261.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 262: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/262.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 263: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/263.jpg)
. . . . . .
Step 4: Graph
f(x) = cos x− x
..x
.y.......
f(x)
.
shape
..
−π/2
.
π/2
.
IP
..
π/2
.
−π/2
.
IP
..
3π/2
.
−3π/2
.
IP
..
5π/2
.
−5π/2
.
IP
..
7π/2
.
−7π/2
.
IP
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 48 / 55
![Page 264: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/264.jpg)
. . . . . .
Logarithmic
ProblemGraph f(x) = x ln x2
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 49 / 55
![Page 265: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/265.jpg)
. . . . . .
Step 0: Intercepts and Asymptotes
I limx→∞
f(x) = ∞, limx→−∞
f(x) = −∞.
I f is not originally defined at 0 because limx→0
ln x2 = −∞. But
limx→0
x ln x2 = limx→0
ln x2
1/xH= lim
x→0
(1/x2)(2x)−1/x2
= limx→0
2x = 0.
So we can define f(0) = 0 to make it a continuous function on(−∞,∞).
I Other zeroes?
ln x2 = 0 =⇒ x2 = 1 =⇒ x = ±1
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 50 / 55
![Page 266: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/266.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e
.+ .↗
.− .↘
. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 267: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/267.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+
.↗
.− .↘
. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 268: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/268.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+
.↗
.−
.↘
. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 269: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/269.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+
.↗
.−
.↘
. −
.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 270: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/270.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+
.↗
.−
.↘
. −
.↘
. +
.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 271: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/271.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.−
.↘
. −
.↘
. +
.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 272: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/272.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −
.↘
. +
.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 273: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/273.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +
.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 274: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/274.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 275: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/275.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 276: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/276.jpg)
. . . . . .
Step 1: Monotonicity
If f(x) = x ln x2, then
f′(x) = ln x2 + x · 1x2
(2x) = ln x2 + 2
This is not defined at 0 and is 0 when
ln x2 = −2 =⇒ x2 = e−2 =⇒ x = ±e−1
Use test points ±1, ±e−2. Here is the sign chart:
.. f′(x).f(x)
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 51 / 55
![Page 277: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/277.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0
.−− .⌢
. ++.⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 278: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/278.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−−
.⌢
. ++.⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 279: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/279.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−−
.⌢
. ++
.⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 280: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/280.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−− .
⌢. ++
.⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 281: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/281.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−− .
⌢. ++.
⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 282: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/282.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−− .
⌢. ++.
⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 283: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/283.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−− .
⌢. ++.
⌣
.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 284: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/284.jpg)
. . . . . .
Step 2: Concavity
If f′(x) = ln x2+ 2, then f′′(x) = 1/x2 · (2x) = 2/x. Here is the sign chart:
.. f′(x).f(x)
..×.0.−− .
⌢. ++.
⌣.
IP
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 52 / 55
![Page 285: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/285.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 286: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/286.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 287: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/287.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 288: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/288.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 289: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/289.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 290: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/290.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 291: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/291.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
.
"
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 292: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/292.jpg)
. . . . . .
Step 3: Synthesis
.. f′(x).mono
..0 .−1/e
..×.0.. 0.
1/e.+ .
↗.− .
↘. −.↘
. +.↗
.
max
.
min
.
f′(x)
.
conc
..
×
.
0
.
−−
.
⌢
.
++
.
⌣
.
IP
.
f′(x)
.
shape
..
0
.
−1/e
..
×
.
0
..
0
.
1/e
.
"
.
.
."
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 53 / 55
![Page 293: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/293.jpg)
. . . . . .
Step 4: Graph
.. x.
y
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 54 / 55
![Page 294: Lesson 21: Curve Sketching (Section 041 slides)](https://reader034.vdocuments.site/reader034/viewer/2022052316/5597df921a28ab58388b47ba/html5/thumbnails/294.jpg)
. . . . . .
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?
V63.0121.041, Calculus I (NYU) Section 4.4 Curve Sketching November 17, 2010 55 / 55