The increasing/decreasing test and test for concavity allow us to spread out all the information we need about a function to reconstruct its graph.

1. Section 4.4
Curve Sketching I
V63.0121, Calculus I
March 30, 2009
Announcements
Quiz 4 this week (Sections 2.5–3.5)
Office hours this week: M 1–2, T 1–2, W 2–3, R 9–10
.
.
Image credit: Fast Eddie 42
. . . . . .

2. Office Hours and other help
In addition to recitation
Day Time Who/What Where in WWH
M 1:00–2:00 Leingang OH 718/618
3:30–4:30 Katarina OH 707
5:00–7:00 Curto PS 517
T 1:00–2:00 Leingang OH 718/618
4:00–5:50 Curto PS 317
W 1:00–2:00 Katarina OH 707
2:00–3:00 Leingang OH 718/618
R 9:00–10:00am Leingang OH 718/618
5:00–7:00pm Maria OH 807
F 2:00–4:00 Curto OH 1310
. . . . . .

3. CIMS/NYU professor wins Abel Prize
Mikhail Gromov, born
1943 in Russia
contributions to
geometry and topology
discovered the
pseudoholomorphic
curve
Abel Prize is the highest
in mathematics
. . . . . .

4. On the problems assigned from Section 2.8
Announcements were made in class but not online
Resubmit your Problem Set 6 on Wednesday, April 1 with
Problem Set 8
We will pick two additional problems to grade from Problem
Set 8
If the scores on the makeup problems from PS 8 exceed the
scores of the Section 2.8 problems from PS 6, the makeup
scores will be substituted.
This also takes care of the problematic problem 2.8.28.
This offer is only good this week.
. . . . . .

5. Outline
The Procedure
The examples
A cubic function
A quartic function
. . . . . .

6. The Increasing/Decreasing Test
Theorem (The Increasing/Decreasing Test)
If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b),
then f is decreasing on (a, b).
Proof.
It works the same as the last theorem. Pick two points x and y in
(a, b) with x < y. We must show f(x) < f(y). By MVT there exists
a point c in (x, y) such that
f(y) − f(x)
= f′ (c) > 0.
y−x
So
f(y) − f(x) = f′ (c)(y − x) > 0.
. . . . . .

7. Theorem (Concavity Test)
If f′′ (x) > 0 for all x in I, then the graph of f is concave
upward on I
If f′′ (x) < 0 for all x in I, then the graph of f is concave
downward on I
Proof.
Suppose f′′ (x) > 0 on I. This means f′ is increasing on I. Let a and
x be in I. The tangent line through (a, f(a)) is the graph of
L(x) = f(a) + f′ (a)(x − a)
f(x) − f(a)
= f′ (b).
By MVT, there exists a b between a and x with
x−a
So
f(x) = f(a) + f′ (b)(x − a) ≥ f(a) + f′ (a)(x − a) = L(x)
. . . . . .

8. Graphing Checklist
To graph a function f, follow this plan:
0. Find when f is positive, negative, zero, not defined.
1. Find f′ and form its sign chart. Conclude information about
increasing/decreasing and local max/min.
2. Find f′′ and form its sign chart. Conclude concave
up/concave down and inflection.
3. Put together a big chart to assemble monotonicity and
concavity data
4. Graph!
. . . . . .

9. Outline
The Procedure
The examples
A cubic function
A quartic function
. . . . . .

10. Graphing a cubic
Example
Graph f(x) = 2x3 − 3x2 − 12x.
. . . . . .

11. Graphing a cubic
Example
Graph f(x) = 2x3 − 3x2 − 12x.
First, let’s find the zeros. We can at least factor out one power of
x:
f(x) = x(2x2 − 3x − 12)
so f(0) = 0. The other factor is a quadratic, so we the other two
roots are
√
√
3 ± 32 − 4(2)(−12) 3 ± 105
x= =
4 4
It’s OK to skip this step for now since the roots are so
complicated.
. . . . . .

12. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
.
. . . . . .

13. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
. . . −2
x
2
.
. x
. +1
−
.1
.′ (x)
f
. .
− 2
.
.1 f
.(x)
. . . . . .

14. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
. x
. +1
−
.1
.′ (x)
f
. .
− 2
.
.1 f
.(x)
. . . . . .

15. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
. .
− 2
.
.1 f
.(x)
. . . . . .

16. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
..
+ .
− 2
.
.1 f
.(x)
. . . . . .

17. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. .
+ .
− 2
.
.1 f
.(x)
. . . . . .

18. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
− 2
.
.1 f
.(x)
. . . . . .

19. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
↗− 2
.
. .1 f
.(x)
. . . . . .

20. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
↗− ↘ 2
.
. .1 . f
.(x)
. . . . . .

21. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
↗− ↘ ↗
2
.
. .1 . . f
.(x)
. . . . . .

22. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
↗− ↘ ↗
2
.
. .1 . . f
.(x)
m
. ax
. . . . . .

23. Monotonicity
f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)
We can form a sign chart from this:
− −
. .. .
+
. . −2
x
2
.
−
.. . .
+ +
x
. +1
−
.1
.′ (x)
f
−
.. . .
+ +
.
↗− ↘ ↗
2
.
. .1 . . f
.(x)
m
. ax m
. in
. . . . . .

24. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
. . . . . .

25. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
.
f
. .(x)
1/2
. . . . . .

26. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
−
.− .
f
. .(x)
1/2
. . . . . .

27. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
−
.− .+
+
.
f
. .(x)
1/2
. . . . . .

28. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
−
.− .+
+
.
.
⌢ f
. .(x)
1/2
. . . . . .

29. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
−
.− .+
+
.
. .
⌢ ⌣ f
. .(x)
1/2
. . . . . .

30. Concavity
f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
−
.− .+
+
.
. .
⌢ ⌣ f
. .(x)
1/2
I
.P
. . . . . .

31. One sign chart to rule them all
.
. . . . . .

32. One sign chart to rule them all
.′ (x)
f
− −
. . .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
. onotonicity
. . . . . .

33. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘. ↗
2
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. . . .
⌢ ⌢ 1/2 ⌣ ⌣ c
. oncavity
.
. . . . . .

34. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. ⌢ 1/2 .
. .
⌢ ⌣ ⌣ c
. oncavity
.
− f
. 61/2 .(x)
−.
. 20
7
.. .
− 2
.
.1 . . hape of f
s
1/2
m
. ax I
.P m
. in
. . . . . .

35. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. ⌢ 1/2 .
. .
⌢ ⌣ ⌣ c
. oncavity
.
− f
. 61/2 .(x)
−.
. 20
7
.. .
..1
− 2
.
. . hape of f
s
1/2
m
. ax I
.P m
. in
. . . . . .

36. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. ⌢ 1/2 .
. .
⌢ ⌣ ⌣ c
. oncavity
.
− f
. 61/2 .(x)
−.
. 20
7
.. .
. . 1 . 1/2
− 2
.
. . hape of f
s
m
. ax I
.P m
. in
. . . . . .

37. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. ⌢ 1/2 .
. .
⌢ ⌣ ⌣ c
. oncavity
.
− f
. 61/2 .(x)
−.
. 20
7
.. .
. . 1 . 1/2 .
− 2
.
. . hape of f
s
m
. ax I
.P m
. in
. . . . . .

38. One sign chart to rule them all
.′ (x)
f
− −
.. .. . .
+ +
.
↗− ↘ ↘ ↗
2
.
. .1 . . . m
.′′ onotonicity
f
. (x)
−
.− −
.− . .+ .+
+ +
. ⌢ 1/2 .
. .
⌢ ⌣ ⌣ c
. oncavity
.
− f
. 61/2 .(x)
−.
. 20
7
.. .
. . 1 . 1/2 . .
− 2
.
. . hape of f
s
m
. ax I
.P m
. in
. . . . . .

39. Graph
f
.(x)
. −1, 7)
(
(√ ) .
. 3− 4105 , 0 . 0, 0)
(
. . . x
(. )
√
. 1/2, −61/2)
( 3+ 105
. ,0
. 4
.
. 2, −20)
(
. . . . . .

40. Graph
f
.(x)
. −1, 7)
(
(√ ) .
. 3− 4105 , 0 . 0, 0)
(
. . . x
(. )
√
. 1/2, −61/2)
( 3+ 105
. ,0
. 4
.
. 2, −20)
(
. . . . . .

41. Graph
f
.(x)
. −1, 7)
(
(√ ) .
. 3− 4105 , 0 . 0, 0)
(
. . . x
(. )
√
. 1/2, −61/2)
( 3+ 105
. ,0
. 4
.
. 2, −20)
(
. . . . . .

42. Graph
f
.(x)
. −1, 7)
(
(√ ) .
. 3− 4105 , 0 . 0, 0)
(
. . . x
(. )
√
. 1/2, −61/2)
( 3+ 105
. ,0
. 4
.
. 2, −20)
(
. . . . . .

43. Graph
f
.(x)
. −1, 7)
(
(√ ) .
. 3− 4105 , 0 . 0, 0)
(
. . . x
(. )
√
. 1/2, −61/2)
( 3+ 105
. ,0
. 4
.
. 2, −20)
(
. . . . . .

44. Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
. . . . . .

45. Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
We know f(0) = 10 and lim f(x) = +∞. Not too many other
x→±∞
points on the graph are evident.
. . . . . .

46. Monotonicity
f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. .. . .
+0 + +
. x2
4
0
.
− −
. . .. .
0+
. x − 3)
(
3
.
.′ (x)
f
−0 −
. .. . .. .
0+
↘0 ↘ 3↗
.. ..
. f
.(x)
m
. in
. . . . . .

47. Concavity
f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
−0
. .. . .
+ +
1
. 2x
0
.
− −
. . .
0
.. +
. −2
x
2
.
.′′ (x)
f
−
.−
. + .. .+
+0 0
.. +
.. . .
⌣0 ⌢ ⌣
2
. f
.(x)
I
.P I
.P
. . . . . .

48. Grand Unified Sign Chart
.
.′ (x)
f
−0 − −0+
. .. . . .. .
↘0 ↘ ↘3↗
.. . ..
. m
.′′ onotonicity
f
. (x)
−
.−
. + .. .. . + . +
+0 0+ +
.. . . .
⌣0 ⌢ ⌣ ⌣
2
. c
. oncavity
f
.(x)
− −.
. .6 . 17
1.
.0
0
. 2
. 3
. s
. hape
I
.P I
.P . inm
. . . . . .

49. Grand Unified Sign Chart
.
.′ (x)
f
−0 − −0+
. .. . . .. .
↘0 ↘ ↘3↗
.. . ..
. m
.′′ onotonicity
f
. (x)
−
.−
. + .. .. . + . +
+0 0+ +
.. . . .
⌣0 ⌢ ⌣ ⌣
2
. c
. oncavity
f
.(x)
− −.
. .6 . 17
1.
.0
..0 2
. 3
. s
. hape
I
.P I
.P . inm
. . . . . .

50. Grand Unified Sign Chart
.
.′ (x)
f
−0 − −0+
. .. . . .. .
↘0 ↘ ↘3↗
.. . ..
. m
.′′ onotonicity
f
. (x)
−
.−
. + .. .. . + . +
+0 0+ +
.. . . .
⌣0 ⌢ ⌣ ⌣
2
. c
. oncavity
f
.(x)
− −.
. .6 . 17
1.
.0
.. .
0 2
. 3
. s
. hape
I
.P I
.P . inm
. . . . . .

51. Grand Unified Sign Chart
.
.′ (x)
f
−0 − −0+
. .. . . .. .
↘0 ↘ ↘3↗
.. . ..
. m
.′′ onotonicity
f
. (x)
−
.−
. + .. .. . + . +
+0 0+ +
.. . . .
⌣0 ⌢ ⌣ ⌣
2
. c
. oncavity
f
.(x)
− −.
. .6 . 17
1.
.0
.. . ..
0 2
. 3 s
. hape
I
.P I
.P . inm
. . . . . .

52. Grand Unified Sign Chart
.
.′ (x)
f
−0 − −0+
. .. . . .. .
↘0 ↘ ↘3↗
.. . ..
. m
.′′ onotonicity
f
. (x)
−
.−
. + .. .. . + . +
+0 0+ +
.. . . .
⌣0 ⌢ ⌣ ⌣
2
. c
. oncavity
f
.(x)
− −.
. .6 . 17
1.
.0
.. . ...
0 2
. 3 s
. hape
I
.P I
.P . inm
. . . . . .

53. Graph
y
.
. 0, 10)
(
.
. x
.
.
.
. 2, −6)
(
. 3, −17)
(
. . . . . .