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 31, 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
Day Time Who/What Where in WWH
M 1:00–2:00 Leingang OH 718/618
5:00–7:00 Curto PS 517
T 1:00–2:00 Leingang OH 718/618
4:00–5:50 Curto PS 317
W 2:00–3:00 Leingang OH 718/618
R 9:00–10:00am Leingang OH 718/618
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. Outline
The Procedure
The examples
A cubic function
A quartic function
Worksheet
. . . . . .

5. 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.
. . . . . .

6. 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)
. . . . . .

7. 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!
. . . . . .

8. Outline
The Procedure
The examples
A cubic function
A quartic function
Worksheet
. . . . . .

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

10. 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.
. . . . . .

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

12. 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)
. . . . . .

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)
m
. ax
. . . . . .

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 m
. in
. . . . . .

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

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

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
I
.P
. . . . . .

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

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

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

33. 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
. . . . . .

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

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

44. 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.
. . . . . .

45. 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
. . . . . .

46. 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
. . . . . .

47. 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
. . . . . .

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

53. Worksheet
.
Image: Erick Cifuentes
.
. . . . . .

54. Worksheet Problem 1
Problem
Graph f(x) = x3 − 3x2 + 3x
. . . . . .

55. Worksheet Problem 1
Problem
Graph f(x) = x3 − 3x2 + 3x
5
1 1 2 3
5
. . . . . .

56. Worksheet Problem 2
Problem
Graph f(x) = x4 − 3x2 + 2x
. . . . . .

57. Worksheet Problem 2
Problem
Graph f(x) = x4 − 3x2 + 2x
8
6
4
2
2 1 1 2
2
4
. . . . . .

58. Worksheet Problem 3
Problem
Graph f(x) = cos x − x
. . . . . .

59. Worksheet Problem 3
Problem
Graph f(x) = cos x − x
5
5 5 10
5
10
. . . . . .

60. Worksheet Problem 4
Problem
Graph f(x) = x ln x2
. . . . . .

61. Worksheet Problem 4
Problem
Graph f(x) = x ln x2
6
4
2
3 2 1 1 2 3
2
4
6
. . . . . .

62. Worksheet Problem 5
Problem
2
Graph f(x) = e−x
. . . . . .

63. Worksheet Problem 5
Problem
2
Graph f(x) = e−x
1.0
0.8
0.6
0.4
0.2
3 2 1 1 2 3
. . . . . .