1. Section 4.4
Curve Sketching
V63.0121.002.2010Su, Calculus I
New York University
June 10, 2010
Announcements
Homework 4 due Tuesday
. . . . . .

2. Announcements
Homework 4 due Tuesday
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 2 / 45

3. Objectives
given a function, graph it
completely, indicating
zeroes (if easy)
asymptotes if applicable
critical points
local/global max/min
inflection points
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 3 / 45

4. Why?
Graphing functions is like
dissection
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45

5. Why?
Graphing functions is like
dissection … or diagramming
sentences
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45

6. Why?
Graphing functions is like
dissection … or diagramming
sentences
You can really know a lot about
a function when you know all of
its anatomy.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45

7. 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).
Example
Here f(x) = x3 + x2 , and f′ (x) = 3x2 + 2x.
f
.(x)
.′ (x)
f
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 5 / 45

8. 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 concave
downward on (a, b).
Example
Here f(x) = x3 + x2 , f′ (x) = 3x2 + 2x, and f′′ (x) = 6x + 2.
.′′ (x)
f f
.(x)
.′ (x)
f
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 6 / 45

9. 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!
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 7 / 45

10. Outline
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 8 / 45

11. Graphing a cubic
Example
Graph f(x) = 2x3 − 3x2 − 12x.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45

12. 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 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.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45

13. 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:
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

14. 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:
. . . −2
x
2
.
. x
. +1
−
. 1
.′ (x)
f
. .
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

15. 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:
−
. −
. . . .
+
. −2
x
2
.
. x
. +1
−
. 1
.′ (x)
f
. .
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

16. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
.′ (x)
f
. .
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

17. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ .′ (x)
f
.
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

18. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .′ (x)
f
.
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

19. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
−
. 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

20. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
↗−
. . 1 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

21. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. 2
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

22. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. 2
. ↗
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

23. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. 2
. ↗
. f
.(x)
m
. ax
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

24. 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:
−
. −
. . . .
+
. −2
x
2
.
−
. . .
+ .
+
x
. +1
−
. 1
. .
+ −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. 2
. ↗
. f
.(x)
m
. ax m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45

25. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

26. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
.′′ (x)
f
.
./2
1 f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

27. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
−
. − .′′ (x)
f
.
./2
1 f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

28. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
−
. − . +
+ .′′ (x)
f
.
./2
1 f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

29. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
−
. − . +
+ .′′ (x)
f
.
.
⌢ ./2
1 f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

30. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
−
. − . +
+ .′′ (x)
f
.
.
⌢ ./2
1 .
⌣ f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

31. Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart: .
−
. − . +
+ .′′ (x)
f
.
.
⌢ ./2
1 .
⌣ f
.(x)
I
.P
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45

32. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45

33. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
−
. . . .
+ −
. .
+ .′ (x)
f
.
↗− ↘
. . 1 . ↘
. 2
. ↗
. m
. onotonicity
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45

34. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . +
+ . +
+ f
. (x)
.
⌢ .
⌢ 1/2
. .
⌣ .
⌣ c
. oncavity
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45

35. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . + + . +
+ f
. (x)
.
⌢ ⌢ ./2 .
. 1 ⌣ .
⌣ c
. oncavity
7
.. −
. 6 1/2 −.
. 20 f
.(x)
.
−
. 1 .
1/2 2
. s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45

36. Combinations of monotonicity and concavity
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45

37. Combinations of monotonicity and concavity
.
decreasing,
concave
down
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45

38. Combinations of monotonicity and concavity
. .
increasing, decreasing,
concave concave
down down
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45

39. Combinations of monotonicity and concavity
. .
increasing, decreasing,
concave concave
down down
I
.I I
.
.
I
.II I
.V
.
decreasing,
concave up
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45

40. Combinations of monotonicity and concavity
. .
increasing, decreasing,
concave concave
down down
I
.I I
.
.
I
.II I
.V
. .
decreasing, increasing,
concave up concave up
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45

41. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . + + . +
+ f
. (x)
.
⌢ ⌢ ./2 .
. 1 ⌣ .
⌣ c
. oncavity
7
.. −
. 6 1/2 −.
. 20 f
.(x)
.
. . 1
− .
1/2 2
. s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45

42. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . + + . +
+ f
. (x)
.
⌢ ⌢ ./2 .
. 1 ⌣ .
⌣ c
. oncavity
7
.. −
. 6 1/2 −.
. 20 f
.(x)
.
. . 1 . ./2
− 1 2
. s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45

43. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . + + . +
+ f
. (x)
.
⌢ ⌢ ./2 .
. 1 ⌣ .
⌣ c
. oncavity
7
.. −
. 6 1/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45

44. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
. .
+ −
. . −
. .
+ .′ (x)
f
.
↗−
. . 1 ↘
. ↘ .
. 2 ↗
. m
.′′ onotonicity
−
. − −
. − . . + + . +
+ f
. (x)
.
⌢ ⌢ ./2 .
. 1 ⌣ .
⌣ c
. oncavity
7
.. −
. 6 1/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45

45. Step 4: Graph
f
.(x)
.(x) = 2x3 − 3x2 − 12x
f
( √ ) . −1, 7)
(
.
. 3− 4105 , 0 . 0, 0)
(
. . .
. 1/2, −61/2)
( ( . x
√ )
. . 3+ 4105 , 0
. 2, −20)
(
.
7
.. −
. 61/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45

46. Step 4: Graph
f
.(x)
.(x) = 2x3 − 3x2 − 12x
f
( √ ) . −1, 7)
(
.
. 3− 4105 , 0 . 0, 0)
(
. . .
. 1/2, −61/2)
( ( . x
√ )
. . 3+ 4105 , 0
. 2, −20)
(
.
7
.. −
. 61/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45

47. Step 4: Graph
f
.(x)
.(x) = 2x3 − 3x2 − 12x
f
( √ ) . −1, 7)
(
.
. 3− 4105 , 0 . 0, 0)
(
. . .
. 1/2, −61/2)
( ( . x
√ )
. . 3+ 4105 , 0
. 2, −20)
(
.
7
.. −
. 61/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45

48. Step 4: Graph
f
.(x)
.(x) = 2x3 − 3x2 − 12x
f
( √ ) . −1, 7)
(
.
. 3− 4105 , 0 . 0, 0)
(
. . .
. 1/2, −61/2)
( ( . x
√ )
. . 3+ 4105 , 0
. 2, −20)
(
.
7
.. −
. 61/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45

49. Step 4: Graph
f
.(x)
.(x) = 2x3 − 3x2 − 12x
f
( √ ) . −1, 7)
(
.
. 3− 4105 , 0 . 0, 0)
(
. . .
. 1/2, −61/2)
( ( . x
√ )
. . 3+ 4105 , 0
. 2, −20)
(
.
7
.. −
. 61/2 −.
. 20 f
.(x)
.
. . 1 . ./2 .
− 1 2
. . s
. hape of f
m
. ax I
.P m
. in . . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45

50. Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45

51. Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
(Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many other
x→±∞
points on the graph are evident.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45

52. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

53. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

54. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
0
..
. x2
4
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

55. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0
. x2
4
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

56. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+
. x2
4
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

57. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

58. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
0
..
. x − 3)
(
3
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

59. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. 0
..
. x − 3)
(
3
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

60. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. 0
..
. x − 3)
(
3
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

61. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

62. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
0
.. 0
.. .′ (x)
f
0
. 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

63. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. 0
.. .′ (x)
f
0
. 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

64. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. 0
.. .′ (x)
f
0
. 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

65. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. .. .
0 + .′ (x)
f
0
. 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

66. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. .. .
0 + .′ (x)
f
↘ 0
. . 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

67. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. .. .
0 + .′ (x)
f
↘ 0
. . ↘
. 3
. f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

68. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. .. .
0 + .′ (x)
f
↘ 0
. . ↘
. 3 ↗
. . f
.(x)
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

69. Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
. ..
+ 0 .
+ .
+
. x2
4
0
.
−
. −
. .. .
0 +
. x − 3)
(
3
.
− 0
. .. −
. .. .
0 + .′ (x)
f
↘ 0
. . ↘
. 3 ↗
. . f
.(x)
m
. in
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45

70. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

71. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

72. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
0
..
1
. 2x
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

73. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. ..
1
. 2x
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

74. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. .. .
+
1
. 2x
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

75. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. .. .
+ .
+
1
. 2x
0
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

76. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. .. .
+ .
+
1
. 2x
0
.
0
..
. −2
x
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

77. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. .. .
+ .
+
1
. 2x
0
.
−
. 0
..
. −2
x
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45

78. Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
− 0
. .. .
+ .
+
1
. 2x
0
.
−
. −
. 0
..
. −2
x
2
.
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45