2. Second-chance Midterm: Tomorrow in Recitation
12 free-response questions, no multiple choice
Covers all sections so far, through today
Your score on this exam will replace your midterm score
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 2 / 47
4. Quiz 3 tomorrow in recitation
Section 2.6: implicit differentiation
Section 2.8: linear approximation and differentials
Section 3.1: exponential functions
Section 3.2: logarithms
Section 3.3: derivatives of logarithmic and exponential functions
Section 3.4: exponential growth and decay
Section 3.5: inverse trigonometric functions
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 4 / 47
5. Outline
The Procedure
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 5 / 47
6. Objective
Given a function, graph it
completely, indicating
zeroes
asymptotes if applicable
critical points
local/global max/min
inflection points
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 6 / 47
7. Objective
Given a function, graph it
completely, indicating
zeroes
asymptotes if applicable
critical points
local/global max/min
inflection points
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 6 / 47
8. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 7 / 47
9. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 8 / 47
10. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 9 / 47
11. Outline
The Procedure
Simple examples
A cubic function
A quartic function
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 10 / 47
12. Graphing a cubic
Example
Graph f(x) = 2x3 − 3x2 − 12x.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 11 / 47
13. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 11 / 47
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:
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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)
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
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
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
25. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 12 / 47
33. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
34. Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
−
. . . .
+ −
. .
+ .′ (x)
f
.
↗− ↘
. . 1 . ↘
. 2
. ↗
. m
. onotonicity
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
35. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
36. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 14 / 47
37. Combinations of monotonicity and concavity
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
38. Combinations of monotonicity and concavity
.
decreasing,
concave
down
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
39. Combinations of monotonicity and concavity
. .
increasing, decreasing,
concave concave
down down
I
.I I
.
.
I
.II I
.V
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
40. Combinations of monotonicity and concavity
. .
increasing, decreasing,
concave concave
down down
I
.I I
.
.
I
.II I
.V
.
decreasing,
concave up
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
41. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 15 / 47
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
− .
1/2 2
. s
. hape of f
m
. ax I
.P m
. in
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
45. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 16 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
50. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 17 / 47
51. Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
. . . . . .
V63.0121, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 18 / 47
52. 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, Calculus I (NYU) Section 4.4 Curve Sketching April 1, 2010 18 / 47
