We graph some more complicated functions than on the first day.

1. Section 4.4
Curve Sketching II
V63.0121, Calculus I
April 2, 2009
.
.
Image credit: svenwerk
. . . . . .

2. 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!
. . . . . .

3. Outline
More Examples
Points of nondifferentiability
Horizontal asymptotes
Vertical asymptotes
Trigonometric and polynomial together
Logarithmic
. . . . . .

4. Example
√
|x|
Graph f(x) = x +
. . . . . .

5. Example
√
|x|
Graph f(x) = x +
This function looks strange because of the absolute value.
But whenever we become nervous, we can just take cases.
First, look at f by itself. We can tell that f(0) = 0 and that
f(x) > 0 if x is positive. Are there negative numbers which
are zeroes for f? Yes, if x = −1 then
√ √
x + |x| = −1 + 1 = 0. No other zeros exist.
. . . . . .

6. Asymptotic behavior
Asymptotically, it’s clear that lim f(x) = ∞, because both
x→∞
terms tend to ∞.
What about x → −∞? This is now indeterminate of the form
−∞ + ∞. To resolve it, first let y = −x to make it look more
familiar:
( √) √ √
lim x + −x = lim (−y + y) = lim ( y − y)
y→∞ y→∞
x→−∞
Now multiply by the conjugate:
√
y − y2
y+y
√
lim ( y − y) · √ = −∞
= lim √
y + y y→∞ y + y
y→∞
. . . . . .

7. The derivative
First, assume x > 0, so
d( √) 1
f′ (x) = x+ x =1+ √
dx 2x
This is always positive. Also, we see that lim f′ (x) = ∞ and
x→0+
′
lim f (x) = 1. If x is negative, we have
x→∞
d( √) 1
f′ (x) = x + −x = 1 − √
dx 2 −x
√
Again, this looks weird because −x appears to be a negative
number. But since x < 0, −x > 0.
. . . . . .

8. Monotonicity
We see that f′ (x) = 0 when
√
1 1 1 1
1= √ =⇒ −x = =⇒ −x = =⇒ x = −
2 4 4
2 −x
Notice also that lim f′ (x) = −∞ and lim f′ (x) = 1. We can’t
x→−∞
x→0−
make a multi-factor sign chart because of the absolute value, but
the conclusion is this:
.′ (x)
f
0 −∓ .
.. . . ∞
. .
+ +
↗ −4 ↘ 0 ↗
. .1 . .
. . f
.(x)
. max cusp
. . . . . .

9. Concavity
If x > 0, then
( )
d 1 1
′′
1 + x−1/2 = − x−3/2
f (x) =
dx 2 4
This is negative whenever x < 0. If x < 0, then
( )
d 1 1
′′ −1/2
= − (−x)−3/2
1 − (−x)
f (x) =
dx 2 4
which is also always negative for negative x. Another way to
1
write this is f′′ (x) = − |x|−3/2 . Here is the sign chart:
4
f′′
. . (x)
−
.− −.
.∞ −
.−
.
. .
⌢ ⌢
0
. f
.(x)
. . . . . .

10. Synthesis
Now we can put these things together.
.′ (x)
f
0 −∓ .
.. . . ∞
. .
+ +
↗ −4 ↘ 0 ↗
. 1. .
. . m
.′′ onotonicity
f
. (x)
−
.− −. .
.−∞
− −
.−
. .. .
⌢ ⌢0 ⌢ c
. oncavity
.1 f
.(x)
0
.. 0
..
4.
. 1. . .
− 0
.1 s
. hape
−
. .4
. .
zero max cusp
. . . . . .

11. Graph
f
.(x)
.−1, 1)
( 44
. −1, 0) .
(
. . x
.
. 0, 0)
(
.1 f
.(x)
0
.. 0
..
4.
. 1. . .
− 0
.1 s
. hape
−
. .4
. .
zero max cusp
. . . . . .

12. Example
2
Graph f(x) = xe−x
. . . . . .

13. Example
2
Graph f(x) = xe−x
Before taking derivatives, we notice that f is odd, that f(0) = 0,
and lim f(x) = 0
x→∞
. . . . . .

14. Monotonicity
Now
( )
2 2 2
f′ (x) = 1 · e−x + xe−x (−2x) = 1 − 2x2 e−x
( √ )( √) 2
= 1 − 2x 1 + 2x e−x
2
The factor e−x is always positive so it doesn’t figure into the sign
of f′ (x). So our sign chart looks like this:
√
−
. .. .
0
.
+ +
√. .−
1 2x
. 1/2
√
−
. . .
0
.. + +
√ 1
.+ 2x
−
. 1/2
′
f
. (x)
− −
. . .
0
.. 0
.
+
√.
√
↘ ↗ ↘
. − 1/2 . . . f
.(x)
.. . 1/2
max
min
. . . . . .

15. Concavity
Now we look at f′′ (x):
( )
2 2 2
f′′ (x) = (−4x)e−x + (1 − 2x2 )e−x (−2x) = 4x3 − 6x e−x
2
= 2x(2x2 − 3)e−x
− −
. . . .
0
.. + +
.x
2
0
.
√ √
− − −
. . . .
0
. +
√. . 2x − 3
. 3/2
√ √
−
. . . .
0
.. + + +
√ . 2x + 3
− 3/2
.
.′′ (x)
f
−
.− −
. − ..
.. . + .+
0+ 0
.. 0 +
√ ⌢ √3
. . . .
⌢ ⌣ ⌣
0
. f
.(x)
− 3/2 .
.. . . /2
IP IP IP
. . . . . .

16. Synthesis
.′ (x)
f
− −0 + +.− −
. . .. . . . √. . .
0
√
↘ ↘ 1/2 .
↗ ↗ ↘ ↘
. .. . . 1/2. . m
. onotonic
−
.′′ (x)
f
−
.− + 0−
. + .. . − −0
. − ..
.. . + .+
0+ +
√. . √3
. ⌣. .
. .
⌢ ⌣ 0⌢ ⌢ ⌣ c
. oncavity
− 3/2
. . /2
√ √
− 2e3 . √1 .√1 . 2e3
3 3
− 2e
. f
.(x)
0
.. 2e
. . . √.
√. √ √.
. − 1/2 . .. .
0 s
. hape
−
.. .. . 1/2 . 3/2
3/2
. . .
max IP
IP min IP
. . . . . .

17. Graph
f
.(x)
(√ )(
√)
√
. 1/2, √1 3
2e . 3/2,
. 2e3
.
. x
.
. 0, 0)
(
.
( √)
√ .
. − 3/2, − 2e3 ( √ )
3
. − 1/2, − √1
2e
√ √
− 2e3 √1 .√1 . 2e3
3 3
−
. 2e
. f
.(x)
0
.. 2e
. . √. √.
√√
. − ..
. . 3/2 1/2 . . . . . .1/2.. 3/2 .
0 s
. hape
−.
. max IP
IP min IP
. . . . . .

18. Example
1 1
Graph f(x) = +2
x x
. . . . . .

19. Step 0
Find when f is positive, negative, zero, not defined.
. . . . . .

20. Step 0
Find when f is positive, negative, zero, not defined. We need to
factor f:
x+1
1 1
f(x) = + 2 = .
x2
x x
This means f is 0 at −1 and has trouble at 0. In fact,
x+1
= ∞,
lim
x→0 x2
so x = 0 is a vertical asymptote of the graph.
. . . . . .

21. Step 0
Find when f is positive, negative, zero, not defined. We need to
factor f:
x+1
1 1
f(x) = + 2 = .
x2
x x
This means f is 0 at −1 and has trouble at 0. In fact,
x+1
= ∞,
lim
x→0 x2
so x = 0 is a vertical asymptote of the graph. We can make a sign
chart as follows:
−
. .
0
.. +
. x
. +1
−
.1
. .
0
..
+ +
.2
x
0
.
−
. .. . .
∞
0+ .. +
f
.(x)
− 0
.
.1
. . . . . .

22. For horizontal asymptotes, notice that
x+1
lim = 0,
x2
x→∞
so y = 0 is a horizontal asymptote of the graph. The same is true
at −∞.
. . . . . .

23. Monotonicity
. . . . . .

24. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .
0
.. +
.3
x
0
.
. . . . . .

25. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .
0
.. +
.3
x
0
.
.′ (x)
f
− −
. . .
∞
0
.. ..
+
−
.2 0
. f
.(x)
. . . . . .

26. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .
0
.. +
.3
x
0
.
.′ (x)
f
− .. −
. . .
∞
0 ..
+
−
↘ .2 0
.
. f
.(x)
. . . . . .

27. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .
0
.. +
.3
x
0
.
.′ (x)
f
− .. −
. . .
∞
0 ..
+
−
↘ .2 ↗ 0
.
. . f
.(x)
. . . . . .

28. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .. .
0+
.3
x
0
.
.′ (x)
f
− .. ∞−
. . .. .
0 +
−
↘ .2 ↗ 0↘
..
. . f
.(x)
. . . . . .

29. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .. .
0+
.3
x
0
.
.′ (x)
f
− .. ∞−
. . .. .
0 +
−
↘ .2 ↗ 0↘
..
. . f
.(x)
m
. in
. . . . . .

30. Monotonicity
We have
x+2
1 2
f′ (x) = − − 3 =− 3 .
2
x x x
The critical points are x = −2 and x = 0. We have the following
sign chart:
−
. .
0
..
+ . −
. (x + 2)
−
.2
−
. .. .
0+
.3
x
0
.
.′ (x)
f
− .. ∞−
. . .. .
0 +
−
↘ .2 ↗ 0↘
..
. . f
.(x)
m
. in V
.A
. . . . . .

31. Concavity
. . . . . .

32. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .
0
..
+ +
.4
x
0
.
.′ (x)
f
∞
0
.. ..
−
.3 0
. f
.(x)
. . . . . .

33. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .
0
..
+ +
.4
x
0
.
.′ (x)
f
−
. − .. ∞
0 ..
−
.3 0
. f
.(x)
. . . . . .

34. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .
0
..
+ +
.4
x
0
.
.′ (x)
f
−
. − .. .+ ∞
0 ..
+
−
.3 0
. f
.(x)
. . . . . .

35. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
−
.3 0
. f
.(x)
. . . . . .

36. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
.
⌢ .3
− 0
. f
.(x)
. . . . . .

37. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
. .
⌢ .3 ⌣
− 0
. f
.(x)
. . . . . .

38. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
. . ..
⌢ .3 ⌣ 0⌣
− f
.(x)
. . . . . .

39. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
. . ..
⌢ .3 ⌣ 0⌣
− f
.(x)
I
.P
. . . . . .

40. Concavity
We have
2 6 2(x + 3)
f′′ (x) = + = .
x 3 x4 x4
The critical points of f′ are −3 and 0. Sign chart:
−
. .
0
.. +
. . x + 3)
(
−
.3
. .. .
0+
+
.4
x
0
.
.′ (x)
f
−
. − .. .+ .. . +
∞+
0 +
. . ..
⌢ .3 ⌣ 0⌣
− f
.(x)
I
.P V
.A
. . . . . .

41. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
. . . . . .

42. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
H
.A
. . . . . .

43. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A .
H
. . . . . .

44. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . I
.P
H
. . . . . .

45. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . .P .
I
H
. . . . . .

46. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . .P . . in
I
H m
. . . . . .

47. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . .P . . in .
I
H m
. . . . . .

48. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . .P . . in . 0
.
I
H m
. . . . . .

49. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . .
− −+
.2 0
. +
.A . .P . . in . 0.
.
I
H m
. . . . . .

50. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . 0.
− −+.
.2 +
.A . .P . . in . 0 . .A
.
I
H m V
. . . . . .

51. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . 0.
− −+.
.2 +
.A . .P . . in . 0 . .A .
.
I
H m V
. . . . . .

52. Synthesis
.
.′
− .. ∞−
. . .. .
0 f
+
−
↘ .2 ↗ 0↘
..
. . m
. onotonicity
.′′
−
. − .. ∞−
.. . −
.+
0 f
+
. . ..
⌢ .3 ⌣ 0⌣
− c
. oncavity
− −
. 2/9 . 1/4 ∞
0
. 0
.. 0f
..
..
. .
∞s
. . hape of f
−
.∞ . .3
−− .1 . 0.
− −+.
.2 +
.A . .P . . in . 0 . .A . . A
.
I
H m V H
. . . . . .

53. Graph
y
.
. x
.
. .
. −3, −2/9) . −2, −1/4)
( (
. . . . . .

54. Problem
Graph f(x) = cos x − x
. . . . . .

55. Problem
Graph f(x) = cos x − x
5
5 5 10
5
10
. . . . . .

56. Problem
Graph f(x) = x ln x2
. . . . . .

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