(a) Can
f
(3) ever be negative? Give a reason for your answer.
(b) Can
f
(

3) ever be negative? Give a reason for your answer.
(c) How large can
f
(3) be? How large can
f
(

3) be?
(d) Must
f
have a critical point between

3 and 3? Either show your reasoning or
give a counterexample.
86
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Math 180 Review Worksheets
R3
8. Di
↵
erentiate the following functions, you do not need to simplify your answers.
(a)
f
(
x
) =
5
p
x
8
(b)
f
(
x
) = ln(
x
2

4)
(c)
f
(
t
) = arctan
✓
2
t
2
◆
(d)
f
(
x
) =
x
2
tan(2
x
3
+ 1)
(e)
f
(
x
) =
x
3
x
87
Math 180 Review Worksheets
R3
9. Use the Intermediate Value Theorem to show that the polynomial
f
(
x
) =
x
2
+
x

1
has a zero in the interval (0
,
1).
10. Find the shortest distance from the point (4
,
0) to a point on the parabola
y
2
= 2
x
.
Hint: Minimize the distance squared.
88
Math 180 Review Worksheets
R3
11. Consider the function
f
(
x
) =
x
1
/
3
.
(a) Determine the absolute extrema of
f
on [

1
,
8].
(b) Find the intervals where
f
is increasing and where it is decreasing.
(c) Find the intervals where
f
is concave upward and where it is concave downward.
(d) Sketch the graph of
f
on the grid below.
(e) Why does the Mean Value Theorem fail for
f
on [

8
,
8]?
89
Math 180 Review Worksheets
R3
12. Let
f
(
x
) =
x
2

x

2
x
2

2
x
+ 1
=
x
2

x

2
(
x

1)
2
,
f
0
(
x
) =
5

x
(
x

1)
3
and
f
00
(
x
) =
2(
x

7)
(
x

1)
4
.
(a) Find the following limits.
i.
lim
x
!1
f
(
x
)
ii.
lim
x
!1
f
(
x
)
iii.
lim
x
!
1

f
(
x
)
iv.
lim
x
!
1
+
f
(
x
)
(b) Does
f
have any asymptotes?
(c) On what intervals is
f
increasing? decreasing? At what values of
x
does
f
have
a local maximum? local minimum?
90
Math 180 Review Worksheets
R3
(d) On what intervals is
f
concave upward? concave downward? Does
f
have any
points of inflection?
(e) Sketch a graph of
f
using information from the previous parts.
(f) Does
f
have an absolute maximum value? absolute minimum value?
91
Math 180 Review Worksheets
R3
13. Find the linear approximation
L
(
x
) of the function
f
(
x
) = ln(
x
) at
a
= 1 and use it
to approximate ln(9
/
10).
14. Find the equation for the tangent line at the point (1
,

1) of the curve given implicitly
by:
x
2
y

3
y
3
=
x
2
+ 1
.
92
Math 180 Review Worksheets
R3
15. Suppose
f
is a function that is continuous and di
↵
erentiable everywhere with the
following properties:
•
f
has a horizontal asymptote at
y
= 1
•
f
0
(
x
)
>
0 on the interval (
1
,

2) and (0
,
1
), and
f
0
(
x
)
<
0 on the interval
(

2
,
0)
•
f
00
(
x
)
>
0 on the interval (
1
,

3) and (

1
,
1
), and
f
00
(
x
)
<
0 on the interval
(

3
,

1)
•
f
(

3) = 2,
f
(

2) = 3, and
f
(

1) = 2
Sketch the graph of
f
on the grid below.
16. A paper cup shaped like a cone with height 20 cm and top radius 4 cm springs a leak
and water begins to pour out through the bottom. If the water is flowing out at a rate
of 2 cm
3
/second when the water level is at the halfway point, how fast is the water
level decreasing at this time?
93
Math 180 Review Worksheets
R3
17. Consider the integral
Z
4

4
1

x
2
dx.
Use a right Riemann sum approximation with 4 subintervals to estimate the value of
this integral.