1210-90 Final Exam
Name
Fall 2012
K Y
Instructions. Show all work and include appropriate explanations when necessary.
work in the space provided. Please circle your final answer.
Please try to do all
1. (2Opts) Find the following derivatives. Show your work below and circle your final answer.
(5pts)D 43 x+1)
(a)(>
(b) (5pts) D(x
3 sin
(c) (5pts)
(sin x
1
(5pts) D
2. (lOpts) Compute the following limits. Show your work below and circle your final answer.
(a) (5pts) jim
x-2
(b) (5pts)
x 2 -x-2
x-2
.
4
cos(ir--h)—cos(ir)
h
Hint: Think derivative!
F
1
1L4.4..
1
(j+i)
3. (lOpts) Find the equation of the tangent line to the curve determined by
x+zy 3 =y+5
9
at the point (2, 1).
974.
—----—-
3xj’
j”
B
+3
3
2 of cardboard. The tube
4. (l2pts) A manufacturer wants to make a closed cylindrical tube out of 671 ft
has length L feet and the radius of the end caps is r feet (see picture below). Find the values of L
and r such that volume enclosed is at a maximum. Note: You must use calculus to get full credit!
2irv-L
2
ir —2’Irr
LLr
r
VCr)wrC+ -r)
F’
cva1It5
==Z• 3
V)T(3 3r)
4’ a
No I-c- / “L’)
bzl
C
2
—
5. (l8pts) Consider the function
f(x)
=
(a) (3pts) Find f’(i).
(b) (3pts) Find the critical point(s) of
6x
—
3
f.
12-?c tLX(X
Ox
—
3
(c) (4pts) Fill in the blanks: f(x) is decreasing on the two intervals
Note: ± are acceptable answers.
—
-t
0
(
).
) and (
s-i.
13
(d) (3pts) F
(e) (3pts) Find the inflection point(s) of
o
=
L2-
2
(f) (2pts) Fill in the blanks:
acceptable answers.
L
f.
f(x)
4f
is concave down on the interval
-I-
(L
L .
Note:
±00
are
I
—
*15 on the interval [0, 6] using the partition of 3
6. (8pts) Evaluate the Riemann sum for f(x)
subintervals of equal length with the sample points being the midpoints of each subinterval.
1—(-’
0
(
‘<()(l
’f5
2
3
f)x
-
-
3
(i)C)
(X)
t()C)
7. (l5pts) Use both versions of the Fundamental Theorem of Calculus to evaluate the following:
(a) (5pts)
-1)dx
2
f (x
Jo
(
I)
sin(t3) dt
(b) (5pts)
Note: You do not need to know an an
(c) (5pts)
d
dx
—
C
/i
i
erivative of sin (t
) to answer this question or (c) below.
3
L e-k-
) dt
3
sin(t
tM’%ê gLaRh-L
-Ftt) • 12
-)
—
cs-f-
(2.)
sI
8. (l5pts) Find the following antiderivatives. Remember: +0!
(a) (5pts)
fxcos(
2
) dx
(b) (5pts) 1
—
±
v)
tO
-i-C
ctU- 2-y. o1)t
(c) (5pts)
f
2 (3x) cos (3x) dx
sin
t*
k-
OlAA 3tos(x
4
(3x)+C.
3
t
3)
9. (l6pts) Consider the region R bounded by y
a rough sketch of the region R.
=
3
—
x, x
=
1, z
=
2 and the i-axis. Figure A below is
(a) (8pts) Find the volume of the solid obtained by rotating the region around the i-axis.
V
(3-tx
i
3
X
II
-+3
1
3
(b) (8pts) Find the volume of the solid obtained by rotating the region around the p-axis.
V
)
zirx
‘
-x 1
j Ac
I3ir
‘1.
3
3)
3
-;.
(.)
t\
•—_(3
-
2 and p
2 = 27x sketched in Figure B below.
10. (8pts) Consider the region S bounded by the curves p = x
Each integral below is the volume of a solid obtained by rotating S around a particular axis. Match
the correct axis with the expression for volume by writing the appropriate letter in the blank provided.
Each answer is used exactly once.
L((’°
)
2
—i
—(10—
2x(1 + x)(
A
f
—
)2)
) dx
2
x
A. i-axis
dx
B. p-axis
.
C.x=—1
dx
(27x_x
)
4
f(y)2)dy
D.y=10
2
I
Figure B
Figure A
5
(I.
11. (8pts) Find the length of the arc of the curve p
fL
t_
(x-z.)
L
(x +
2)3/2
from x
=
—2 to x
=
1.
(
f
-
3/i.. 14)
I I
_
2
12. (l0pts) A tank in the shape of an inverted square pyramid is filled with water. The tank is 10 feet tall
and has sides of 5 feet at its rim. How much work is required to pump the water over the top edge of
.
3
the tank? Use p to denote the density of water in lbs/ft
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