Class ID:
Name:
_on
MATH 1210008
U
FINAL
Foil 2012
December 11, 2012
•
•
•
•
You must show all your work to get full credit,
Make sure your answers are clear and legible.
Simplify all answers unless otherwise indicated.
Clearly indicate your final answers.
No calculators (or other electronic devices) allowed.
You may use a 5x8 index card with handwritten notes (on both
sides, if you wish) for reference. It must be a 5x8 index card and
NOT something that resembles a 5x8 index card.
bo NOT write on this table!
Part I:
Port II:
Part III:
ALL
Choose 2
Choose 4
Prob. # Points Prob. # Points Prob, # Points
1
/20 A2
/10 A3
/10
2
/40 B2
/10 B3
/10
3
/30 C2
/10 C3
/10
4
/10
/10
b3
5
/5
E3
/10
6
/20
F3
/10
7
/25
EC
Total
/210
PART I: bo all of the following problems.
1. Find each limit, if it exists.
—4x+1
3
x
a) Iam 2
x—2X
—X—O
o
—
(2 c-i)
‘
?
I
.
—-1..
-
co
(0
urn
—4x+1
3
x
—x—6
2
x22x
=
tan(5x)cos(x)
x-O
sin(4x)
b)lim
IrL I
S
•••.
n
(Os
CoS(’%)
• tan(Sx)cos(x)
urn
x—O
sin(4x)
=
0
—
3
-
3
-4.
0
-4.
g
0
I
0
-
by-.
k
—..
‘C
P
k
J
.,<
=
-
3
0
+
0
—.
-e.
0
-h
N
>-
E
0
4-
4-
0
o
e.
I,—’
—
I
÷
II
-o
c)
z
4-
4—
0
-o
4-’
-
-
40’)
%d
0
s—’
r
.
C
3. Evaluate these integrals.
2
1
6
+
3
x
x
3
a)
-
x
—
f
3
—
6
2x
+
x
3
v
dx=
—
b) f1Ocost(sint)dt
ic’
0
2
5
0
(- —
snt
3,r
fiOcost(sin4 t)dt
c)
=
f x3 cos(2x)dx
3
coS(2,<)
3f x3 cos(2x)dx
=
3 Jx + 8 at x 0.
4. Find the equation of the tQngent line to the curve f(x) = 4x
—
:
—
0—
-
—-—
3(q)
Equation of tangent line:
5. befine what it means f or a function, f(x), to be continuous at x:c?
I)
ii)
iii)
c:(c:)
6. Setup (you bO NOT have to evaluate) the integrals. Given the region in the first
2 and x=7
quadrant bounded by x=2+y
2
a) Setup the integral to find the solid generated by revolving the region about the
y-axis.
L.
V(2%2à.
J
b) Setup the integral to find the solid generated by revolving the region about
y
=
—2
c’
-(‘)
-
()
7. For f(x)
— 3x(x+2)
(given
(x_1)2
3
(x—1)
and f”(x)=
6(4x+5)
4
(x—1)
a) Find the asymptotes.
-
—
—-
—..————..—
4
-
2.
Vertical asymptote(s):.
Horizontal/oblique asymptote:
b) Fill in the sign line for f’(x).
‘,
c) Find the x-values of all local minimum and maximum.
Local Max.:
Local Mm.:
or
(--
d) Fill in the sign line for f”(x).
0
c”c=
-
e) Find the .ues of all inflection point(s).
x-value(s):
f) Sketch the graph of f(x).
o
cç
2-
Part II: Choose 2 out of the next 3 questions to do, Indicate dearly which
proWems you want gradedmu
A2. (Grade: Y or N) Find the arc ‘ength of the curve given by x = t6 +2 and
y=t
—forOst1.
)
0
O’)’ (j
-
AriswerA2:
U’u
B2, (Grade: Y or N) Given the natur& ‘ength of a spring is 2 meters and that it
takes a force of 12 newtons to keep the spring extended to 8 meters, find the
work done in stretching the spring from its natur& length to a length of 5 meters.
(bont forget units.)
Answer B2:
C2. (Grade: Y or N) Find the surface area generated by revolving y
the x-axis for —1x2.
z
%
\4_
Answer C2:
I
L4-”
\J
—‘
=
iJ4 — x
2 about
Part III: Choose 4 out of the next 6 questions to do. Indicate dear’y which
mswant
AS. (Grade: Y or N) Use the definition of derivative to find f’(x) given f(x)
L
m
F
i
—
—
\‘b
-
c
‘
-‘
)
i
fv(x)
=
x2
+8
c4
rO
o
L.
‘40
0
+-
‘4-o
c)
D
2
o
0
(Y
C
II
H
0
—
p
H
&
—
F
H
p.
+
4
U
-c
V)
II
0
0
0
01
‘I-.
>
(I)
2
-g
II
-
4-
_c
I3
(j
U
t
1
-
—
I_fl
—
k
II
—
L.
c)
fl
C
b3: (Grade: Y or N) Use differenti&s to approximate the decrease in vo’ume of a
chewing gum bubb’e when its radius decreases from 20 mm to 19 mm. (bon’t forget
your units.)
-
3
20w
Answer b3:
20
\
E3, (Grade Y or N) A student is using a straw to drink from a right conical paper
cup, whose axis is vertical, at a rate of 3 cubic centimeters per second. If the
height of the cup is 10 centimeters and the radius of its opening is 3 centimeters,
how fast is the level of the liquid fatling when the height of the liquid is 5
centimeters? (bon’t forget your units.)
-
cr/s
it,
71:
-
4
3
0
2
-I
(3L
2
-
Answer E3:
-
I
ir
F3. (&rade Y or N) A long rectangular sheet of metal, 12 inches wide, is to be made
into a rain gutter by turning up two sides at right angles to the sheet. How many
inches should be turned up to give the gutter the greatest volume? You can assume
that the length of the gutter is some fixed value of L. (bon’t forget your units.)
‘j
d
0
Answer F3:
Extra Credit (10 pts)
a) In your own words, describe/define a linear operator. (Hint: There are two
key properties.)
L (‘ j)
Linear Operator:
ccLc
coo
b) For each of the following mathematical operators, answer true if the
operator is linear and the answer false if the operator is not linear. For each
operator that is NOT linear, give an example to show how it fails the
definition of a linear operator.
a. Multiplication
(circle one)
F
If false, explain why.
b. Limit
If false, explain why.
F
(circle one)
c. berivative
If false, explain why.
F
(circle one)
d, Square Root
If false, explain why.
T
®
(circle one)
e. Sine
If false, explain why.,
T
(:F:
(circle one)
F
(cirde one)
3
f. Integral
If false, explain why..