Math 1210 Final Exam
Name
-.
Spring, 2016
Franco Rota
uid:
Instructions:
• Please show all of your work as partial credit will be given where appropriate,
and there may be no credit given for problems where there is no work shown.
•
All answers should be completely simplified, unless otherwise stated.
•
You may use a scientific calculator for this exam. No other electronics are
allowed on this exam. Make sure all cell phones are put away and out of sight. If
you have a cell phone out at any point, for any reason, you will receive a zero on
this exam.
•
You may ask for scratch paper, but please transfer all finished work onto the
proper page in the test for us to grade there. I will
grade the work on the
scratch page.
•
Notice that the space left for each question is sufficient, but possibly not
necessary, to answer the question.
(This exam totals 210 points, I will enter the grades as a total of 200 to allow for some
room for arithmetic mistakes)
1. (10 points) Compute the limit. Write DNE if the limit does not exist.
urn
x3
x
+
2
4x—21
2x—6
p.
-,
Iw
—
2(X3)
5
Solution:
2. (10 points) Compute the limit. Write DNE if the limit does not exist.
urn
x +3x
2
2—ix
2l\
—
-
3
-3x
3
2/x.
—
-
-
J
—p()
3j
Solution:
—
I
3. (10 points) Compute the limit. Write DNE if the limit does not exist.
urn
xO
3 x tan (x)
sin(x)
L
x
—)
3
ci
.
Solution:
0
if x<2, x0
f
4. Consider the function
(x)= 2,
1,
ifxto
if x 2
a. (10 points) Determine all points at which f(x) is not continuous. Justify your
answer
using the definition of continuity.
4%
?
?if’f’-
zzjo
x=o
LAC.
ro
&-:
s
Zr
x2
+
-
eQ tN2QJU-.
ii
i&—-) L
2
n
Solution:
b. (15 points)Draw the graph of f(x). Label at least two points on the graph.
—
I
I
I
I
I
J_ _L _L _L_ L
—
I
I
L 1_ 1
I
-fr—:—-b—i-—F— F—F—
I
It
I
I
I
-
I
Ii
_L_
I
I
—
Il
tH
i-— t—t— ---f-----
lI
Il
F--F--b-b-F-F
--b-b-F--F-F
I
F-b-F- 4- F--F-
—
-
44
-r-rr44
:‘
-I—-I
—
—r
LL_L_L_1A —L —L _L_ 1—1—1—1—1— £4_1
—
— —
F4—F—F—F—F—F—t—t—
1. A. -L A.I.
-F—F—-F—-f
L
L
L
F
1.
-
JJ
J
J.IJ
L_L_L_L_
b-b-b-F-F-F
—
—
I
A.
—
F
—
F
—
—
—
—
F_
£_
-J —J
-
J
-
A. A. J
F—F—F—F—F—F—F—H-fH
I
LLL
——1——F-———F—F—F—F—F-+—+—4—4— -4-4--f--i
—————-—-—f-—i-—i-—f-—+—4—4— ,1 9 9—9
C A.
4_F F
-
-
F
-
LL
r r rr
C
how do
c. (5 points) If you found some discontinuities, are they removable? If
at
those
points
to
make
it
continuous?
we need to define/redefine f
so,
e
e
zQ
raL
0
2
5.(1O points) Find the equation of the line tangent to the curve described by
2
x —xy+y
=3 at the point (0,3) Show proof of your work.
2
.
2x.
-
=0
-
—x
\f),
1
a (o
i\T1
z
—
Solution:
2
+
u
6. (10 points) Compute the following derivative. (Do not simplify the final answer, just
do the differentiation.)
/
3
3x +sin2x\/
2)(i)
(33 +rn’X)(X)
2
3
D(3x
x—1
(xi(x
2c.)(K)
(xL
)L
(3x2x)(2)
z:.
Solution:
D
4J
Ln
a)
C
C
0
U)
D
a)
.1—i
ov
-0
cOD
LI)LL)
‘—‘C
I
4J
-
4-
0
0
0
C
-
a)
-
—
0
C
-I-J
E
a)
-c
U)
C
+
-,.---—-,
c—J.
-4-
CD
—
C
+
:
,<
/‘
-.
‘—
-z;—--
D
0
CD
jZ
G)
-c
•i
E
a
o
C
o
-
-
C
o
o
4-
4J
•-
00
4_I
C
.0
‘
e
-
%__
c
0
D
0
9. (20 points) A flower bed will be in the shape of a sector of circle (a pie slice) of
radius r and vertex angle 0
Find the values of r and 0 such that the
area is ninnurn, assuming that the perimeter is constant and equal to P
(Hint: This is similar but not equal to the problem we did in class during review. Recall
that the perimeter of a circular sector is given by P=2r+rO and the area is
,
A=0r2
)
2
-
(
A
IC
2,
Z
)
1,
ci
) z
Solution:___________________________
10. (15 points) The volume of a sphere is given by
3
r
where
r is the radius
of the sphere. The volume of the sphere is changing, increasing by 10 cubic
centimeters per second. Therefore, its radius changes along with the volume. What is
the rate of change of the radius of the sphere, when the radius is 100 centimeters?
V=
LflycL
1
\J
2
‘111 )C
;rv
AO
if
/
1.
(AOQ)
OM-”
—
Solution=
tiTi
-1()
CW/
7
A
-.
11. Compute the following antiderivatives:
a.(1O points)
X
-—
_1)dx
4
f(X
c
C4)c
3
x
Solution:
b.(1O points)
I (x
)sth
4
±
2
I
Il
J
j
-
I
—
x
—+— 4c
X
3
1-
-
I
—
S
Solution:
,Ut
I
C—
3
r
12. (10 points) Compute the following definite integral:
io
lOx
dx
2
2
(x +3)
2.
(
4Q
J
xc
->
A.I,L
)
—
3
ii.
Solution:________________________
13. Consider the function
f ‘(x)
,
2
(x1)
a. (8 points) Where is
OjrCkjv:
its second derivative is
f (x)
.
Its first derivative is
f “(x)
defined? Where is it increasing/decresing?
xc1R1
(x)
f (x)=
w+
>
<0
d
ci.
b. (7 points) Where is it concave up! concave down?
x)
<) (X —
C)rc.
)rrC.
( S-
ftf X
,
1
OM.
Solution:
> 0
1
r\Lc1
o
>
<
>0
o
/
c
*
0
x
(-i,
efv
(- CX’
4) (i,-oc)
c. (5 points)Draw the graph of f(x). Label at least two points on the graph.
_L _L _L
—
A.
1
A. _L_ A.
—
—
T —f--I
A. 1_I—
9
T9
1fl
I
I
I
_J_J_L_L_LL_A..L.A._A.
9C
Ldfl_J
—
—
—
—:-
—
-—
-
—
-
b—F I- —F F F F— 4—4—
—
—
—
—
-
I
-
I
—b--
L_L
A.
A.
r
r
r
;____
—
-II
JJ__1
F F—I-— I-—F F —1- —4—4—
—
-
—J
—
-
-
I
-----
I
I_
r—,-r—r—Th—r—
I
I
1
-t
I
7
7
-TtYYTTTnrflrJr
I
I_i
I
t-r—tt—
I
I
ft
I
I
: :__A
I
I
I
I
I
I
I
I
I
I
I
I
I
I
tr—r’r—r—r-r—r
I
I
::* -:
:“
JJJJiJ
-
:: : :
_\J J J_i. hi
I
I
ft I I
_L _L
—
A.
—
A.
—
A. _.L
—
A.
—
£ _L_ 1_I
_J
—
J
-
LI _L _LI _L_
L LI
I
-
-
I
I
I
I
-
I
-
I
LLL
-
—‘
I
L
A.
A.
—
F-F--h-h-F-H
F—F—F—F—F—F—t—4—4L_L_LL_LA._A.A._i
L..LI...
_J_J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
—
I
I
I
L_L_L_L_A._A.
-! ZLL -
-
—,
I
JJJJ
7
9
-
II
II
11111
I
r
II
II
I
I
LI
I
I
I
I
I
I
I
I
I
I
I
I
I
Ii
I
I
I
I
I
I
I
I
I
I
I
7777
14. Consider the region R bounded by the curve
2 and y=x—1
x=3—y
a. (7 points) Sketch the region R
I
I
I
I
I
1
I
1
I
I
I
I
_J__LL_L_L_A._A._A._iI_.L_J.
I
I
I
I
I
I
I
I
I
I
I
I
_J__L_L_L_L_A._A._A._A._i_S_J
I
I
I
I
II
I
I
I
I
II
11111111
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
L_J_
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
L_L_L._L_A._A..
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I I
A.._L_L_L_A._A..
I
I
I
I
I
I
I
_J_J_J_J_J_J
I
I
I
I
I
I
I
I
I
I
I
I
I
I
II
I
I
I
I
1
I
I
I
1I
2.J_J_J__I_J_J
I
I
I
I
I
I
I
I
I
-T-r-rr-r-r-r-rrrT-T-
,‘l
I
I
I1
I’
I
I
--r-r-r-r
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
II
I
I
I
I
I
I
I
I
r
1)1
I
I
rr
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
;1t!ht1”thhflt -t
i i I I I I I
I
I
‘I
——I——I-—I——I-—I-— I-— I-— I-—4-—4—4—4—
I
I
I
I
I
I
II
I
I
I
I
I
-
hhthk
Ti
II
-
I
1
I
1111
I
I
I
I
I
II
I
I
I
I
I
——I——I——I——I-—I-—I-—I-— I-—ft-—ft-—+—-f—
_J__L_L_L._A._A._A._L_
I
—
I
I
I
I
I
I
—
—
I
I
I
I
I
I
I
I
A.
A.
—
A.
I
A.
—
—
I
—
A.
A.
I
A.
—
—
I
—
A.
A.
I
A.
I
—
A.
—
A.
I
—
A.
I
—
I
———
—
A.
—
-
-
I
I
ttthhhhF
J_ _L _L A.
A.
I
I
A.
A.
—
£
—
I
A.
I
I
I
I
I
I
I
_JJ_J_
-
J- -I
I
I
-—
I__A. A. A.
A. A.
I
I
I
I
I
I
I
I
I
LA.ILA.A.A.A.A..
-
-
-
—
I
A.
I
I
—
—
.1 _J_
.1
I
—
.1
_J_J/J_J_Jfl
—
L _A.
L
-
A.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
11111
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
II
II
I
I
I
I
I
I
I
I
I
I
t
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
I
I
I
- 7”17”l”I
I
-
I
A._A._A._A..
I
—
-F-4-41
--I--F-F-F-F--
A.A.
I
I
I
I
I
r
I
r
-
-
b. (8 points) Set up an integral which computes the area of the region R.
4c21.
J((3’) -(j
‘Ct,:
3L
x
4
15. Let R be the region bounded by
,(Z)(jo
y=4x and
2
y4x
a. (5 points) Sketch the region R
rIr
r
r
R
>
—
.1
: :4
LL_L.Lfl_L_LI_L
L
:
.I_J_
_J__J_J_J
L.L_L_L_L.
b. (10 points) Revolve the region around the y-axis. Set up an integral which
computes the volume of the solid of revolution thus obtained. Use either one of
washers or shells, state which one. You don’t need to compute the integral.
x=\r
woIAL:
AV 21 X
v
‘(x
-
S
ti?)Ax
i.
/
L
0
jj.
)
STUDENT—PLEASE DO NOT WRITE BELOW THIS LINE. THIS TABLE IS TO BE USED FOR
GRADING.
_Problem
Score
1
/10
2
/10
3
/10
4
/25
5
/10
6
/10
7
/10
8
/10
9
/20
10
/15
11
/20
12
/10
13
/20
14
/15
15
/15
ETotal Score:
/210