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Friday, April 18, 2008
EXAM III - MATH 135, SPRING 2008
READ EACH PROBLEM CAREFULLY! To get full credit, you must show all work!
The exam has 7 problems on 4 pages! Turn in all pages! NO GRAPHING CALCULATORS ALLOWED!
• Problem 1.
The half-live of Cesium-137 is 30 years. Suppose we have a 100-mg sample.
(a) Find the mass that remains after t years.
(b) How much of the sample remains after 100 years?
(c) After how long will only 1 mg remain?
(ex)
rm (-t) .=. '11'\0.e. let . ~ \ (/ . 6L..--~
__
_
~)
CbJ
(c..)
¥-
0
&t-)--=o
\
~
")
\....~!-~t.~> le-~ ~
+\--'=50
00 .e
~
\
• Problem 2.
'l. 'loa ~
J(IA.
'"30
t- _/')
\0
I'}
\2', '"\
-~*J
...L -:-.e.
1<>
..••• )
~
n
-X-..JA..loo=
-
(00
Use l'Hopital's Rule to compute the following limits.
><-
:::
'l.-
-.-, SL-
(b)
K~70
.(\
kL
-tv..
kt\-..
r-.=.
-=
..Q..
'- ----k'L
I
'30
==
"",,-"\
d
2....
x@ ~
lim - = ~
X--->OO
K
I
-
~o
(a)
X
.t
o e~·l-. -::-'/
.R..•.'1 1:-
-mC\~}~ 100.e
t,.,,t
d 0 ..e. - '""i""o
-==-
,
\
~~-:::W
X
lim
--->0+
-:-}
--2.
><- ~
(c)
-
'JC.:, (>0
eX
x2
In x =
L
"":'0
x--->O
."sm x
..e.)C
)c"L
k\'-..L2- X'?.'L
;=
~
'>c-70
..L
~i D-, C),.31
- ~~ \~O ,....
~.1
'>0 ~
\
~x
-L
n ...)<.Jv--k-)o"\-
~n
lim. 1-COS~.l.-
t =)"L =
~~ 1...
~{O]
~
=eJ~
D ~\~r~~k,<
O.
5v.:
)C
C-<S"1><
o~\Q\
W.
r
• Problem 3.
Find the local and absolute extreme values of the function on the interval specified.
lnx
I
,.~
~ (~):::
= -2'
f(x)
E [l,e]
X
x
)( ,)( - ~x . '2~ .~
X~
X -
2.)( ~
l- 'L~
~.
xc,
X
-
~
'3>
t
l
Cr\~
~ciM--\-S :
k~
;:::=)
~(>C/~
t"'~
At-L
~
• Problem
4.
,IJ
1-2
~X=<1
K -:;::.Q,1.
k C~
-
A('j", ~:
® oJ
At--
-e .1.
1-.
~
,2...)
(~?
'¥1
-=
~
'? ')
~K
II
~-=
L
.-€.
I ~ ,-=-...bJ- -:::..-(ill
(g:')
-e
2~
1-
~-e.)~
~;-f
~
.
Find the maximum value of the function
W<-'
~
2e
\
~ A~s. tyN:x ><
~(~)=
f(x)=x(l-x),
)<._)<:.L
XE[O,l]
---2 X
~C>CJ:;::' t
.,)
;
J:
r(~) ~
K ~
1'L
)(:=:
II
L () -')
"'--"') ~ ()()-~ -'2.
z
-I«:.~
1 ~~",0."~J.
\\\- w..l
X
=} =1
c....
• Problem 5.
'\M.D.x.)
>c
I
1.-
L-.l...
e ~
4
-
2.e.
==0
~cJ
~LV,
x: -:z-
X
'=-
~S()iIM\.v....
(
~X
\M.ax
k~ ~"",
r;d)
"
'>C' VI
~
r(~)-:::
l~-=--.l']
Af")
= 0 = ~ I)
-
~
~
Find all the inflection points (if any) for the function
= x(x + 1)3.
f(x)
t(I )=
:7
I,
)C
()c {-I)
+
+I")
-==
.
(xI-==-
2(X-+-\)(~~+I)'-t' (X+I) (4):::: '2(K+'JL0x+\)+ '2(K4-I)]
2()<+I)(b'X-t3)~
6C>C+I')(7-)c-+\)
i./
.~ (~1 /-0
k~~
~\~
'L.
+/) (4
(>c f- ,) C Ie + I + 3 X ) ::::D~
L
,II
~
L
'L
X, 3()C
K .----- i
\N~
~~
-\
-
=\;:>
~L
~oo
~""""-'-"
n
bokl
w-<
v __ \
'--
c'+J:,,,,,
k~ ~
r'~.,J
s\
r\~
K~
I)
f
• Problem 6.
Sketch the graph of the function
x
= 1 + x2
f(x)
Determine: (1) the domain, (2) symmetries (if any), (3) the critical points, (4) intervals where
the function increases or decreases, (5) concavity and inflection points, and (6) limits at infinity.
Remember, a sign chart is always of great help!
CD ~~~
( '2.)
I
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'1..
t+ X
-?:x
(14..x?-)
'1.....
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,
o +-+ a -
A- ~
(\v,.
~'J ~ (-"\)
,S-~ ~~d
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((,
ao )
c;u...~ (~¢O I
(5)
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(, k
~ ('t.)::
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Q -f
><7..)
.
i.t
(3 _~l.-)~
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~.
o
--0
0+
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(
(
-G
I •
-\
--L
.'-
+
D
+ 0- -\J
-:-'lx('--f x'"') [ L \+ xn
+ + -t
-
(')
-\-'2. C l ~ X- B
(\.of )(l-) Lj
• Problem 7.
Consider the function
= (X2
f(x)
-
1)2.
(a) Find the critical points for f. Find relative maxima and minima points.
(b) On which intervals(s) is f concave upwards? Are there any inflection points?
I
(~)
f(K) -
'2 ()( '2 -
i).
t
k()C/~
U~ ~
~eJ
J
Q.
X ()( '-)_ I
\.x-:=
~ r .
t('><I'~4>C
-4x
'),
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J
'71
ff
.
(X)'=
il
(..,').:=
-4
L.o
x-=
-;PJ
f20cJL
0
V)
~ (I)-:=' \L -~ /0
.,) 1\.'2 1
~
\\
A
C - ()-'/
17. ~ ~
0
k(K]-==-
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r-
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~~
f'N.A
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.
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+ ''0
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t'Ucs
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~) 1 ~.
(2)<:
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II
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cw... 01.
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, o (- - (-0-+
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ch 1-
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