# 7~_

```7~_
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..)
&yen;-
0
&amp;t-)--=o
\
~
&quot;)
\....~!-~t.~&gt; le-~ ~
+\--'=50
00 .e
~
\
• Problem 2.
'l. 'loa ~
J(IA.
'&quot;30
t- _/')
\0
I'}
\2', '&quot;\
-~*J
...L -:-.e.
1&lt;&gt;
..••• )
~
n
-X-..JA..loo=
-
(00
Use l'Hopital's Rule to compute the following limits.
&gt;&lt;-
:::
'l.-
-.-, SL-
(b)
K~70
.(\
kL
-tv..
kt\-..
r-.=.
-=
..Q..
'- ----k'L
I
'30
==
&quot;&quot;,,-&quot;\
d
2....
x@ ~
lim - = ~
X---&gt;OO
K
I
-
~o
(a)
X
.t
o e~&middot;l-. -::-'/
.R..•.'1 1:-
-mC\~}~ 100.e
t,.,,t
d 0 ..e. - '&quot;&quot;i&quot;&quot;o
-==-
,
\
~~-:::W
X
lim
---&gt;0+
-:-}
--2.
&gt;&lt;- ~
(c)
-
'JC.:, (&gt;0
eX
x2
In x =
L
&quot;&quot;:'0
x---&gt;O
.&quot;sm x
..e.)C
)c&quot;L
k\'-..L2- X'?.'L
;=
~
'&gt;c-70
..L
~i D-, C),.31
- ~~ \~O ,....
~.1
'&gt;0 ~
\
~x
-L
n ...)&lt;.Jv--k-)o&quot;\-
~n
lim. 1-COS~.l.-
t =)&quot;L =
~~ 1...
~{O]
~
=eJ~
D ~\~r~~k,&lt;
O.
5v.:
)C
C-&lt;S&quot;1&gt;&lt;
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&gt;
t
l
Cr\~
~ciM--\-S :
k~
;:::=)
~(&gt;C/~
t&quot;'~
At-L
~
• Problem
4.
,IJ
1-2
~X=&lt;1
K -:;::.Q,1.
k C~
-
A('j&quot;, ~:
&reg; oJ
At--
-e .1.
1-.
~
,2...)
(~?
'&yen;1
-=
~
'? ')
~K
II
~-=
L
.-€.
I ~ ,-=-...bJ- -:::..-(ill
(g:')
-e
2~
1-
~-e.)~
~;-f
~
.
Find the maximum value of the function
W&lt;-'
~
2e
\
~ A~s. tyN:x &gt;&lt;
~(~)=
f(x)=x(l-x),
)&lt;._)&lt;:.L
XE[O,l]
---2 X
~C&gt;CJ:;::' t
.,)
;
J:
r(~) ~
K ~
1'L
)(:=:
II
L () -')
&quot;'--&quot;') ~ ()()-~ -'2.
z
-I&laquo;:.~
1 ~~&quot;,0.&quot;~J.
\\\- w..l
X
=} =1
c....
• Problem 5.
'\M.D.x.)
&gt;c
I
1.-
L-.l...
e ~
4
-
2.e.
==0
~cJ
~LV,
x: -:z-
X
'=-
~S()iIM\.v....
(
~X
\M.ax
k~ ~&quot;&quot;,
r;d)
&quot;
'&gt;C' VI
~
r(~)-:::
l~-=--.l']
Af&quot;)
= 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&quot;)
-==
.
(xI-==-
2(X-+-\)(~~+I)'-t' (X+I) (4):::: '2(K+'JL0x+\)+ '2(K4-I)]
2()&lt;+I)(b'X-t3)~
6C&gt;C+I')(7-)c-+\)
i./
.~ (~1 /-0
k~~
~\~
'L.
+/) (4
(&gt;c f- ,) C Ie + I + 3 X ) ::::D~
L
,II
~
L
'L
X, 3()C
K .----- i
\N~
~~
-\
-
=\;:&gt;
~L
~oo
~&quot;&quot;&quot;&quot;-'-&quot;
n
bokl
w-&lt;
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
'&lt;A ~\;.'&quot;
'1..
t+ X
-?:x
(14..x?-)
'1.....
'L
,
o +-+ a -
A- ~
(\v,.
~'J ~ (-&quot;\)
,S-~ ~~d
CS1-\
((,
ao )
c;u...~ (~&cent;O I
(5)
\\
-1..X
(, k
~ ('t.)::
&gt;(~
-(\-X) 2Ct+K}'?-X
Q -f
&gt;&lt;7..)
.
i.t
(3 _~l.-)~
_ LX
(J -\- t&lt;-) '}
~.
o
--0
0+
-O
\
(
(
-G
I •
-\
--L
.'-
+
D
+ 0- -\J
-:-'lx('--f x'&quot;') [ 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('&gt;&lt;I'~4&gt;C
-4x
'),
,J1
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]-==-
'1-) '7..0 --v1
~D~
X'-
~~C
r-
I
IV\M
~~
f'N.A
Lf
)c ,~
+.1~3 )
0rv'uw-e
iA
'1...
X
Lv-...
.
IV'-
-T
\
+ ''0
I' '&quot;'&quot;
(~oo
fJ.~
t'Ucs
OV\
~) 1 ~.
(2)&lt;:
&quot;?
-f-Do
1&middot;+ .~
II
f(
-&quot;'-1&quot;'&gt;C~
cw... 01.
~J
f3 -+
, o (- - (-0-+
x\-~
+
tll (r:&gt;x'l_!)
I(
Chj
')
&quot;,1J r ,,;J-s
Q
q
'&quot;7 )
&plusmn;0 ~
ch 1-
tvk ~
~~~h-f
~ \t.
&lt;t(X!::::
4&quot;
.
J
G =~
~
-=
'2. X
~~
I.
I - ~3 ')
&quot;J&quot;c* ,+00)
-
4
•
-=-
4(3X-1 2.
)
```