-
Friday, Mar 14, 2008
EXAM II
MATH 135, SPRING 2008
READ EACH PROBLEM CAREFULLY! To get full credit, you must show all work!
The exam has 7 problems on 3 pages! Turn in all pages!
NO GRAPHING
CALCULATORS
ALLOWED!
• Problem
1
• Find the derivative of the following functions'
.( )
(a) f(x)
=
(b) g(x)
= In(2x + 1),
esm 3x,
f'(x)
= .e
g'(x)
=
~~('x)
>~ (~)()
G.::,~l7:)·1<..)
.1-
~.'2-::.
2)(+\
'2x. +-\
• Problem 2
Find the equation of the tangent line to the curve
y=
x2ex-1
at the point where :r = 1.
'2 X
e
)(-1
2. ,,-I
.-t)(-€.
'1..
'/--- It
..
;\\1\
~j
::... .~)(
~
( ,
i-I
-
.)-= /2 I + (L) € 1-' -:::~.
3
.
.3-
\
'3
e
l-V"I ("
x) .
• Problem 3
,
(a) Find the derivative of the function
•
f(x)
(b) Determine
.,1
( L\)
r (,\:.)
=
(-- ~
.
.,
x
)
\ ')
':J
7'
the points (if any) where the tangent line to the graph y
rx
("+1)1. -
.,,\
·1
=
<- ~)\
".
( X
( X
z
~,~
of-
/ )
'2.x.
-=
is horizontal.
'2.xJ
-
( X -t ,
1- X
--------
(I - x)
+ \)
= f(x)
·1 (X*I)[x-tf.-
.'2.(X+I)
'i
0
l
zfX (X+I»
rX. ( X.+I)~
i
\J \J..,k
~." ~in\ ~
JCx)':-D
:=:)
\
-
X'
:::-0 ';'- ) ,--.
x-:=.
-.-.J I
2{X (r:.+/J
• Problem 4
Using implicit differentiation,
.
2X
find ~, given that x2
+ 2y2 = 3.
.-
-ex
. ~y
+-
It '-I
Jx -::;;.O'Z- J
I .:.-1-
4y
• Problem 5
(a) Find the linear approximation of f(x) = /f+3X near x
(b) Write down the differential dy, where y = VI + 3x.
~l x)
(0.)
~ ,{co)
.
-t-
Je) ()(- 0)
I
(-'
",
..
-2.
-\(,x)
~ '-(1
t-;,JC) .;.
"2.
~". X
~
'3
:=
(
-,:::-)
2. •.•
1-+ Jx
--
(
= O.
- , L/~!.-f.-\" '3 x:\.
Jy
t)c!xl·
'3
J ;(v')~
_'7? ~
~,
-t ).\)
-:=
3
7-
1
• Problem 6
If a (spherical) snowball melts so that its surface area decreases at a rate of 1 cm2/min, find
the rate at which its diameter decreases when the diameter is IDem. [Surface area of a sphere
is S = 47fr2, where r is the radius.]
c.\\~~-r
I
(
I
Su. <:+u
\
)
':;
c;,,-Jl,::..,
) ~.~-
..
"
.-=. ._
..:Xa'
!!.
't _
- _
. ry
,1
~5
~-;
(.\
-t
,,~c;i
c\
c\
.~
2
if
._.~------..
r
J·t
01
J).
c~
--
.~~
.
del _
10
J1
.~ .. ) d-*
1;(
\~ I
• Problem 7
Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and it forms a pile
in the shape of a cone whose base diameter and height are always equal (see picture). How
fast is the height of the pile increasing when the pile is 10 ft high? [Volume of a cone is
V
= ~7fr2h.]
.-
.:' ')
'\ -:--. 12-h.>:::2-
-v /
AP;.•..........•.
.,' ..-c;'-...
'-.;y.' ~.r
c~
i -.9;7;
1~
.
'
jt;
-
l~\ ">
'
._
s~\F_I~
V
It
-Il h
-:L
~ '7 ) ..
\-l
-=-...
\
." '2-
~
'(
(
)
I
~
.-
k '(
I~~
3>
It
2...
-
'\.
-:::
12
'~
.-
Ie
;1.-
i
4-·~··v_
~
r~('I-,
__ .--." ~
') ~\ V
"-
._~
-
(e..
.
Jt~V~
..
-=
_-.......•/
R'*:::-
'L70
r.
\~
~k
A ~,
.. ,J-t
\ .- {~
~
Jt
-:J(.t~
L
lO
t '2--0
100lC
c.tt
'4 ((
V)
Jh
elk
toTe
CN
,
I