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Math 1161 Test
April 7, 2009
MAKE SURETHAT YOU READ AND FOLLOW ALL INSTRUCTIONSCAREFULLY,
CALCULATORS ARE ALLOWED.
This exam consists of EIGHT questions (some with multiple parts), and an addendum to exam 2 consisting of four problems at the end.
IMPOHIANT: You are to choosewhich of ff7 or fr8 to do. You can try both, but you MUST INDICATE WHICH ONES YOU WANT GRADED.
The addendum to Exam 2 appears on the last two pages. These problems are optional-they will not a-ffectyour exam 3 score; they can only help your exam 2 score.
Correct answers given without adequate justification will not receive full credit.
P r o b l e m 2 - L 2 p o i n t s Problem 3 -^ 8, 10 points P r o b l e m 4 - , 6 , 8^ p o i n t s Problem 5 -^ 21 4, 6, 8 points P r o b l e m 6 - 1 6 p o i n t s Problem (^7) /8 -^ LO points
Exam 2 addendum -^ 6, 3, 3, 3 points
- The graph of the first derivafivef'(r) of a function f (r) is shown.
(i) /(") has a local maximum at r :3.
TRUE GAffi)
(ii) /(r) hasan infl;;6h polt at r :8.
(iv) /(r) is decreasingon the interval(3,5).
TRUE < rffi->-"--.-*-/
(v) As z increases from 0 to 5, the concavity of f (r) changes from concave down, to concave up, then back to concave down.
z-l-REE') FALSE
2. Sketchthe graphof a functionthat satisfiesall of the givenconditions:
( 1 )f ( - 2 ) : 1 , / ( 0 ) : 3 ,^ a n d/ ( 1 ) : 5.
(ii) f'(-2): //(1) :^ 0.
( i i i )/ ' ( r ) < 0 i f r < - 2 o r r > 1 ,a n df ' ( * ) > 0 i f - 2 < r < I.
( i v )f " ( r ) ' 0 i f r < - 0 o r n > 2 a n df " ( * ) € 0 i f 0 < r < 2.
,IgL f @): m and,lgg/(r)
: o
li*
(/nt o o
,.-zlad
F.-.---f-r,-{!*JP.^ --^ tt""nr''/ot<4.t^
a/u44 n!?
- Completethe statement of the Mean Value Theorem by writing down the secondhy- pothesison / and the conclusionof the theorem.
(i) The Mean Value Theorem: (^) Let f be a function that satisfiesthe following hypotheses: i. / is continuouson the closedinterval [a,b], and
ii. r is dl^tlt"r^*r^l'U^ d^^
(u,b)
Then, there is a numberc in (a,b) suchthat:
U - f u \ , 4 ' 1 t \
b - A
! (ii) Use the given diagram to illustrate the conclusion of the Mean Value Theorem. Label it appropriately, including the points a,b,c,/(a), and /(b),
h)
ee
(iii) Find all values of c that satisfy the conclusionof the Mean Value Theorem for f (r) :^ rs -^ 6r f^ 1 on the interval^ [-3,0].
o { - l \ j^3
( ' / x l ' - 5 v ' - b^ Q o 4 o l e
X * t,[T
fu'-h*a
'3lY' -- ol
a. n o 3 )
1l^t" ."4^ -'ft^ i4 i^ (^) Ll,ol, & * L t u e - -- G
- You've beenhired by a pharmaceuticalcompanyto designa pill. The pill has (^) the shape of a cylinder, with hemisphericalcaps on each end. The pill must have a volume of n12ml. The material usedto manufacturethe hemisphericalcaps costs0.75 centsper squarecentimeter,and the material for the cylindrical part costs 0.5 cents per square centimeter.Find the value (^) of the radius of the pill (correctto two decimalplaces)that minimizesthe cost, find the cost (in cents) per pill, and use the secondderivativetest to show that your answeris a minimum.
l.-t
tL+k^
-- hz,;u/' (. 1A//^
I t E f a a ( ' , u u t o (^ P t ( t
ils(- (^) br^7fta'^'{ {t^ ?uf t o7l
+ l A ' e
t?
n r ' h
1! r 7 t ( ' '^
tu
! 7
l - I r r ]
a 7 r a
) r 2
+.]'ncz=
7 d t (^) f ' =
T + M
2 r 3
5 o
c'1c=
z , o
5o fi@^ ^o,/en,(^ s^ Ld| t' n b"y'^
4,
t/ a a4^ pitl
gzr^^//nnnlw L"l')'- ?t+
'5 (^) ; Pa5:/';(''
'htn ( 4
ru
T,+)
5 r ' r h t z
t (^) or,f (o,
V o l u r r e" ( g i U '^
/= Tf
tA
a - t t a
2 r r r h ' * ' i , 1 "
^ n L ) ^ 1e A Z, i t t
' " 5 ( z r r r h )
t ' 1 5 1 4 n " \ = n r l ^ + 3 - t r t L
) n f e t * 5
I -^ L r r T Z 4
1r lz
/ l t
o f f^ o n t \ iJ
, E.LJ,
'
16
L','
?^'
1o
'?t* (^) a;* - 0',!"'* ytu&e'
& fntntYnltnj
Addendum (^) to Exam 2
- A baseballdiamond is a squarewith side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ftls. At what rate is his distance from secondbase decteasingwhen he is halfway to first base? Give your answercorrect to one decimal place.
ler s =ol'(^ P" tvnw +" zd
bnv
l"/ (=^ rvutwtLd(it^ /co*^
l"'"< Pblc
1"4'^ *t^
= 11
uh^ *=(^5
I
e Xrun'A'n'
c 4 (c*L4^ '
wL, n (^) y - - r t 5 ,
{
&o
-x' r 1t'
z hs
*r)(-t)xio
' / ^ ' &
z = 5
= 2 5 s t
5 &
a(
?r)
w#,)#
qo'*(9" 5' =-)^ 3 = tlSG^ s (^) 1Dd,b, 1
5 s^ - ( 1 0 - t i l , ,a I (^) & ) o ,7 ft(szc
l orJ,^ hz
Addendum to Exam 2
(ii) Find dyldr by writing ridg) -^ I^ and using implicit differentiation. Your answer should be a function of r'only. Include an appropriatelylabelled right triangle.
s i a I
6 t A
z { t ' 4
(iii) Do whatever simplifying is necessaryto show that your answersto i) and ii) are equal.
t-^t (^)?
5 t
x
/oo^ 4 '),
4,
x',[t- f-
!o
''
^J ii\ (^) a,\L u(&
