### Solutions Midterm 3 Spring 2014

```MATH
7:15-8:15 pm,
205
EXAM
Thursday April,
3
17, 2014
N
Instructor:Section time:
No books or notes are allowed. Simple calculators are OK, but no sma,rtphones.
Show atl work to get full cred,i,t. Arguments must be clear.
K-state honor code: "On my honor, as a student, I haue neither gi'uen nor recei'ued
u,n,an,thori,zed, aid, on thi,s academic work."
r
The second derivative test for two variables:
point (a,b),
D:
f,,fyy
- fln. lt at critical
D > 0, f,, > 0, then (o, b) is a local min,
D > 0, f,, < 0, then (o, b) is a local max,
D <0, then (o, b) is neither local min or local max,
D :0, then test is inconclusive.
o Linear approximation in two va,riables:
f (a + Lr,b
r
Average va,lue of
/
+
LE)
x
f (a,b) + f"(a,b)Lr + [email protected],b)Ls.
on the interral from a to
b: * I! teV".
o Logistic growth function:
P(t\:-I + Ue-k"
,-L.
f :l:*1.
lsapl
o Equation of the tangent iine: y: f'(")(r - a) + f(a).
r
The elasticity of demand:
o Forml
as
for derivatives:
[email protected]@))': f'(g("))s'@).
U @)[email protected])),
/
f(")\'
:
f, @)[email protected]) + f (r)s, (x).
f'@)[email protected])- f(r)s'@)
\n6):-----TW
Good luckl
SCORE
Total:
100
Max
#1
-un
ffJ
#4 #5 #6 #7 #8 #e
10
10
10
12
10
72
12
10
1.4
1 (1o pts).
The total cost of production, in thousands of dolla"rs, is C(q)
where q is in thousands and 0 < q < 8.
Determine
g\$g4llth"
:
q3
-
I2q2
+
60q,
exact value of q at which average cost is minimized.
&rf) = 9_!D = Xs - tz ?z +eoT =
- tzL+ 60
7,*
q.
q
Cfil-c^l- 1'o.n+ro{ awotanLcf co>* ! atry)=O L? .,":
ctilh rd'!
Q=
l/
6
enl-poin+s )-"2 n"
Ufu* +L
q =O
t
,r=
'T=&
,fu:y"-t
AL|C^>--6O
a (O : 36
7ai--/J
u-ti''<' wa'L^t
- 1L+6o [email protected]
a-(al"= 6v -q6 +60 -23
a(?) is *ri'tit*ti 2ed a'-t ?'=6
2 (LO pts). The demand for yams is given by S: 5000of yams a,nd p is the price of a por.urd of yams.
10p2, where q is
in pounds
a) Find the elasticity of the dema,nd at the price \$2.
P=z
F = I P l?.1
'Vrr)= ,o(x) - rD'Q -- +s6a
t'?7rl
\/
:
t 1/
rt
(z?
I
I
{-v0)l!
-_-YU
- l\s60'
4g=
1'= -'o'P
dp
l,
L
I
P=-
I
o117
b) Is the demand elastic of inelastic?
E <1- t So *fu
oln^"a'- al
iE inelaah'c-
c) What is the effect on the quantity dema.nded of a 570 price decrease?
T
ht- cll*. a--a( wi tl
i
nuva's4 -h' aJou+ lx
Q.a1 6t
I
o'o& %
(10 pts). The rate of sales of an automobile anti-theft device are given in the
table.
Months
Sales per month
1
2
140
520
680
4
5
6
750
700
550
a) When is the point of diminishing returns reached?
+4t ya6 y,*'t^ , :-itc-?7
. .t
I
rat<-t
tl
tt'fu.r
o*e
*t,a 6:"tan* 4-l +1"is Poi'db) What are the total sales at this point?
/11
O+t:-c)r 680+79o = zoqo
J-ezticoa
c) Assuming logistic sales growth, use your a.nswer to part b) to estimate total
potential sales of the device.
-76+rt.(. ta.t{).rLil',r
a
(1"2
v t?a dLvlgs
= L := 2' (zoco)'=
pts). Compute erpticitly the integral tZ t @a, of the following function.
lz
( f r"i = (cx,vo+ E {D :o
r+s')
)
It
laj
u
-6
\
-/
t
I3
-(o-^o*"
6lta,^ \'= nr*6
u
r-,'-Pa.E
6
9
l5
12
r5
t0
r
4
Itt
-z\-s 4). to
A
-60
q
=- 49/ -- 13;
t /< -\.re q*o -Z"G'')',-
=\l = 3'{
m
\2
\\@t
-
4
)I
?<+ t3.{
-f
(x)4x -
- 60.
[email protected]
5 (10 pts). Estimate ff2
In
dz using
-J-
Make the tabiee of values that vou use
Points
o ,7_ Y (" a to (rz)
L
1
c+r
,s
>s
l'L
(
)
o
*-1, "
,
+
,q
n.
:6.
that left-hand sum:
uol 8.e u^a/ iu &\l'ha"d evua
/('
\
n [s
( t+ \z=+
!+)"+
l.*
l) '" x
3"tv
Given the demand curve p : 35 - q and the supply cl ve p: 3 + q,
find the the producer surplus when the ma,rket is in equilibrium. The a,nswer
must be the exact number.
6 (12 pts).
3s-t = 3++
€q,. i\ ibri r^'"'' 9 o in-f :
3z-*zY
?
4b \ nouiltbr;",.=
I
f
9t
\
'r
a+15=t4
^_
\r-"
l1
t
,-aini
I
l6
Prodttclr
sLL'cp\us
= LL 'tt *" -
--
shaAtA
(ii)
7 (12 pts). compute the indicated pa,rtial derivatives for the
following functions:
a) /" for f (x,A) :2r2 + BAr
ulsLLur_4,
ln= 4x *
3t
(")
3x2+zgx
Cu
'_'
.3= 2t
(81
9'2'x
-zt
+, -- L2'u
:
b)
f,
for
f(x,y)
:
az"zu
't'3-i,3X
/r\
(-r
lZ= LrLe26
.z
n
l>='/,
I*u
''a --- -2-
,_{I_
:2ev
a) f,y for f (r. u\
rX- -
t
.y'b
O
(1q
Lr
/'\
:
\,
l-r \/
(-t Jt
= "(u')t =-?
a function f(r,s), we are given
f(10,20):
lts).20)For
:
fr(ro,
3i **
300,
-100. Estimate /(0, 2b).
a-PYloYi'"raJnr'l''r
l(o,zs-)
=
=
3
:
200,
:
!{w,-ut) -ro.
sO
/"(i0,20)
4y(lor 2-o)
I s f(
(
tort-o) =
(-te-o) =Fzz;o-]
- 19 'zoo + 9'
\L
9 (14 pts). Find all critical points. Determine whether each is a local
maximum,
local minimum or neither.
f (',g)
:
"
+
a2
+
6r-
1og
OA+al goira{at to --o
*
B'
f"><+8
-o- I =-=
41--o L-U-to--o Z-t
I'b c{oferrni'q
Ix>= 2r
lXa= 2n^
tYl=,
k4+
'r"d d-o-Ju4{-v-a
+t^a tsfff t uN .14^-e
9=L'L-O = 4>o 1I
J
t"" -- 7 -zo
B'a
f'a
-k-^+
C-z,s) is
{-oco'l
u't'i
ni n'u*'a^:
```