Math 1100
Practice Exam 2
24 October, 2011
1. Be able to define/explain/identify in a picture/draw the following
(a) Absolute minimum/maximum
o~0I4e
~1v~
~~‘—
[~MF ~ o
~~c’ 5c~-c
0”
Afr&~I
-,
—
lov<sFpzitf
oVC’
Cf
4 ,tvol
(b) Critical point
~
h-~
4
(c) Relative minimum/maximum
d,At~k-f
‘5
~:≠c
/~
s
≤,&c
5—i(
0p-en
i~tean4/.
(d) Pofi~ctioi,,,~
c
(e) Concave up/down
Coj~Cc.’<.
c.’p:
Fuviqe”F
(~oiy1J
)
l~’~s IOe(J’~’
I.,.-.. Cj,’c’vc
~U~e
oh
C~CQk1Iy
c~,l Ca’-c
cc~/vC
a
You should be able to deal with the geometry of circles, rectangles, right triangles, and
boxes. (If any other shapes come up, I will provide the relevant formulas on the test.
2. Find the elasticity of the demand function p t4q
P
—
tL;z
f.
‘ji
P
dv
=
80 at the point (10,40)
%
dp
—‘to
—l
(
JOP’<
3. Find
(a)
y=lnz
V
(b)
y
—
=
6 log7 x
-
K
(c)
2
y~
-2 ~
-
(x3,~b)(tn&f ~
(d)
16x4+1N
~
in~ x—3
)
—)~ (y~)
y;
(e)
,Y~ 2
dy
~
2
7
~-
6
ln((x+3)2(x2 -6)~)
(f)
=
4]
I
•3~y’2~
log5 x3
3
__
InS
(g)
(x-1N
log6 ~x+i)
)/;;
dy
io~~
_J—~.
V-I
(xi-!)
—
—
~i-j
-jTh
fn~
~
h1b
(h)
2x_x
y=e
(Ix
Lc2~
(I)
y
=
e~2
=
2X
ZxeA
V2
ix
(j)
dx
(k)
1
Y
e
-
C
_2-x C
(1)
dx
=
y
(Zx-2) ~7t_z~ .Jn
(m)
=
9
y
~
4X~2X
=
=
y=(x2—1)lnx
ax
~
xy2—y2=1
~
Zx,y %i
dx
-
Zy~I
(p)
x2 + 2y2
4
—~
0
=
/
c1A
(q)
-x
dX
x4 + 2x3y2
=
x
—
-)
~
‘-~
~
~
~lx3~I
I
-
-
-
~
~Lr
~
~2yt
~X3yfr3Y2
4. A study showed that, on average, the productivity of a worker after t hours on the job
can be modeled by the function P(t) = 27t + 6t2
t3 for 0 ≤ t ≤ 8, where P is the
number of units produced per hour.
—
(a) For which values oft isP increasing?
F/n
2
77 ~ 12~ -3/’
~‘{j=
P1i) z3(~
-~;I
—
3
27k
I-
U
—
bq)
7~12
/97z36
fi—
2 h7~
0
jJ
~
‘4~II6
~q
2-
~t
~-
2 -fI~ <0,
2FJ7~ ,~ b-i ~ ~vi-~•~ nkc fr-f
,Y1’krC)13
c-S.
(b) When are workers most productive?
4~ b~
~
(y7C_Cc)/~5
(o,.
2t(7~)
OY~\
5. Find the absolute maximum of
P(x)= ~ ~x
~%Z
=
—
2z2
—
4x + 2 on the interval [—1,3]
-~
0
%~—~
y~
f(z)
-‘1±
a
Es’- 2(f)tb W-’)
~
~a
-~
-
77
H3)
(‘3,i)
~,ayii~v.~:
—1
47
t~
(z,
~(‘fl~
-;)
6. The total cost function for a product is 0(x) = 100 + x2 dollars for x units. How many
units will result in a minimum average cost per unit?
31
~ K
3
~
‘flN
—ret
—eTc
/
-LOQ
~e
-wO
-(00
-
)-
R-~
~2CC~~<
(=0
U
Vt,>0
~-iia-ifn-,v1~
cut-
(P
<0
Cr44-
C~’-CJV,~
tw,
0c
X~fr
IC ~‘y HO ~
~
~
7. Find all relative minima, maxima, and horizontal points of inflection of
14
~
2
_~x3+~x2_6
(J~
0
1
yI~i
&) C~-~~’
&1~~’~
borr?c~k4
yflu-e)
poMt
-1—2
If ‘~z
(2
o
2—
~-?~—r
—l
-i---~
‘i <0
:-
4cc)
j
at
/12
~ (o~ —6)
it~irn~’
I
at
(~
,i~)
8. Suppose the demand for a product is given by (p + 1)~4Ti
(a) Find the elasticity when p
~ (pi () ~t~(
r_4jO0G
(‘3~H)
(OOD
i1’T
~
(pfl)
cc,F(~
CZ5~
‘itt
c14
-—
lvi
2RPT
4
1000
$39.
=
(pit)
~F(
=
~_!.
1’
‘ilLJ
d~-
~
—
~3q
_z(62f)
a
~
(pre)
(b) What type of elasticity is this?
(c) How would a price increase affect revenue?
~_~•I1
l~—(u-c~juz
cLocr.ec.ce
9. Is the function f(x) = 2x3 + 4x
~
K
12
C1
£
So
P
I)
I>
10. Find the critical values of y = 2x3
yf~_
6xt
~K~2C1
~z ~
O
o~
8 concave up or down at z = —1? At x = 4?
—
y~L4
Co~(~
cv~a~
—
12x2 + 6
Jo~ ~
n-i
(,
15—s)
11. An agency charges $10 per person for a trip to a concert if 30 people travel in a group.
For each person above the 30, the charge per person will be reduced by $0.20. The
agency cannot take more than fifty people. How many people will maximize revenue for
the agency?
(~j
y..
nv~Ler
(so
R(A)
of rap4
posfr
~O.
fz)(/O
+Llx
fr~OO
~(4y,~ -Lj
Xz lo
4,
U F
j~ (x)~
—.
s
q
~JI/
feaf’~
U{0
C-,n cLt-<
d0~.
-d
r~ (0
S
cc
frflOXf’1141Z4 r-ev~evIL~(.
12. A 30-foot ladder is leaning against a wall. If the bottom of the ladder is pulled away
from the wall at 1 ft/sec, at what rate is the top of the ladder sliding down the wall
when the bottom is 18 ft from the wall?
3 0
?y~j
dk f
y
Yt:
13
y~~i3ot_
‘C
2- (~ ~
cit
Z
4-
2.Z’~
%IS*
—
~1/3
~/nh.1
—6
dx
!gZ
2Uj
13. A rectangular field with one side along a river is to be fenced. Suppose that no fence is
needed along the river, the fence opposite the river costs $20 per foot and the fence on
the other sides costs $5 per foot.
(a) If the field must contain 45,000 square feet, what dimensions will minimize the cost?
xdY=
r
j-
~
(Oxi- ZOy
F~d
(0 x
-~
lQx~-
r
/
t
=
rCUOQOO
t~
5-600
x
ç0
b-c
~/k1(j7~...
qooooo
vs
~iJ
._~n_,
i.Jl-~~~1
x--300
q~a
ID
qooovo
At
yS
,—
p’~(3uo)
DCOV~
ix)
÷
~(/
~x
ico.
—
-
qooo°
ri.
x.~-i,DO
(b) How large an area can be fenced for $6,600?
20)1+
x
10X2
~6OQ
A~xy
Zo(7~s-) HO .x=-≤660
* (0 x ‘-6600
≤&O-2y
f04-_
2300
x-~- 230
AI1~
(~6O-2y)y
i
AD
S° b—
0
“go
14. Ifxy=x+3 ‘cit
~=—1 andx=3, find
V
CL(
~
~yt~ 3t3
C~
Ti-
-)
(Ii
~‘fn~ ic
t”-~ á~~~~cnrs
~~(qq
16c x z~ U
6av~cc’-<
15. When $700 is invested at 9% interest compounded continuously, its value after t years
is 3(t) = 700e09t. At what rate is the money in the account growing when t = 4? When
t = 10?
4)
7°O•~1
c
~3e
15L1_
C3e~
~-~—
16. At what points does the curve defined by x2 + 4y2
Vertical tangents?
ZA
‘~)‘4~
44-
fl7~
dx
I4o~rt~H(
—
4
=
0 have horizontal tangents?
D~
K
€,y~
Fa~~A/≤
~a~y~n~Z
ye/h-C1!
~ ~yt~ ‘a
K~
~(O)2_~t~
x~-t
si
honr4Iw( 4-a.~h
c.f
(a,,)
.iJ
17. Find the relative maxima and minima of f(x)
~.4
~(x)
t~
-~
1
4
(o~-i)
=
(74)
x In x.
l~x
€‘(~~)
(nx~
In A
f
2
I
-(
--(
+
p..Jat-r~-c
n’v,vY7Q9j-~
L~i
91
(~)
~a4-
-
-
‘>0
( ~ —4)
27O~
18. Make a sign diagram for y =
minimum points of the function.
/1=
—
3~2
+ 6x + 1 and find all relative maximum and
3xt~~ ~
•k~—~y4- L ~O
± 1’t—’1-2I
2
2
2±
fVc
i—cd
t-€i°S
04
ho
~,
~(
flfl1/~’~’
3
,pDIA
19. Suppose that air is being pumped into a spherical balloon at a rate of 5 in3/min. At
what rate is the radius of the balloon changing when the radius is 5 inches? (Recall that
the volume of a sphere is given by V = frr3)
V~
yQ(vn—c
V~
fl~7t
ç~ (OOq-
sLc
ô4
i:~-;
rn/~,~
20. Use the second derivative test to find all maxima and minima of f(x)
f~(X~~
1x~
X~
Uk
~Y~—ZOx~
(x
-
a~a y~
—
~sk
4-
c~-c
=
E-
~-r~( v~/U~
5x4.
I-
I
~y~(
(L~l)
—2c6)
a
—
-
—~
C4
2c
alcxti—et-
t_ol~~c
~ -qØ~ 32~
7
—
0
ZOK~tOK2
-
~
~
>c
~c~M
o+
(°,o)
21. A rectangular box with a square base is to be formed from a square piece of cardboard
with 24 inch sides. A square is cut out of each corner and the box is folded from the
remainder of the material. (If this description isn’t clear, see problem 27 on page 740.)
How large a square should be cut from each corner to maximize the volume of the box?
2 ~y-(qL
V(~
~j~(Lf)~
~x3
>0
~
~
Xt
I~2~ ~
~
~
C
=0
—
(t- t7)(Y-~)
>~
22. A firm has total revenues given by R(x) = 2800x 8x2
x3 dollars for z units of a
product. Find the maximum possible revenue for sale of this product.
—
g’Cx)
_~t
z~oO-
j~L~
F
-
9,y~Y
dx
dx
a,
~-Zg
Cf
&K
I
y
~
—~
I
I
r017&
2-
~
hi’s
23. Find the equation of the tangent line to the curve
(2,2).
2x
i&
-
/
—‘
~
—700
// jç12u(ñ=
(Gy ~i~OO ~D
(6 t J~3.2tø~
—b
L6±
—
/
~,<
~4
<0
—
~ /2x~ -171A ~
—
~I~-(’~2
—
7-2
__
{5
4x + 2y2
0
—
4
=
0 at the point
~
24. Between the years 1960 and 2002 the percentage of women in the workforce can be
modeled by
W(z) = 2.552 + 14.5691nx
when x is the number of years past 1950. If this model holds, at what rate will the
percentage of women in the workforce be changing in 2015?
w’( Lc)
2~77≤
25. A firm can produce up to 100 units per week. If its total cost function is 0(x) =
500 + 1500x dollars and its total revenue is R(x) = 1600x x2 dollars, how many units
will maximize profit?
~Z “(x)
,
f2(x-~ ,j ~
—
PG)= PLx)
~
C(x)
a
—coo~~cooK
l60Ofl~
LO0x —YOO
~(x)
4— (O~
p({~)~ -—2y4m1~$
S~’
—oytv~”~’” occvt3
~
‘—
x~-
SQ
26. The amount of the radioactive isotope thorium-234 present after t years is given by
Q(t) = lOOcO2825t. Find the function that describes the isotope is decaying
Q(t)~
~
e~2~’~