pgl
Fall2014 @ Drost
MATH I42
Exam 2 REVIEW
Fall2014, Mrs.I)rost
4.1 Derivatives
Section
1) Find the derivative of f (x) = o'*'
9'k)
*+.x
JZ
{Vx
x+3lx
J;
f '{*1=
dy
dx
for
y-nxo
.
Ztfx +L * ?'!
=
2) Find the derivative of "f (x) =
3) Find
+..6 +2lnx
= x't' * 3{36
t;"- $*'tr'
+7nx2
,' r*t + 2'4
X
4gr
dT, = 'V
4) Find the equation of the tangent to
,*t (x)=
r7L=
Zzc
f (*)=x'+e'
+ex
+1 at
r=0.
gt -2: t
f ' @)= 2'a =/
fQ)*o+t t'r=Z
+eo
(x'a)
t-zs"x
g=x*Z
(arz) rtz= I
5) Estimatethecostofthe 35'ftitemwherethecostfunctionis C(x)=Ji(x+12)+50.
C
(x)= xe/' * rzxf.
r'a
c'(u)=-2-*'/'+ei/'to
2
C'(y)=
Section
4.2
-
Cr(sq )=
Jhl
z--''6+ e.
*i7 + #
Products and Quotients
6) Find "f'(x) tor
f (x):(r'-
3tnx)(ze'+3x)
f' = lstf-$Xrf
rc;,)+ (rt - 3.r-til(zt* s)
Fatl2014 @ Drost
pc2
MATHI42
Find
/'(x)
for
f (x) =x"!t*
*ro)'^
-x-
.
+ 5x
Do not simplify.
2
t
,f -- (,.3 -7- t- 3u).(z)(sx"a}3)
Q3-v"
8)
+3f
Suppose the demand equation for a product is given
- ( s **'tf (3 x" - Z' .s)
by p
:
=I-,
l+x''
number of items sold and p is the price in dollars. Find the marginal
&.*..p
where x is the
revenue. r
kz:+*-,t-?* = l-t: - fno^e{rt-t
(+ x= )= (*"7 Aeoefiua
tl
L
R-=r/t
)=
\t+!- j
It*
R'= Q:{Xr) -?.(3r-)
( **)'
9) Find
it
and h aredifferentiable.
*tt r.h) .f , s,
3= r"Q. h)
h'n h'J' ) ,-9. h) .S '
"J'= |SN +$j'1.+ $'.3.F.
,4 =
*(,
10) Given the cost function: C(x) =
2x3
Ae' c
- 4x2 + 10x + 50 ; find the marginal average cost.
I
rtff + *#
Acei = r*r-r/*\/a+so{t
Avxro,Se cosr
/Wkc = et
Section
4.3
11) Find:
= 4x
"
-
^4 -Sb,ra,
AAC= 4x-
Chain RuIe
(s*n-,,)"
# h/-;--) = g;
r t f8:,f -'d'b (su"u)
E-\
'I/g
= rtrb (r J -ti
4-
SQ
XZ
pg3
Fall2014 @ Drost
MATH I42
12) Find:
!(-tsr;;=)
=$y
F'' V.i'4Yt)
:r,=,,Jg#g =-8x. ei-)*"
13) Find:
= ot.*'2(r'" *)q)
+ @n
*V''unxl'z)
,a3.t3 prodtu-tnt t{., avr}t C'ha'r;" }r,d'r"
"; ':'
.5
for
tees is given by V =l 0 + '007 (d - 5)'
certain
of
feet
board
in
V,
volume,
14) The
respect to
Find the rate of change of v with
d >10, rvhere d is the diameter in inches.
dwhen
d=12. {= tO t.oo.l(d-Sf
V,o , O?-t (a-Sf (r)
V' = .ozr(4- 5)'
2
ls) rnafrwhere r=#
Section 4.4
16)
t7)
*r:")
ac'
$''
Logarithmic Functions
Derivatives of Exponentials and
2 f".b."g,n3
z t^Sxll'. t,w3
*r{'"rnr)
=
"t''t
*
A^oJ .2x
Fail2014 @ Drost
pg4
MATH142
18)
Find the derivative of y -(fn1Sx;)'
U'= 3 (
19)
I.^^"
'*
tuof
Suppose the price and demand are relatedby
revenue is
zero. R.=
fvt (_=
x. p = *
N=
ftlR:
d'lt
d4"
t+*(+)+ d'r(,)
.
".(4X+t) =6
Q--
x+
qz, +s)t +'!.,w
\ '- 2--(m (zr+5 ) -
i=
,J
*=+
4 -4ur,rli*
20) Find the derivative of y- = rn [(2
t
g = -ln
p(x)= e4' . Find where the marginal
s)' e-.
''1
4"*'-. )
tt
X'--
-' J,.^
,zxy1
-+.- .
'a'^/u Uzx+l
t?s+b_ - Il- {:i.z.t"'t = -1 *tt -29,,^+
e
+zx+t
b.**S
Section 4.5 Elasticity of Demand
2I)
Given: x + 4 p = 20, find the elasticity of demand at $3, and state whether the
demand is elastic, inelastic, or unit elastic.
:e+.lp = Zb
X= zo-4p
xt =-4
3l = 3.
- 2o-{g
s-'Y
io 4p
6=-+Gq)
EG;=
22)
*
=7= t.5 :,Lasrrrc.tp*a;ltr,5)
Given the price elasticity of air transportation is 1 10 , if the price is decreased by
l0o/o , what will be the approximate change in demand?
10
\
dV
lo'/o
to"/o( r.ro)
dV \\'lo
Fall2Ol4 @ Drost
MATH I42
pg5
23) Given x = 30-10 p ,ftndthe price which maximizes revenue,
rop+rc,so fr =x+
revenue.
lDp=-x+36
'('
f =-fi.xtn
and find the maximum
xf*xr3) = -Ixz+3>c f=*(tS)+3
i{=-?y+B--!*t}=cc p=-JEtb
=
z
ft=-kg;:€fi?:66
3= tLrc
=$t' 50
f
,s = 1
24) The demand function for beef consum-p'ffi-per capita in pounds, x, arrd p is the
price of beef divided by disposable income per capita, then x =126.5
-1800p. Find the
p=0-05. r=
y= lz(o.5 - l8oory s -fC rgoo) -,lg.oa f 126.5-l8ooP 1265'l8eP
y,= rgoo '
elasticitvofdemand at
-
FQ.o€)=
= 2'16sB
fu=*
25) A restaurant owner sells 100 dinner specials for $10 each. After raising the price to
$11, she noticed that only 90 specials were sold. What is the elasticity of demand, and
what price maximizes revenue?
(x, +)
(too, io)
ECf)=3-(-rC -K\4Zoolop Le,o-lW
tD4; -a-+?,oo7? 200- lof
(1or I r)
F(fi=-#
' Zo-Q
yl e--to
nIro= -L x+zo
Section 5.1 The
26)
l"
f
?o
Derivative
:l
2()- f
7A3
Find the critical values and determine the intervals where f(x) is increasing and
is decreasin
gif f(x)= I + i *
g -- ( +?{t *
$'=
- b{.2
+
-Bx-4 =o >i114
-vx-- 4
.
z}'*
*=-jL
3
-'{{3
5'' k-*=o vn^*b3vi
27) f' (r) = 2(x +t)'
(x
- t)' (* -2) ,
graph.2- Xr-t rr=t x=L
Gti
t
b,=
JP
x3
-.3x
:
f(-r)=-2, f(r)=3, "f (2)=a Sketch
J
4
(x-rJ5 lxiz) l3r-*s
| ^ tf
(x)
i
V
(-+la,o)
) k.;?'),eg
pg6
Fall2014 @ Drost
MATH 142
Find the domain, critical values, intervals where the function is increasing/decreasing,
and all relative extrema.
2S)
f (*)=4x6 -6xo +5
Sf
domo* B,
O.nitr'col-
V
\
vo\r,tcs
)o =
v 2\
-2{f =o
-Af (t+r)=6
=-ztxs
-t
x=o a
o
t
(- m.,o)
foroo)
n€t. rna4.
af X=O
9t=eI-a-x =o
29) f(*)=e'+e-'
Lf=
ex-J^tg
vt- \'
gl
donnorh R,
Cc.trlat values y=o
,.1 (D
ox=l
v
ex
,Do)
gt"= I
2x=o
\ ("e,o)
I€l-. vhin of
X=O
2v+4> o
Jomath R, v >'7
v2
7
(-z
r*)
NoN
a7
"t
d'-b,
f-e
flr3
t
d -!e
Po5
x7.(j^
30) f (*)=2x+tnl2x+al
cr;t(ia"t Voluqt t
\
2y, -4
s
-L
fr '= 2* L=o
zx+4
thutt b3 2-r't4
Ztzx+q)*]: o
-2. r'7
@o
tr*
4re+8+2=o
4v=-lo
)c=$-t5
-a 2-
flOt tn d"rna.ra^,
Fall2014 @ Drost
pc7
MATH I42
Section
3l)
5.2 The 2'd Derivative
Find all inflection points for f (x)= ya -10x3 +24x2
E = 4't
-3a;+4bx
\J f n*,r.J
+3
z
V= n;-6oy+4t3b
)z(Xt-Sx+rf)=O
S',
tzLx-'{Xx-D=o
*=1 > X=l
32) Find the second derivative.o t
fc*)
=
{'t'
f (t)=
+ lt'
+3x+5.
=
rzCx -
L ttst
+Lx- t)
#*G
vx
:- -l-i't:- ,
t;"'
Y'g) = z;7'-+;tu =
h*t/'(:-.,c)
9'oO
33)
Find the inflection points for
$' . 3b yL -
f'(x)
=12x3 -48x2
16y- ,1
V^t
,,1
S"- 12-Y-qG=o %u
1Lx = Qb
w\$\ec't ion pofnt at v- *h
x= L
3
34) Find the intervals where /(x)
given
f" (*): x(x-z)'(*+3). The domain is all reals- t
)(=^3
Y=
x$
x4 z
J:1o i >, 1u
'^z
'a
-4
\-/
n
t
(- n,-3).,(o rz)rQ
(^3
is concave up or down and any inflection points.
ro)
urdlecttbru gotnb
G-d { 1x*311 a-ta
fes|
-+
)" (
.{*
+
a
,b) I
\
at r= *3 rx= o
'l
+
3
+
ga
t
+ )r
+
+
pc8
Fall 2014 @ Drost
MATH I42
35)
Find the intervals where
given .f (*)=(x+a)'
/
/(.r)
is concave up or down and any inflection points.
x. Thedomain is all reals, X f
O
I tr; = (lc* qJ'
S,r= Zzff. =
Y\
++) - (x++)=.\
x(zxr
9, =
3'=@t
5,=
Section
t
:
@!y:C
*t
5.3 Limits
36) Find:
&- =o 4
at
U
'b)
f1.Gero)
v3-v. [*rbf2
=
CD
l-lo rngtechirn forirb
Injinity
ti^s!'r*3*-]
x+a $16'-X+7
c0lnpartt, &eSttes
5/,
37)Find:1'*FT
38) Find:
,{q(s
={4; = L
-2t') ; 5- zgsl :5
-&1u-)o
-)s'e"(
&s x+
r
39) Find: x+
[- vt + I
'X+5
40) Find:
lim
xo
Covnfq^t Aegees
-xt +x-1
x3 +1
t-ornpqrrs d€qtte9
xjT
t*>u-!!y"+ =-N
X-) lrnattr,t!O
Q43
Fall 2014 @ Drost
pg9
MATHI42
Section 5.4 Additional Curve Sketching
=x"?* ^t' ,
4r) Find vertical asymptotes for f (x)
-3x -4
.
,}cr) = 3tK- 4)
ho0e.
af x= L&
Qe r)1x+ r)
tl trtga-r" 4.65m gtote,
42)
Sketch the graph of
f (x) =
2x2
+5x-3
. lnclude a
Y=- t
of all asymptotes.
Q*-tf,x+ a)
.
v
Bb
o,
-l
;\:*>
G -a)Cx+3)
UA x=3
ftA )=L
hota *-= -3
(rrfttcspb .[ct f=o @>'b)
-0-ct y=o Vz>o)
43) Sketch the graph ot f (x) =;j
{^z L-= q.a-/*z.-*
.
r-\
$'=&-q)-!--{e'V
Lr Lx+rf*l
VA y=-\) x-- I
.
Co.,o)
)=o
trfrrcepfs
Hr.,
*, - ",rS"-*-L
-'=ffffif
- e zo d
.$,='-n
G":E*
"tl
rt&t
44) Sketchthegraph
otf(x):**E. :
b= !tf25fl
q'= [ -zs.yfz= t-25- =o
-73'
q r/
tr -s(a\
l=5l-v t
Er"
-6
3'G
u:'-F
ra glty
(l'r = so x3=
*
g
*=z-S
X=
t-
nelofuir u .'' Aecruasrn5
5;S
7-z+2S
Y
qp
X,=
O
HA n6f\eektrrfu.s--
5=*
pg l0
Fatl2014 @ Drost
MATH I42
45) Sketch a graph of the function:
*a 1'("ft o o" (-*'o)
f'>o
on
7 ,U
(o'oo)\
-
f"(t)>o
'\'/
f@)=4
f
/(o)=0,,!ry* $)=-1, Jiq
and
f'(*\<o
Section 5.5 Absolate Extrema
Locatethevalue(s)whereeachfunctionhasanabsolutemaximum,anabsolute
over the given intervals'
minimum' if they
"*isi
Ot-6y' o
*z*28 -rLrL= -/5 C") =
3v (r- z)
3(4)=<.q-49t2=lt
46) f (*)=xt -3x'+z on [o'+]
S
=
Yz(> Y=Z
$
to)'a
-D
47) f (t)= 4-2x-x' or.-(--'*)
3.t
J
= -2'2-y-=a
:=3
5'/ =-z n
48) f (t)='tf .r
on
[-l'+l
ii'V= Cr'*
D" ['-)
iir*
n- X
*,'=
,rl
v
v,
V
xr+l
=()
"rD
-E
rl
to
{
,ifr
g6
li?l[=
9ce1I to) = rlil= \
5(.\)= fiCt-r= tiF
Fall2014 @ Drost
MATH I42
pg
ll
49)
Find two numbers whose sum is 42, and for which the sum of their squares is a
mmlmum.
'rrr,".q't'
x+3 = 4z
S=
5=
U=tz-x
S'=
aR: S'/=r{ U
2xr Z&?t:*X)
*=*
ti; ';;;;
y with x+y- 60 forwhichtheterm xzy
'z
tr = x5
T= tex+GO
and
y+D -60
;;r
t=2e
4x-84= o
"'. cnrt,irrrwrq
50) Findtwonon-negativenumbers x
maximized.
t+qL
**[+r-p-t
b = ->a+t.'o
is
-f
+60 t
*3rt+tz"Ox =O
qt *3x(ut;
4 o)=o
Y=O X= r[O
-T"
Section 5.6 Optimization
=
51) A trailer rental agency rents l0 trailers per day at arate of $30 per day. For each $5
increase in rate, one less trailer is rented. At what rate should the trailers be rented to
produce the maximum income? How many trailers will be rented?
(*'?)
T:lP
0o,ao
T= x(-Src*So)
(q
T=
)3s)
*St
Jtqt€-
+o
gra/.,l.c-<-
lY:lp.-X, r,,r+c,onnC,
+tox
S'{O
*tro-r"b\s reltfkL I
=
f = "sv+80 T'= - IDY -l-8o
gl't>: tOK
f = -s€ )+ro
8=X
fl =s4o
52) Arancher wants to build
to enclose rectangular areaof
e>
-
1,800 m' for cattle.
a fence
a
The fence a long one side is to be made of heavy-duty material that costs $15 per meter.
The material along the remaining three sides costs only $5 per meter. Find the
dimensions and total cost of fencing for the field that is least expensive to fence.
l{oorri
x
b
N3 U)id)h tm
b'
ft=
x[: tSoo
ux t8e
tr
zl =
rW*ts
x:1 3o YntAgr.s
g = bo wrefils
C= lSxr
5
5+sPtSg
(,2 Lavllog
1889)/-r
Q = 2ox-?'o UY
i,{
C = zox + t8rooo X
Ot =Q+-- I ? ooO =t>
?nf=,fo.o
qo6 Y=3O
*=
pg12
Fall2014 @ Drost
MATH I42
53) A company is planning to manufacture
After conducting extensive
market surveys, the ressarch department estimates a weekly demand of 600 blenders at a
price of $50 per blender and a weekly demand of 800 blenders at apnce of $40 per
a new blender.
blender. Assuming the demand equation is linear, use the research department's estimates
to find the price that maximizes revenue.
(r., ?)
R=
((oD6, 5o)
L'f
R= p6-Sr+xo)
(Soo, 4"J
fl=-J-t+8ox
?D
f = -*)cf80
fri=
{ ='+o
*l--Y +80=c
to gD
= i.,x
8oo = /'
54) A box is to be made oul of a piece of cardboard that measures 12 inches by 12
inches. Squares of equal size will be cut out of each comer and then the ends and sides
will be folded up to form a rectangular box. What size square should be cut from each
corner to obtain a maximum volume?
t7,t
L:z=
l/t
Vr =
0 st^,.!r- n|C'
fha-ryirn,ats
\AlLtr-'7Y,"n.
(rz-z,x)trzax)
!= )c(l z-Zx)"
V' = v' CXz -zx) frz)+ ( z- z>) [ D
V= x
x
*he
Qz-zx)L- 4r + z-2v,J
V,= i,r -zx)Lrz-6x)
l7-Zx=Q
lZ'bN=O
,t::"
tz=zi
Q=Y
nct
uru dornq-i'"t