ChaoterTest 631
Thenumberof unitsr of a productthat is suppricep is givenbY
p=-1
,:{p-t,
ricep is $10,what is the rate of changeof the
with respectto the price, and what does it tell us?
Supposean object moves so that its
to a sensor,in feet, is given bY
s(r)=16+140t+8\n
at
t is the time in seconds'Find the acceleration
= 4 seconds.
profit (in dollars)is given
a company's
Suppose
P(x)=70x-0.Ix2-5500
x is the numberof units.Find andinterpret
104.RevenueThetotal revenuefunctionfor a commodity
is givenby R : 80x - 0.04x2.
(a) Find the marginalrevenuefunction.
(b) What is the marginalrevenueat x : 100?
(c) Interpretyour answerin part (b).
1 0 5 .RevenueIf the revenuefunctionfor a productis
R(r) :
find the marginalrevenue.
106. Profit Afirm hasmonthlycostsgivenby
C:45,000*100x*x3
wherer is the numberof unitsproducedper month'
Thefirm cansellits productin a competitivemarket
for $4600per unit. Find the marginalprofit'
hasweeklycostsof
107.Profit A smallbusiness
) andP"(:oo).
100-107,cost' revenue,and profit arein dolthe number of units.
Ifthe costfunctionfor a particulargoodis
: 3x2+ 6x * 5oo,whatis the
inal costfunction?
costif 30units areProduced?
of your answerin part (b)?
If the total costfunctionfor a commodityis
= 400 * 5x I x3,whatisthemarginalcost
4 units are produced, and what doesit mean?
Thetotal revenuefunctionfor a commodity
the number
40x- 0.02*, with.r representing
the marginalrevenuefunction.
At whatlevelof productionwill marginalrevenue
0?
Ifthe total revenuefunctionfor a productis
by R(x) : 60xandthe total costfunctionis
byC = 200* 10x* O.lx2,whatisthemarpiofit at x : 10?What doesthe marginalprofit
= 10predict?
the following limits, if they exist. Use algebraic
lim
2+x-5X-
x2-5x-14
lim --;----:
5x-25
lh*
r+-)
X -f ,
wherex is the numberof unitsproducedeachweek'
prodThecompetitivemarketpricefor this business's
profit'
marginal
the
uct is $46per unit' Find
108. Cosf,,"ueiue, andprofit Thegraphshowsthe total
revenueandtotal costfunctionsfor a company'Use
the graphto decide(andjustify) at which of pointsA'
B, and C
(a) the revenuefrom the nextitem will be least'
(b) the profit will be greatest'
(c) the profit from the saleof the nextitem will be
greatest.
(d) the nextitem soldwill reducethe profit'
S
c(x)
(a) Write the limit definition for f' (x).
'(x) for
(b) Usethe definitionfrom (a) to find f
3 . Let f(x) : --
4x-8
8x2-4x+l
lim -:-.-x+6
X'-bX-
C : 1 0 0+ l O x + f i
f(*):3x2-x*9'
4x-x2
t+-2
r+7
6ox'
2x*l
/
4r
is
-. Identify allr-values where/(x)
nof continuous.
4. Use derivative formulas to find the derivative of each of
the following. Simplify, exceptfor part (d).
(a) B: 0.523W- 5176
(b) p:gtlo-6t7-l7t+23
9 Derivatives
CHAPTER
532
T.\
tL,
3x3
v -
2x' -l II
@) f(x) : (3xs 2x * 3)(4xr0+ rox4 17)
(.) g(") :1Q*' + 7x3- s)12
( f ) y : ( x 2+ z ) ( z x + s ) 6
tl0
G) flx):r2Yx
-
? +17
'?
1
5. tmo
u y'
..
-y3_X-r,
*trory:)
Let f(x) : x3 - 3x2 - 24x - I0.
(a) Write the equation of the line tangent to the graph
ofy:f(x)atx:-r.
(b) Find all points (both r- and y-coordinates)where
f
'(') : o'
7. Find the averagerate ofchange of f(x)
13. Use the graph to perform the evaluations(a)-(0
answer (g)-(i). If no value exists,so indicate.
(a) /(t)
(u)lS/(,)
(.) ,ttll_/(')
(d) ,[ry,/(")
(.) ,1q-/(r)
(f) Estimate/'(4).
(g) Find all r-valueswhere/'(r) doesnot exist.
(h) Find all x-valueswhere/(x) is not contin
(i) Rankfrom smallest
to largestf' (-Z),f' (Z
rateofchangeoff(x) over[-2,
theaverage
: 4 - x - 2x2
overIt, o].
Use the given tablesto evaluatethe following limits, if
they exist.
t") |gi/(x) (u)l'41g(')(c),ltiB-g(')
4.999 * 5 *
2.01 2.001 * ? -
4.99
rw)
5.001 5.01
1.999 1 . 9 9
9. Use the definition of continuity to investigatewhether
g(x) is continuous at x : -2. Show your work.
-*
g,\ x\ / : f e
lx3
ifx<-2
i fx > _ 2
In Problems l0 and I l, suppose a company has its total
cost for a product given by dt) = 20Ox * 10,000 dollars and its total reyenue given by R(x) = 250x - 0.0112
dollars, where r is the number of units produced and
sold.
10. (a) Find the marginal revenuefunction.
(b) Find R(72) and R'(72) and tell what each
representsor predicts.
1 1 . (a) Form the profit function for this product.
(b) Find the marginal profit function.
(c) Find the marginal profit when x : 1000,and then
write a sentencethat interprets this result.
12, Supposethatf(x) is a differentiable function. Use the
tableofvalues to approximatef' (3) as accuratelyas
possible.
Giventhat the line y -- ?x - 8 is tangentto t
ofy: f(x) atx:6,find
(a) f'(6).
(b) /(6).
(c) the instantaneousrate of changeof f(x)
respecttoxatx:6.
1 5 . The graph showsthe total revenueand total
functions for a company.Use the graph to dec
justify) at which of points A, B, and C
(a) profit is the greatest.
(b) there is a loss.
(c) producing and selling another item will
profit.
(d) the next item sold will decreaseprofit.
$
696
10
CHAPTER
of Derivatives
Applications
46. Productivity The number ofparts produced per hour
by a worker is given by
N-4+3t2-t3
where f is the number of hours on the job without a
break. If the worker starts at 8 e.u., when will shebe at
maximum productivity during the morning?
47. Populatiorc Population estimatesshow that the equation P : 300 + 10t - f2 representsthe sizeofthe
graduating classof a high school, where f representsthe
number of yearsafter 2010, 0 < f < 10.What will be
the largestgraduating classin the next 10 years?
48. Night brightness Supposethat an observatoryis to be
built between cities A and B, which are 30 miles apart.
For the best viewing, the observatory should be located
where the night brightnessfrom these cities is minimum. If the night brightnessof city A is 8 times that
of city B, then the night brightness b between the two
cities and x miles from A is given by
,8kk
o
:7-1:o-ry
where k is a constant.Find the best location for the
observatory;that is, find x that minimizes b.
49. Product desigLnA playpen manufacturer wants to
make a rectangular enclosurewith maximum play area.
To remain competitive, he wants the perimeter,of the
baseto be only 15 feet. What dimensions should the
playpen have?
50. Printing design A printed pageis to contain 56 square
inches and have a | -inch margin at the bottom and
l-inch margins at the top and on both sides.Find the
dimensions that minimize the size of the page (and
hence the costsfor paper).
5 1 . Drug sensitivity the reaction R to an injection of a
drug is related to the dosex, in milligrams, according
to
R(x):"'(to-i)
Thesensitivityto the drug is definedby dR/dx.
the dosethat maximizessensitivity.
52. Per capita health carecosfs For the yearsfrom
and projected to 2018,the U.S.per capita out
cost for health care C (in dollars) can be
the function
- 2.5u2+ 40.2t+ 677
c(r) : 0.11813
where f is the number of yearspast 2000 (Sourcc,
Centersfor Medicare and Medicaid Services).
(a) When doesthe rate of changeof health
per capita reach its minimum?
(b) On a graph of C(t), what feature occurs at
f-value found in part (a)?
53. lnventory costmodel A companyneedsto
288,000itemsperyear.Productioncostsare$l
preparefor a productionrun and$30for each
produced.Inventorycostsare$1.50per year
item stored. Find the number of items that
oroduced in eachrun so that the total costsof
tion and storageare minimum.
54. ,\verage cosf Supposethe total cost of
units ofa product is given by
C(r) : 4500+ l20x + 0.05* dollars
(a) Find anyasymptotes
ofthe average
cost
e61: c(x)lx.
(b) Graphthe average
costfunction.
percent
55. Market slare Suppose
a company's
the market(actualandprojected)for a new
quartersafterits introductionis givenby
M(t):
3.9t2+ 3
o . L t 2+ |
(a) Find the company'smarket sharewhen
uct is introduced.
(b) Find any horizontal asymptoteof the
M(t), and write a sentencethat explains
ing of this asymptote.
Find the local maxima,local minimar points of inflection,
and asymptotes,if they exist, for eachof the functions in
Problemsl-3. Grapheachfunction.
t.f(*):x3+6x2+gx+3
2.Y:4x3-x4-ro
x2-3x+6
3 . 'r , :
x-2
6 . Find the relative maxima and minima of this
,7
Find the absolutemaximum and minimum for
In Problems4-6, usethe function | = 3xs - 5x3 + 2.
4. Overwhatintervalsis the graphof this functionconcaveup?
5. Find the pointsof inflectiooof this function.
9. Use the graph of y : f(x) and the indicated
f(*)
:
2f - 15* * 3 ontheintervall-2"
Find all horizontal and vertical asrrmptotesof
function
flxl :
200x- 500
x*300
to completethe chart.Enter *, -, or 0,
whetherf, f ' , andf " arepositive,negative,or
eachpoint.
ChapterTest 69i
Point
1 5 . A firm sells100TV setsper month at $300each,but
A
B
c
the figure to completethe following.
"!iq.f(,)
: i
) What is the vertical asymptote?
Find the horizontal asymptote.
6):
I } , f ' ( e ) : O ,a n d f " ( 6 ) : - 3 , w h a t c a n
aboutthepoint on the graphof y : f(x)
market researchindicatesthat it can sell I more set per
month for each $2 reduction of the price. At what price
will the revenuebe maximized?
L6. An open-top box is made by cutting squaresfrom the
cornersofa pieceoftin and folding up the sides.Ifthe
pieceof tin was originally 20 centimeterson a side,
how long should the sidesof the removedsquaresbe to
maximize the resulting volume?
1 7 . A company estimatesthat it will need 784,000items
during the coming year.It costs$420 to manufacture
eachitem, $2500to preparefor eachproduction run,
and $5 per year for each item stored. How many units
should be in eachproduction run so that the total costs
of production and storageare minimized?
1 8 .Modeling The table givesthe number of women age 16
years and older (in millions) in the U.S.civilian work
force for selectedyearsfrom 1950and projected to
2050.
(a) Usex asthe number of yearspast 1950to createa
cubic model using thesedata.
(b) During what year doesthe model indicate that the
rate of changeof the number of women in the work
force reachedits maximum?
(c) What feature of the graph of t\e function found in
part (a) is the resultfound in p4rt (bX
Year
Womenin th€ Work
Force(in millions)
1950
18.4
x : 6? Explain.
1960
23.2
U.S.CensusBureaudatasince1870,the average
1970
31.5
of acresper U.S.farm can be modeled with the
: -0.000377t3
+ 0.I02Lt2- q.ntt + 174.3
f is the number of yearspast 1870.
the yearswhen the model has a relative
and a relative minimum.
changesin our society and in the economy
the last halfofthe 20th centuryhavecaused
to increasethen?
ue function for a product is R(x) : 164x
and the cost function for the product is
C(*) : o.olx2* 2oxr 3oo dollars
.r is the number of units produced and sold.
many units of the product should be sold to
maximum profit?
is the maximum possibleprofit?
ofproducing x units ofa product is given by
C(x) : 100* 20x-t 0.0rx2 dollars
units should be produced to minimize aver-
1980
45.5
1990
56.8
2000
65.6
2010
75.5
2015
78.6
2020
79.2
2030
81.6
2040
86.5
2050
91.5
Source:
U.S.Bureauof LaborStatistics
742
CHAPTER
11
Derivatives
Continued
a guide line is 7 ft away from a spot directly below the
sign. How fast is the worker taking in the guide line at
the instant the sign is 25 ft from the worker's hands?
Seethe figure.
38. Elasticity A demand function is given by
pq: 27
where p is in dollars and 4 is the number of units.
(a) Find the elasticityof demandat (9, 3).
(b) How will a price increaseaffecttotal revenuC
39. Elasticity Supposethe demand for a product is
p ' ( z q +1 ) : t o , o o o
Guideline
34. Envirownent Supposethat in a study of water birds,
the relationship between the areaA of wetlands (in
squaremiles) and the number of different speciesS of
birds found in the areawas determined to be
S:
kAt/3
where k is constant.Find the percent rate ofchange of
the number of speciesin terms of the percent rate of
changeofthe area.
35. Taxes Can increasingthe tax per unit sold actually lead
to a decreasein tax revenues?
36. Tares Supposethe demand and supply functions for a
product are
p : 2 8 0 0 - 8 q-
q2
i
a n dp : 4 0 0+ 2 q
respectively,where p is in dollars and 4 is the number
of units. Find the tax per unit / that will maximizethe
tax revenue 7, and find the maximum tax revenue.
37. Taxes Supposethe supply and demand functions for a
product are
to,oo.
p: +o* 2oq and.p:
'
q+r
where p is in dollars and 4 is the number of units.
(a) Find the elasticity of demand when p : S20.
(b) How will a price increaseaffect total reven
40. Elasticity Supposethe weekly demand function
product is given by
P = l}}e-orq
where p is the price in dollars and q is the num
tons demanded.
(a) What is the elasticity of demand when the
$36.79and the quantity demandedis l0?
(b) How will a price increaseaffect total
, 4 I ' RevenueA product hasthe demandfunction
fi--T,11
p:100-0.5q
where p is in dollarsand q is the number of units
(a) Find the elasticityq(q) as a function ofq,
graph the function
fQ) : q(q)
\
(b) Findwhere/(C) {, whichgivesthequa
which the producth{s unitaryelasticity.
Therevenuefunctiorffor this productis
/
\L./
R(q): pql Qoo- o.sq)q
_ ,,,/
GraphR(4)andfind the4-valuefor which
maxlmum revenueoccurs.
(d) What is the relationship between elasticitr'
maximum revenue?
respectively,where p is in dollars and 4 is the number
of units. Find the tax f that maximizes the tax revenue
7, and find the maximum tax revenue.
In Problems l-8, find the derivative of each function.
L'Y:,e'-rx2. l : 4 l n ( r 3 + t )
3.y=ln(xa+t)3
4. Jt(\ " '\/ : rnf?2")
5. J:te'7
",
-
lnx
-
6.
v-
8.
g(r):zlogr(+x+z)
9. Find y'1f 3x4 * 2y2 -l 1 0 : 0 .
10. Letx' + yt: 100.
-It. -d x : 2,find;when.r
,dy
: 6 andT : 3.
dt
dt
1 1 . F i n dy ' i f x e Y : l \ y .
12. Supposethe weekly revenueand weekly cost
(both in dollars) for a product are given by
R(") : 300x - 0.001x2and C(x) : 4000 *
respectively,where x is the number of units p
ChapterTest 743
and sold. Find the rate at which profit is changing with
respectto time when the number of units produced
and sold is 50 and is increasingat a rate of5 units per
week.
Supposethe demand for a product is p2 + 3p + q :
1500,wherep is in dollarsand 4 is the number of units.
Find the elasticity of demand * p : ZO.If the price is
raisedto $31,doesrevenueincreaseor decrease?
Supposethe demand function for a product is given
by (p + t)qt : 10,000,wherep is the price and 4 is
the quantity, Find the rate of changeof quantity with
respectto price when p : $99.
The salesof a product are given by S : 80,000e-04r,
where S is the daily salesand / is the number of days
after the end of an advertising campaign.Find the rate
ofsalesdecay10 daysafterthe end ofthe ad campaign.
The table givesthe total U.S.expenditures(in billions
of dollars) for health servicesand suppliesfor selected
yearsfrom 2000 and projected to 2018. Given that the
model for thesedata is
|
:
l3I9eoo62t
wheref:0in2000.
(a) Find the function that models the rate of change.
(b) Find the model'svalue for the rate of changeof
U.S.expendituresfor health servicesand supplies
in 2005 andin2020.
Year
Health
Expenditures
Year
2000
1264
2 0 10
2458
2002
1498
2012
2746
2004
1733
2014
3107
2006
1976
2016
3556
2008
2227
2018
4086
Health
Expenditures
U.S.Centers
Source;
for Medicareand MedicaidServices
t7. Supposethe demand and supply functions for a
product arep : 1100 - 5qand p : 20 -l 0.4q,
respectively,wherep is in dollars and 4 is the number
of units. Find the tax per unit f that will maximize the
taxrevenueT : t4.
1 8 . Nlodeling Projections indicate that the percent of U.S.
adults with diabetescould dramatically increase,and
alreadyin2007 this diseasehad cost the country almost
$175billion. The tablegivesthe percentof U.S.adults
with diabetesfor selectedyearsfrom 2010 and projectedto 2050.
(a) Find a logarithmic model,T : f (x), for thesedata.
Usex : 0 to represent2000.
(b) Find the function that models the rate of chanseof
the percentof U.S.adultswith diabetes.
(c) Find and interpret f(25) and f'(25),
Yehr
Percent
Year
Percent
2 0 10
15.7
203s
29.0
2015
18.9
2040
31.4
2020
21.1
2045
32.1
2025
24.2
2050
34.3
2030
27.2
Source.'
Centers
forDisease
Control
andPrevention
1 9 . By using datafrom the U.S.Bureauof Labor Statistics
for the years1968-2008,the purchasingpower P ofa
1983dollar can be modeledby the function
P(f) : 3.4$Kr.046)-t
where f is the number of ylars past 1960.
Use the me44 to irid the function that models the
rate ofchangeofthe purchasingpower ofthe 1983dollar and use that function to predict the rate of change
in the year2015.
7g4
Integrals
Indefinite
12
CHAPTER
Evaluate the integrals in Problems 1=8.
r. !G*'+ 8x- 7)dx
, I(^ + \/; - t)0.
- 7)edx
lsx'(+x'
- 6x-t t)-3(zx- 2)dx
+.
lG*'
z.
'[]a"
o,
6 Iroor-o,o
,. lrr'r','-'d,
8.
i-')*
I(,"*
tuz
9. Evaluatel-:-
JX+
L
a..Uselongdivision.
f
10. IfJ/(x) dx = 2x3- x * 5e'+ C,find/(x).
In Problems1l and 12,find the particular solution to
eachdifferential equation.
! ! . Y ' : 4 x 3 + 3 x 2 , i f . Y ( O )a:
s^,
n. +:
ax
e * , i yf ( o ): 2
differential
13. Find the generalsolutionofthe separable
dy x'f"
equation
-ax *
the rateof growthof the populationof a city is
14. Suppose
predictedto be
dp
: 2000t1'oa
dt
wherep is the population and f is the number of
past2010.Ifthe populationin theyear2010is
what is the predicted population in the year
1 5 . Supposethat the marginal cost for x units of a I
is MC : 4x * 50,the marginal revenueis MR
and the cost ofthe production and saleof 10
$1000.What is the profit functionfor this
the marginalpropensityto saveis
t6. Suppose
4:
dy
(inbillionsof
o.r, - -+
Yo.5y+ |
and national consumption is $6.6billion when
disposableincome is $0. Find the national
consumption function.
17. A certain radioactivematerial has a halflife of
x, satisfies
If the amountof materialPresent,
f;
wheret is in days,how longwill it takefor 9096
radioactivityto dissiPate?
that a liquid carriesa drug with
18. Suppose
0.I glcc into a 160-ccorganat the rateof 4
leavesat the samerate.Find the amountof
organ as a function of time f, if initially there
the organ.
866
13
CHAPTER
of Integration
Techniques
Definitelntegrals:
5 1 . Cosf If the marginal cost for x units of a product is
MC : 3 + 60(r + 1) ln (x + l) dollarsper unit and
ifthe fixed cost is $2000,find the total cost function'
52. Quality control Find the probability that a phone lasts
more than I year if the probability density function for
its life expectancyis given by
/(rJ:1
fJ.
55. Revenue A company has the data shown in the
from the saleof its product.
(r.4r-',' x> o
x(o
o
t
Capital value Find the capital value of a businessif its
income is considereda continuous income stream with
annual rate offlow given by
f(t)
:
hundredsofunits andrevenueR
Ifx lepresents
the total
hundredsofdollars,approxirnate
from the saleof 1000unitsby approximating
L2oeo'o3t
in thousandsofdollars per year, and the current intercomPoundedcontinuously.
est rate is 6010
54. Total income Supposethat a continuous income
0r1
streamhas an annual rate of flow f(t) : 100e-0
(in thousandsof dollars per year). Use Simpson'sRule
with n : 4 to approximatethe total income from this
stream over the next 2 Years.
to
Useleft-handendpointsandn = 4 subdivisions
the areaunder/(r) : \/ q 12on the
approximate
interval[0,2].
: l with
:
Consider
f(*) : 5 2xfromx 0tox
r equalsubdivisions.
(a) If f(x) is evaluatedat right-handendpoints,find ,
a formulafor the sum,S,of the areasof the n
rectangles.
(b) Find lim S.
ExpresstfJJrea in QuadrantI undery : 12 +
4x - x2 (shadedin the figure)asan integral.Then
evaluatethe integralto find the area.
rl0
I MRdx
J6
Rule.
with the Trapezoidal
I
7
UseTable L3.2in Section 13.5to evaluate
following.
t"l Jr"Qx)ax
f-
0)JxVlx-7dx
8. Use the numerical integration feature of a
14 -calculatorto approximateI Vr' + 10dx.
the supplyfunctionfor a productir
9. Suppose
p : 40 * 0.001x2andthe demandfunctioo
x is the numberof
p : 120- O.2x,where
p is the pricein dollars.If the market
surplusand
is $80,find (a)the consumer's
producertsurplus.
Evaluatethe following definite integrals. (Do not
approximate with technolo gy.)
13
,, l, @-+x)AxG)
'' f i-*
-r
| x(sx'? e)-ltz 6*
J0
f*q
(d) | -dx
Jr x-
10. Supposea continuous income stream has an
rate of flow /(f) : 8se-0'01t,in thousandsd
per year,and the current interest rate is 7%
pounded continuously.
(a) Find the total income over the next 12
(b) Find the presentvalue over the next 12
(c) Find the capital value of the stream.