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CHAPTER 4 4 • • • •
292 P2: OSO/OVY 8383-c QC: OSO/OVY tions T1: OSO entia tion o July 4, 2016 2019 15:33 4-82 Unsaved...
Applications of Differentiation
CHAPTER
CHAPTER 4 4
Applica
Applications of Differentiation
• • • • Applications of Differentiation
f
er
4-82
of
4-82
Dif
CHAPTER
292
292
4-82
292
292 CHAPTER 4 • • Applications of Differentiation 4-82
Notice the similarities between examples 9.4 and 9.8. One reason that mathematics
betw
One
ples
ex
am
een
and
9.8.
9.4
hat
similar
hematics
he
otice the similarities between examples 9.4 and 9.8. One reason that mathematics
t
t
eason
ities
No
mat
Notice the similarities between examples 9.4 and 9.8. One reason that mathematics
r
tice
N Notice the similarities between examples 9.4 and 9.8. One reason that mathematics
has such g the similarities between examples 9.4 and 9.8. One reason that mathematics
has Notice
has suc great value is that seemingly unrelated physical processes often have the same
has such great value is that seemingly unrelated physical processes often have the same
such great value is that seemingly unrelated physical processes often have the same
ph
h
ed
seeming
alue
v
t
hat
is
unr
r
elat
l
y
eat
same
sical
v
t
ocesses
t
en
ha
e
he
pr
y
of
has such great value is that seemingly unrelated physical processes often have the same
mathematicaldescription.Comparingexamples9.4and9.8,welearnthattheunderlying
descr is that seemingly unrelated physical processes often have the same
has hematical description.Comparingexamples9.4and9.8,welearnthattheunderlying
matsuch great value
9
hat
9.4
mathematicaldescription.Comparingexamples9.4and9.8,welearnthattheunderlying
e
lying
Com
he
t
ing
par
n
.8,
mathematicaldescription.Comparingexamples9.4and9.8,welearnthattheunderlying
w
iption.
e
under
ples
and
am
t
x
lear
mathematical
mechanisms for autocatalytic reactions and population growth are identical.
mathematical f description.Comparingexamples9.4and9.8,welearnthatthe
mechanisms for autocatalytic reactions and population growth are identical. underlying
identical.
wt
r
e
alytic
o
eactions
aut
or
h
r
g
and
population
mec
mechanisms for autocatalytic reactions and population growth are identical.
ar
ocat
mechanisms
hanisms for autocatalytic reactions and population growth are identical.
ha for autocatalytic reactions and population growth are identical.
We have now discussed examples of eight rates of change drawn from economics
mechanisms e no w discussed e x am ples of eight r at es of c hang e dr a wn fr om economics
We
v
e have now discussed examples of eight rates of change drawn from economics
We have now discussed examples of eight rates of change drawn from economics
W We have now discussed examples of eight rates of change drawn from economics
and the sciences. Add these to the applications that we have seen in previous sections
he have now discussed examples of eight rates of change drawn from economics
and We
A
seen
sciences.
dd
hese
e
t
t the sciences. Add these to the applications that we have seen in previous sections
t
t
evious
and the sciences. Add these to the applications that we have seen in previous sections
he
sections
w
e
applications
ha
v
t
in
hat
o
pr
and
and the sciences. Add these to the applications that we have seen in previous sections
e sciences. Add these to the applications that we have seen in previous sections
and the
and we have an impressive list of applications of the derivative. Even so, we have barely
and w we have an impressive list of applications of the derivative. Even so, we have barely
e
an
so,
im
w
en
E
v
v
applications
iv
v
ativ
and we have an impressive list of applications of the derivative. Even so, we have barely
t
of
bar
t
he
e
der
of
ely
lis
ha
ha
e
e
pr
e.
essiv
and
and we have an impressive list of applications of the derivative. Even so, we have barely
and
o have an impressive list of applications of the derivative. Even so, we have barely
begun to scratch the surface. In any ﬁeld where it is possible to quantify and analyze the
begun we
t to scratch the surface. In any ﬁeld where it is possible to quantify and analyze the
t
scr
and
analyze
In
he
an
uantify
q
e
t
o
surface.
begun to scratch the surface. In any ﬁeld where it is possible to quantify and analyze the
y
c
it
wher
at
is
t
he
ﬁeld
h
possible
begun to scratch the surface. In any ﬁeld where it is possible to quantify and analyze the
begun
properto
properties of a function, calculus and the derivative are powerful tools. This list includes
begun ties scratch the surface. In any ﬁeld where it is possible to quantify and analyze the
operties of a function, calculus and the derivative are powerful tools. This list includes
t
e
he
ar
includes
iv
e
ativ
der
ools.
t
properties of a function, calculus and the derivative are powerful tools. This list includes
of
This
a
function,
and
w
po
t
erful
lis
calculus
pr properties of a function, calculus and the derivative are powerful tools. This list includes
at leastties
at leas some aspect of nearly every major ﬁeld of study. The continued study of calculus
proper t some of a function, calculus and the derivative are powerful tools. This list includes
least some aspect of nearly every major ﬁeld of study. The continued study of calculus
at least some aspect of nearly every major ﬁeld of study. The continued study of calculus
v
e
y
er
y
of
s
l
near
of
.
tudy
s
The
ma
continued
ﬁeld
jor
tudy
calculus
of
aspect
at at least some aspect of nearly every major ﬁeld of study. The continued study of calculus
will give you the ability to read (and understand) technical studies in a wide variety of of of of
will give y ou t he abilit y t o r ead (and under s t and) t ec hnical s tudies in a wide v ar iet y of
will giv you the ability to read (and understand) technical studies in a wide variety
e some aspect of nearly every major ﬁeld of study. The continued study of calculus
will least
at
give you the ability to read (and understand) technical studies in a wide variety
will give you the ability to read (and understand) technical studies in a wide variety
ﬁelds and to see (as we have in this section) the underlying unity that mathematics
ﬁelds give
will and to see (as we have in this section) the underlying unity that mathematics
ﬁelds and to see (as we have in this section) the underlying unity that mathematics of
ﬁelds and you
t the ability to read (and understand) technical studies in a wide variety
t
in
under
hematics
he
t
lying
his
section)
t
e
y
w
(as
hat
see
v
e
mat
o
ha
unit
ﬁelds and to see (as we have in this section) the underlying unity that mathematics
brings to a broad range of human endeavors.
brings
ﬁelds
brings to a broad range of human endeavors. the underlying unity that mathematics
t and to see (as we have in this section)
oad
r
ang
br
ings to a broad range of human endeavors.
o
a
v
or
s.
endea
e
of
human
br brings to a broad range of human endeavors.
brings to a broad range of human endeavors.
EXERCISES 4.9
EXERCISES 4. 9
EXERCISES
EXERCISES
EXERCISES 4.9 .94 4.9
EXERCISES 4.9
discuss the signiﬁcance of this value in terms of the cost of of
discuss the signiﬁcance of this value in terms of the cost
signiﬁcance
he
WRITING E EXERCISES manufacturing. of t his v alue in t er ms of t he cos t of
XERCISES
WRITING EXERCISES
WRITING EXERCISES
manufacturing.
discuss
discuss the signiﬁcance of this value in terms of the cost
WRITING EXERCISES
discuss
discuss the signiﬁcance of this value in terms of the cost of of of
WRITING
t the signiﬁcance of this value in terms of the cost
manufacturing.
manufacturing.
WRITING EXERCISES manufactur ing.
manufacturing.
=
−
6. A baseball team owner has determined that if tickets are
t
t
1. The logistic equation x (t) = x(t)[1 − x(t)] is used to model
used
equation
(
)[1
(
)]
x
)
x
tic
1. The logistic equation x ′ ′ ′ ′ ′ x (t) = x(t)[1 − x(t)] is used to model 6. A baseball team owner has determined that if tickets are
t
1. The logis
model
o
(
is
t
1. 1. The logistic equation x (t) = x(t)[1 − x(t)] is used to model
The logistic equation x (t) = x(t)[1 − x(t)] is used to model
that
priced at
tickets
team10, the average attendance at a game
determined
arewill
owner
baseballAED
if
priced
6. A baseball team owner
A priced at AED 10, the average attendance at a game will be
6. 6. A baseball team owner has determined that if tickets are
has has determined that if tickets are
many im
am
many important phenomena (see examples 9.4 and 9.8).
many
ex
t
′
phenomena
important phenomena (see examples 9.4 and 9.8).
9.8).
(see
ant
ples
and
9.4
many logistic
man important phenomena (see examples 9.4 and 9.8).
1. The y important phenomena (see examples 9.4 and 9.8). 6. Abaseballteamownerhasdeterminedthatifticketsarepriced be
por equation x (t) = x(t)[1 − x(t)] is used to model
27,000
be
attendance
10, if tickets are priced at AED 8, the average atten-
will
a
at
game
average
AED and if tickets are priced at AED 8, the average atten-
27,000 at
the 10, the average attendance at a game will be
at pricedatAED10,theaverageattendanceatagamewillbe27,000
priced
27,000 and AED
at
The equation
The equation has two competing contributions to the rate of of of of
ibutions
com
tw
has two competing contributions to the rate
t
he
o
o
r
contr
t
The equation has two competing contributions to the rate e of at AED 10, the average attendance at a game will be 27,000
The equation has two competing contributions to the rate 9.8).
peting (see examples 9.4 and
The equation has phenomena
many important
at
8,
average
are
attendance
AED
will
the
dance and
priced are priced at AED 8, the average atten-
and if tickets are priced at AED8, the average attendance will
if will be 33,000. Using a linear model,we would then
dance will if tickets
and if tickets are priced at AED 8, the average attendance will be
and
change ′ ′ ′ ′ ′ x (t).Thetermx(t)byitselfwouldmeanthatthelarger 27,000 tickets be 33,000. Using a linear model,we would then
The equation has two competing contributions to the rate
changex x (t).Thetermx(t)byitselfwouldmeanthatthelarger
changex (t).Thetermx(t)byitselfwouldmeanthatthelarger(t).Thetermx(t)byitselfwouldmeanthatthelarger of
change
changex (t).Thetermx(t)byitselfwouldmeanthatthelarger
Usingkets priced at AED 9 would produce an av-
would
that
a
model,we
then
estimate
estimate will be
linear Using a linear model,we would then
be 33,000. Using a linear model,we would then estimate that
estimate that tic 33,000.
33,000. that tickets priced at AED 9 would produce an av-
dance
he ′
population
balanced
he
This
t
)
f
er
y
x(t t)is,thefasterthepopulationgrows.Thisisbalancedbythe be 33,000. Using a linear model,we would then estimate that
s.
t
as
x(t)is,thefasterthepopulationgrows.Thisisbalancedbythe
o
he
g
b
t
r
w
is
t
change
x x(t)is,thefasterthepopulationgrows.Thisisbalancedbythe
is, x (t).Thetermx(t)byitselfwouldmeanthatthelarger
( x(t)is,thefasterthepopulationgrows.Thisisbalancedbythe
produce
would
erage attendance
at
attendance
average
9 30,000. Discuss whether you think the
an
tickets attendance of of
tickets priced at AED9 would produce an average attendance
−
tickets priced at AED 9 would produce an average attendance of
AED 30,000. Discuss whether you think the
estimate that
)g
e
hat
term1 − x(t),whichindicatesthatthecloserx(t)getsto1,the
he
ter rm1 − x(t),whichindicatesthatthecloserx(t)getsto1,the e erage priced tickets priced at AED 9 would produce an av-
(
(
h
h
whic
,t
t
indicat
closer
t
es
o1
x
),
t
st
t
x
t
1 thefasterthepopulationgrows.Thisisbalancedbythe
te term1 − x(t),whichindicatesthatthecloserx(t)getsto1,the
term is,
x(t)
m 1 − x(t),whichindicatesthatthecloserx(t)getsto1,the
of
whether
the
useThen, using the
use of a a linear
Discuss model here is reasonable.
model
linear
think
a
of 30,000. Discuss whether you think the use of a linear model
30,000. linear model here is reasonable. Then, using the
use of attendance of 30,000.
you Discuss whether you think the
erage
o
W
r
he
t
it
slo w thepopulationgrowthis.Withbothterms,themodel of 30,000. Discuss whether you think the use of a linear model
g
model
h
h
t
er
bo
h
slowerthepopulationgrowthis.Withbothterms,themodel
t
is.
population
ms,
he
slowerthepopulationgrowthis.Withbothterms,themodel
wt
t − x(t),whichindicatesthatthecloserx(t)getsto1,the
term
slower 1
er thepopulationgrowthis.Withbothterms,themodel
slower
linear
linear
reasonable.
model,
Then, the price at which the revenue
the
determine
here is reasonable. Then, using the linear model, determine
use
here model, determine here is
linear of a linear model
using reasonable. Then, using
is model, determine the price at which the revenue the
t
he property that for small x(t), slightly larger x(t) means
),
y
slightly
t
t
x
(
(
er
x
)
larg
or
means
f
has the property that for small x(t), slightly larger x(t) means
hat
oper
has t the property that for small x(t), slightly larger x(t) means here is reasonable. Then, using the linear model, determine is is
small
t
has the the
slower
pr populationgrowthis.Withbothterms,themodel
has the property that for small x(t), slightly larger x(t) means
has
at
the price at which the revenue is maximized.
the
linear
price model, determine the price at which
the price at which the revenue is maximized.the revenue is
maximized.
(
t
r
as
h
x
ails
o
oac
appr
wt
)
but
hes
eater growth, but as x(t) approaches 1, the growth tails oﬀ.
1
t
wt
oﬀ.
t
he
g
o
r
,
g
h,
greater growth, but as x(t) approaches 1, the growth tails oﬀ. maximized. which the revenue is maximized.
g greater growth, but as x(t) approaches 1, the growth tails oﬀ.
has
great the
er property that for small x(t), slightly larger x(t) means
r greater growth, but as x(t) approaches 1, the growth tails oﬀ.
maximized.
and
t
population
ation
concentr
h
r
er
o
Explainintermsofpopulationgrowthandtheconcentration
Explain in
Explain in intermsofpopulationgrowthandtheconcentration In exercises 7–10, ﬁnd the production level that minimizes the
wt
of
t termsofpopulationgrowthandtheconcentration
he
g
greater
Explain intermsofpopulationgrowthandtheconcentration
ms but as x(t) approaches 1, the growth tails oﬀ.
Explaingrowth,
lev
es
ercises 7–10, ﬁnd the production level that minimizes the
t
hat
minimiz
el
t
7–1
he
t
production
he
0,
ﬁnd
In In exercises 7–10, ﬁnd the production level that minimizes the
ex exercises 7–10, ﬁnd the production level that minimizes the
of of a c chemical why the model is reasonable. In In exercises t.
t
he
model
easonable.
hemical
r
y
is
a a chemical why the model is reasonable.
of of a chemical why the model is reasonable.
wh ofpopulationgrowthand
Explaininterms
of a chemical why the model is reasonable. theconcentration
erage cost.
age
cos
In
average cost. 7–10, ﬁnd the production level that minimizes the
a average cost.
aver exercises
v average cost.
of a chemical why the model is reasonable.
equently
and
d
eﬁcits
ebt
2. Cor p or at deﬁcits and debt are frequently in the news, but average cost. 2 2 2 2 2
2. Corporate deﬁcits and debt are frequently in the news, but
d
e
fr
a
r
e
h
n
but
ews,
e
Corporate deﬁcits and debt are frequently in the news, but
t
in
2. 2. Corporate deﬁcits and debt are frequently in the news, but
2. Corporate
7. C(x) = 0.1x + 3x + 2000
7. C(x) = 0.1x
7. C(x) = 0.1x + 3x + 2000
7. C(x) = 0.1x + 3x + 2000
.
the
msare
h
t
of
an
o
o
witeach
confused
the terms are often confused with each other. To take an
er
t
en
t
T
her
ake
the
2. Corporate ar eoften confused with h eacother. To take an ex- 7. C(x) = 0.1x + 3x + 2000+ 3x + 2000
t the terms are often confused with each other. To take an
he terms are often confused with each other. To take an ex-
t terms deﬁcits and debt are frequently in the news, but
3 3 3 3 3 + 3x + 2000
2
7. C(x) = 0.1x
ample,
ﬁscal
com
ear
y
example, suppose a company ﬁnishes a ﬁscal year owing 8. C(x) = 0.2x + 4x + 4000+ 4x + 4000
y
ﬁnishes
a
pan
a
wing
oAED
8. C(x) = 0.2x
8. C(x) = 0.2x + 4x + 4000
the terms are often
am suppose a company ﬁnishes a ﬁscal year owing AED
8. C(x) = 0.2x + 4x + 4000
suppose confused with each other. To take
ex example, suppose a company ﬁnishes a ﬁscal year owing
ample,
ple, suppose a company ﬁnishes a ﬁscal year owing an ex-
8. C(x) = 0.2x + 4x + 4000
debt.
Suppose
heir
in
AED5000. is
$5000. That is their debt. Suppose that in the following 8. C(x) = 0.2x 0.02x x
is their debt.
he
wing
5000. That
t
ollo
hat
That That is their Suppose that in the following year
f
debt. Suppose
ample,
$5000. That is their debt. Suppose that in the following year
3
9. C(x) = 10e 0.02x + 4x + 4000
t that in the following
5000. suppose
t a company ﬁnishes a ﬁscal year owing AED
02
0 0.02x
=
( C(x) = 10e
x
)
.
10 0.02x
9. 9. C(x) = 10e
the t company has debt. Suppose that in the following year 9. 9. C C(x) = 10e e
$1
and
ear company has revenues of AED 106,000 and expenses of of
of
xpenses
yrevenues of AED 106,000 and expenses
pan
enues
has
e
of
com
r
06,000
ev
the
year the company has revenues of $106,000 and expenses of of
he That is their
5000.
y year the company has revenues of AED106,000 and expenses
9. C(x) = 10e
√ √ √ √ √ 0.02x
s
AED109,000. The company’s deﬁcit t he y ear is $3000, and 10. C(x) = x + 800
com company’s deﬁcitfortheyearisAED3000,
$109,000. The company’s deﬁcit for the year is $3000, and and 10. C(x) = = =
AED109,000.
AED109,000. The
or
=
deﬁcit
pan
09,000.
y’
x
3 3 3 3 x + 800
)
10.
(
800
C
. C(x)
The The company’sdeﬁcitfortheyearisAED3000,
the
3
f for the year is AED3000,
$1 company has revenues of AED 106,000 and expenses of
x x + 800
x
+ + 800
10.
10 C(x)
............................................................
............................................................
............................................................
t the company’s debt has increased to AED8000. Brieﬂy explain ............................................................
............................................................
s
y’
eased
and the
o
explain
has
√
the company’s debt has increased to $8000. Brieﬂy explain
$8000.
ieﬂy
t
incrincreased to AED 8000. Brieﬂy
Br
debt
comcompany’s debt has deﬁcitfortheyearisAED3000,
and 109,000.
pan Thecompany’s
AED
10. C(x) =
he the company’s debt has increased to AED 8000. Brieﬂy
x + 800
3
wh deﬁcit can be thought of as the derivative of debt. debt. ............................................................
can
debt.
ativ
explainwhy deﬁcit
e
be canbethought
as
y
hought
of
he
explainwhy
der
tofasthe
of
ivderivative
why deﬁcit can be thought of as the derivative of debt. debt.
t debt has increased to AED 8000.
whythe company’s
and
deﬁcit deﬁcitcanbethoughtofasthederivativeof of Brieﬂy
and
function
a
be
v
cos
Let C(x) be the cost function and C(x) be the average cost
cos
t
C
(
e
C
er
)
x
ag
x
he
(
11. (a) Let
11. (a) Let C(x) be the cost function and C(x) be the average costt
t
)
be
t
he
11. (a)
11. (a) Let C(x) be the cost function and C(x) be the average cost
explainwhydeﬁcitcanbethoughtofasthederivativeofdebt. 11. (a) Let C(x) be the cost function and C(x) be the average cost
function. Suppose that C(x) = 0.01x + 40x + 3600. Show
function. Suppose that C(x) = 0.01x + 40x + 3600. Show+ 40x + 3600. Show
11. (a) Let C(x) be the cost function and 2 2 2 2 2
function. Suppose that C(x) = 0.01x + 40x + 3600. Show
function. Suppose that C(x) = 0.01x C(x) be the average cost
function. Suppose that C(x) = 0.01x + 40x + 3600. Show
1. If the cost of manufacturing x items is C(x) = = 3 3 3 3 x 3 function. Suppose that C(x) = 0.01x + 40x + 3600. Show
1. If the cost of manufacturing x items is C(x) = x +If the cost of manufacturing x items is C(x) = x + + + +
1. If the cost of manufacturing x items is C(x)
1. 1. If the cost of manufacturing x items is C(x) = x x
2
′
that
that C (100) < C(100) and show that increasing the produc-(100) < C(100) and show that increasing the produc-
′ ′ ′ ′ C (100) < C(100) and show that increasing the produc-
that C C (100) < C(100) and show that increasing the produc-
that
that C (100) < C(100) and show that increasing the produc-
20x + 90x + 15, ﬁnd the marginal cost function and com-+ 90x + 15, ﬁnd the marginal cost function and com- +
20x + 90x + 15, ﬁnd the marginal cost function and com-
20x the cost of manufacturing x items is C(x) = x
20x + 90x + 15, ﬁnd the marginal cost function and com-
1. If 2 2 2 2 2 3 tion (x) by 1 will decrease the average cost. (b) Show thathat
20x + 90x + 15, ﬁnd the marginal cost function and com-
x (x) by 1 will decrease the average cost. (b) Show that
t
he
ease
t.
w
ag
v
er
(b)
) ′
will
a
Sho
decr
t
e
1
b
cos
tion
tion
tion C (100)
that
tion (x) by 1 will decrease the average cost. (b) Show that
y < C(100) and show that increasing the produc-
( (x) by 1 will decrease the average cost. (b) Show that
=
actual
t
h
pare the marginal cost at x = 50 with the actual cost of man-
pare + he 90x + 15, ﬁnd the marginal cost function and com- C (1000) > C(1000) and show that increasing the produc-(1000) > C(1000) and show that increasing the produc-
cos
man-
wit
cos
of
t
at
50
he
x
t the marginal cost at x = 50 with the actual cost of man-
marginal
t
par the marginal cost at x = 50 with the actual cost of man-
20x
pare 2
e the marginal cost at x = 50 with the actual cost of man-
pare
′
tion (x) by 1 will decrease the average cost. (b) Show that
′ ′ ′ ′ C (1000) > C(1000) and show that increasing the produc-
C (1000) > C(1000) and show that increasing the produc-
it
he
h
ufacturing the 50th item. C C (1000) > C(1000) and show that increasing the produc-
em.
ufacturing marginal
pare
ing the 50th item.
50t cost at
ufacturing the 50th item. x = 50 with the actual cost of man-
ufactur the
t the 50th item.
ufacturing
he
incr
tion
v
b
a
1
v
cos
v-
tion (x) by 1 will increase the average cost. (c) Prove that av-
(c)
ease
o
er
t.
′
y
will
a
Pr
e
x (x) by 1 will increase the average cost. (c) Prove that av-
e
hat
t
ag
t
tion (1000)
) > C(1000) and show that increasing the produc-
tion (x) by 1 will increase the average cost. (c) Prove that av-
C
( (x) by 1 will increase the average cost. (c) Prove that av-
ufacturing the 50th item.
2. If the cost of manufacturing x items is C(x) =
2 2 2 2 2
2. If the cost of manufacturing x items is C(x) = x + 14x ++ 14x + + + + tion ′ ′ ′ ′ ′ C (x) = C(x).
4
4 4 4 4 x + 14x
2. If the cost of manufacturing x items is C(x)
2. If the cost of manufacturing x items is C(x) = x x + 14x
2. If the cost of manufacturing x items is C(x) = = x + 14x
erage cost is minimized at the x-value where
tion (x) by 1 will increase the average cost.
erage cost is minimized at the x-value where (c) Prove that
erage cost is minimized at the x-value where C (x) = C(x).(x) = C(x). av-
erage cost is minimized at the x-value where C C (x) = C(x).
erage cost is minimized at the x-value where C (x) = C(x).
and
he
t
t
ﬁnd
+ + 35, ﬁnd the marginal cost function and compare the
com
cos
he
,
60x + 35, ﬁnd the marginal cost function and compare the +
60x
function
t
marginal
par
35 cost of manufacturing x items is C(x) = x + 14x
60 + 35, ﬁnd the marginal cost function and compare the
4
60x the
2. If x + 35, ﬁnd the marginal cost function and compare the erage cost is minimized at the x-value where C (x) = C(x).
e 2
60x
′
v
he
he
fr
R
om
(
12.
12. Let
Let R(x) be the revenue and C(x) be the cost from man-
r
)
e
x
(
cos
t
enue
C
be
x
t
t
be
and
)
t
cos
he
t
wit
actur
actual
marginal cost
h
50
=
manuf
t at x = 50 with the actual cost of manufactur-
of
marginal cost at x = 50 with the actual cost of manufactur- - 12. Let R(x) be the revenue and C(x) be the cost from man-man-
12.
12 Let R(x) be the revenue and C(x) be the cost from man-
x
. Let R(x) be the revenue and C(x) be the cost from man-
marginal cost
marginal cost at x = 50 with the actual cost of manufactur-
cos at x = 50 with the actual cost of manufactur-
marginal ﬁnd
at the marginal cost function and compare the
60x + 35,
=
−
C
deﬁned
(
ufactur
x
ufacturing x items. Proﬁt is deﬁned as P(x) = R(x) − C(x).
as
Proﬁt
ufacturing x items. Proﬁt is deﬁned as P(x) = R(x) − C(x).).
P
ems.
R
)
x
(
x
is
)
(
em.
he 50th item.
ing
it
ing the 50t cost at 12. Let R(x) ing x items. Proﬁt is deﬁned as P(x) = R(x) − C(x).
h
x x items. Proﬁt is deﬁned as P(x) = R(x) − C(x).
ufacturing be
ufacturing
it the revenue and C(x) be the cost from man-
ing the 50th item.
t the 50th item.
ing
marginal
ing the 50th item. x = 50 with the actual cost of manufactur-
pr
(a) Show that at the value of x that maximizes proﬁt,oﬁt,
alue
maximizes
(a)
of
hat
at
he
t
Show that at the value of x that maximizes proﬁt,
hat
v
x
t
(a) Show that at the value of x that maximizes proﬁt,
ufacturing
(a) Show
t x items. Proﬁt is deﬁned as P(x) = R(x) − C(x).
ing the 50th item.
3. If the cost of manufacturing x items is C(x) = x + 21x ++ 21x + + + + (a) Sho w that at the value of x that maximizes proﬁt,
3. If the cost of manufacturing x items is C(x) =
3. If the cost of manufacturing x items is C(x)
2 2 2 2 2
3 3 3 3 x + 21x
3
3. If the cost of manufacturing x items is C(x) = x x + 21x
3. If the cost of manufacturing x items is C(x) = = x + 21x
eq
marginal
marginal revenue equals marginal cost. (b) Find the max-max-
(b)
he
t
cos
e revenue equals marginal cost. (b) Find the max-
ind
t.
uals
enue
F
marginal revenue equals marginal cost. (b) Find the max-
marginal
v that at the value of x that maximizes proﬁt,
(a) Show
marginal
marginal
ﬁnd
t
par
t
he
com
function
110x
he
+ + 20, ﬁnd the marginal cost function and compare the
and
cos
,
t
110x + 20, ﬁnd the marginal cost function and compare the +
e 2
3
3. If x + 20, ﬁnd the marginal cost function and compare the marginal r revenue equals marginal cost. (b) Find the max-
110x the cost
20 of manufacturing x items is C(x) = x + 21x
110x
110 + 20, ﬁnd the marginal cost function and compare the
imum proﬁt if R(x) = 10x − 0.001x dirhams and C(x) = 2x + +
2 2 2 2 2
imum proﬁt if R(x) = 10x − 0.001x dollars and C(x) = 2x 2x
marginal revenue equals marginal
imum proﬁt if R(x) = 10x − 0.001x dollars and C(x) = max-
imum proﬁt if R(x) = 10x − 0.001x cost. (b) Find the
=
x
t
wit
he
t
cos
1
of
00
marginal cost
actual
h
t at x = 100 with the actual cost of manufactur-
manufactur
marginal cost at x = 100 with the actual cost of manufactur- - imum proﬁt if R(x) = 10x − 0.001x dollars and C(x) = 2x +dollars and C(x) = 2x + +
110x +
cos at x = 100 with the actual cost of manufactur-
marginal 20, ﬁnd
marginal cost at x = 100 with the actual cost of manufactur-
marginal cost
at the marginal cost function and compare the
dollar
s.
5000 AED.
imum
5000 dollars. if
5000 proﬁt
5000 dirhams.R(x) = 10x − 0.001x dollars and C(x) = 2x +
5000 AED.
he 100th item.
ing
ing the 1 cost at x marginal cos t function and com par 2 2 2 2 2 t he In In exercises 1 3–1 6, ﬁnd (a) t he elas ticit 2 y of demand and (b) t he
em.
it
00t
h
marginal
t the 100th item.
ing
ing the 100th item. = 100 with the actual cost of manufactur-
ing the 100th item.
5000 AED.
ing the 100th item.
4. If the cost of manufacturing x items is C(x) = x + 11x ++ 11x + + + +
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education 4. If x + 10, ﬁnd the marginal cost function and compare the - range of of prices for which the demand is elastic (E < −1).
4. If the cost of manufacturing x items is C(x) =
4. If the cost of manufacturing x items is C(x)
3
3 3 3 3 x + 11x
4. If the cost of manufacturing x items is C(x) = x x + 11x
4. If the cost of manufacturing x items is C(x) = = x + 11x
In exercises 13–16, ﬁnd (a) the elasticity of demand and (b) the
ercises 13–16, ﬁnd (a) the elasticity of demand and (b) the
In In exercises 13–16, ﬁnd (a) the elasticity of demand and (b) the
ex exercises 13–16, ﬁnd (a) the elasticity of demand and (b) the
,
ﬁnd
t
he
40x
+ + 10, ﬁnd the marginal cost function and compare the
e 2
40x + 10, ﬁnd the marginal cost function and compare the +
3
40x
40 + 10, ﬁnd the marginal cost function and compare the
40x the
10 cost of manufacturing x items is C(x) = x + 11x
−
(
1).
E
is
f
or
which
ices
<
elas
t
demand
tic
he
r range of prices for which the demand is elastic (E < −1).
pr 13–16, ﬁnd (a) the elasticity of demand and
range of prices for which the demand is elastic (E < −1). (b) the
In exercises
ange of prices for which the demand is elastic (E < −1).
range
=
t
00
1
cos
x
he
actual
of
t
manufactur
marginal cost at x = 100 with the actual cost of manufactur-
wit
h
marginal cost
t at x = 100 with the actual cost of manufactur-
marginal cost at x = 100 with the actual cost of manufactur-
40x + 10,
marginal ﬁnd
at the marginal cost function and compare the
marginal cost
cos at x = 100 with the actual cost of manufactur-
range of prices for which the demand is elastic (E < −1).
13. f(p) = 200(30 − − − −
14. f(p) = 200(20 − − − −
em.
it
ing
ing the
14. f(p) = 200(20
14. f(p) = 200(20 − p) p) p) p) p)
00t
h
he 100th item.
13. f(p) = 200(30 − p) p) p) p) p)
13. f(p) = 200(30
ing the 100th item.
1 cost at x
t the 100th item.
14. f(p) = 200(20
13. f(p) = 200(30
13. f(p) = 200(30
14. f(p) = 200(20
marginal
ing
ing the 100th item. = 100 with the actual cost of manufactur-
ing the 100th item.
13. f(p) = 200(30 − p)
14. f(p) = 200(20 − p)
5. Suppose the cost of manufacturing x items is C(x) = = =
−
5. Suppose the cost of manufacturing x items is C(x) = x − − − −
3 3 3 3 x
3
15. f(p) = 100p(20 − − − −
5. Suppose the cost of manufacturing x items is C(x)
16. f(p) = 60p(10 − − − −
15. f(p) = 100p(20
16. f(p) = 60p(10
16. f(p) = 60p(10 − p) p) p) p) p)
15. f(p) = 100p(20 − p) p) p) p) p)
5. Suppose the cost of manufacturing x items is C(x) = x x x
15. f(p) = 100p(20
5. Suppose the cost of manufacturing x items is C(x)
16. f(p) = 60p(10
16. f(p) = 60p(10
15. f(p) = 100p(20
............................................................
............................................................
............................................................
30x
30x + 300x + 100 dollars. Find the inﬂection point and
2 2 2 2 2
30x + 300x + 100 dirhams. Find the inﬂection point and −
3
5. Suppose the cost of manufacturing x items is C(x)
30x + 300x + 100 dollars. Find the inﬂection point and+ 300x + 100 dollars. Find the inﬂection point and and
............................................................
............................................................
16. f(p) = 60p(10 − p)
30x + 300x + 100 dollars. Find the inﬂection point = x
15. f(p) = 100p(20 − p)
30x + 300x + 100 dollars. Find the inﬂection point and
............................................................
2
313

313
313

Component: ADV_MATH
Program: UAE Component: ADV_MATH
Program: UAE
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