PXL
B. - Marginal Analysis
- analyzing cost, revenue and profit function derivatives
Total Cost : (variable costs) (quantity produced)
Totai Revenue : (price per unit) (quantity sold)
Total Profit : Total Revenue - Total Cost
Recall:
+
(fixed costs)
Marginal Cost, Revenue, and Profit
Marginal
Cost
rate of change of total
Onemoreitem) '^--r-ri'
cost
derivative of total cost- (the approximate cost
&.-
of
eva
: IlLe qlchange of total revenue :
appt.ox imut e i ncome poffiem
".-/\-/ry\F,\
?\i'*/-1A.r--^
Marginal Revenue
derivative of total revenue {the
1
Profit rate of change of total profit
onernoreitem) -.
^--*€
Marginal
4.2
- Derivatives
A. - Derivative Rules
L
derivative of total profit (the approximate profitfrom
)&
of Products and Quotients (and applications)
Lvvlfiortaat *o Wrcwlor5t {hest
J'\J \eS
Product Rule
y
: _f(x): s(x).h{*)
y'
:f'(i :
g,(x).lt(x) + h,{x).g(x)
OR
y': f'{x):
g(x) .h'(*) + h(x). g'(x)
(since multiplication is commutative)
II.
Quotient Rule
),:.f(x):9
/ r,
b(x)
y':.f'(x)
r r\-''' -
b(x)'t'(x-)
- !!x)'b'(x)
[U*)Y
(don't forget that subtraction is not commutative,
so you cannot switch the order in the numerator)
PgI
4.1- Simple Derivative Rules and Applications
A. -
Simple Derivative Rules
derivative formula
I.
g=
Constant
y:f(x):
constant
5t=
tl
o
5=rco
b'=
y': ft(v) :
a
(Recall: derivative is slope and slope of a horizontal line (y - const) is 0.)
(Note: rule of thumb for alphabet: usually the beginning of the alphabet denotes
constants and the end of the alphabet denotes variables.)
II. Porver Rule
y:f (x) - xft, (n eF')
$=? ?c5''
I
m.
s^:ce
?= L*5
nxn-r
d
Sum and Difference Rule
,*
Lx'-5
.)
bxponentral
(a)
and Log
y:
rI
', b'= lo)c'
y:f(x):s(x) +h{x)
ut = Tz' 5Z'
l=
V.
ua X,-'3
tJ
-UI
ut= -31
Constant Times a Function Rule
y : -f {x) : k' g(x}
ry.
y': f'(x) :
ex
(b) y:ttrtx, x>0
y'
q=
u
:f
'(x)
:
klg'(x)7
--L = -2>t3
Y?
-4-- b
*r=.6)c v
y,: f,(x) : g,(x) *
1= {lF
f,= r't,
h,(x)
+q* -#
vlt - ;'/72
*'-i'/'
irn;;;;,=
y': -f'(x) :
y' :
-f'(x)
n'
:_
1
x
Pi
(u)
f(x)
= ex + 2htx
,l'6*;= e+2.J- e+
=
3.
x
*
Al f@) = €''
x2
6
f .'-t*'
$t"1 =
$'t*;= f
rat
r@)
e"-i^
e**2
*ffi
fttr-le
rq f@)="4 + ln.E_
f 'c*)= o + h.
S'u)'2x
= -/-2'
+t{;) z Gr)t -
ir:+
= (zr)t - t*t * a ir,.y -3r,,at
8'67= slzrfe) - t*'(,) *2. -o
*
9'(*)= lo C2")t - g**' * *s160 l-e**'*g
Se
,1
= eT * -9*r x'1. = e4
t*
*!
rn x
Find the derivative
for the fotowing functions
and simpri$z:
rl (*^ rth,q tna pcod..et ,r,rte)
St")= 5 x'(s x") + (f"+z\ro r)
$'G; = 15 x{ + to ,g{ +2o >c
i'C^)' 25 x4 r zo x-
Al -f(x)
rul
= 5x2 €*
f(x) = x3 h1x1
.9'(x)= x3 . UC") + hG).
,.9,(*)=
3t*^ ,- -u- *]
gf
xzf xW{*) + slu(x)]
n3
B.
tf C(x)
- Marginal Ayerage Cost, Revenue, and Profit
is the cost of producing
divided by rhe number of items:
Then Marginat Average
Marginal Average
Profit
:
items, then the average cost
e(") :C(q)
x
:
R'(x)
P'(x)
4.3
+
J}g(;r)l clx
(x)
is the cost of all the items
q Ck)
clx x
and
and
-4clxn(*) - +R(')
clx x
p@)
p@)
4
4
- clx x
- dx
- The (all important) Chain Rule
A funcfion ir is a composite of functions/and g if h{x)
rhen
f
.
: e'@):4 e (r) clx
cost
Marginat Average Revenue
x
-
JTS@)1.
-f'lg@)l.s'(")
General Power Rule:
function
(a)
y-
derivative
f(x): lr{i}o
(Notice that we now have a function
toaporverandnotjust
y'
:.f'(x)
- n.lo@1"-1 .u'(x)
raised q^= [rne>S)Ora-,
. lyleSJ-,
n (nteSS)'
+ =
x raisedtoapower,
EXAMPLES:
*"?;:
;, ;, : ;1";;*:':"*'*
Tl' i',l- la'i' 4 rr-? i'/t no
*z{3
*)=
Ji'c
s'(")=
-#-fu
P5b
4. Find the value(s) of x where -f (")
:
has horizontal tangent lines.
*,
$'(x) =. (>fr rX rl:_(x)_(ex)
(**'f
Lo
2x/-ll
(xt+ t)-
,f 'tx1 =
ff'c*;
=
t-N:
(x'+,)' =o
l'
o
a>t-
5. The price equation ltrr the production of television sets is given by x : 9,000 - 3Ap and the cost
equation is C(x) - 150,000 + 30x, where x is the number of sets that can be s_old at a price Sp
per set and C(x) is the total cost (in dollars) of producing x sets. So LVs $o t )ai
(a) Find the Revenue
X= 9rooo - 7af
3of = lr6oo-;r-
and Profit functions.
K---Y.f
fu r
i--
_t-- ?C)
1e(3oo- 3o
zoa-
6X
p: Rr*c = io ox'-t;if-(rso)ooo t- 3ox)
- lsogoo* 3ox
P = 3ocx-- *^*
30
(b) Find the marginal cost, revenue
and
profit functions.
l=lSo,Doot3t>X
Ntc= 3o
fu=3oox' -L*
5D
Al"R, = 3oo - E*
J-o,trr- 6 Ct ;a-.ttig- ',-€ c'os€
aS ne-Vesu*- Pc,t tc€t(
Ptl\ *s^^nP P =.2-.lo i< -}. x-- l5o )oaO
.-*^- tn"",Yal P'ot"V;t""^
A(F- 2lO -bx
fu :"
'\c,a(-
et
chna"lol-
i
ry5
td f@) = (5 - *2)$3 +2x2)
J't^)= (5-x')G*'+ 4x) *(*, +z*z)pz*)
tlt
&'Cx) = t5:ca + zox-7*{- 4C -1*!- iff$'(*)= - 5xq * 8 x3 ;;;-xq;";--st -f@)=ry
x-
$,(x), =
zl.
U(x)
- ht)c) - Zy
Lt)"zx't't{.0
lz' h'(r\ -
I'C*1 =
x4
(g f(x)
r.
I'J e)-- 2 t,G)
=
Nt
a?li
- *2 -3
$C*l -
t'(
Ixtx'ts-3
x) =
9'c*)
=
(t=DQt\;-\"'/T,zi
(x'- ').14te
2x!/s -ui|" - lzr-
(n'-l):
2
-l-ol!-:lz
-ailu .' ,l/t =
rol*i ;6I-"
$'6*;= :- roxlo
Ytl3
(x'- a1
3. Find the equation ofthe line tangent to the graph of
Stx;
f(x) :
(7
-
=
3x)(x+ 2xz)
at
w
x:2.
' C'l - 3xXx -v 2x")
9tt)= Cl- 6 ) C2+ I ) = l(ro)' lo Cz. )lc>)
S'(x)' ('l- 3p)( t *ax) * (x r 2Y-'X-g)
oY3
$'(*) = T + 28 x-'3Y -lLl( - 3 x- ,.$'cr)
11
+ tb2c -
"
rl+
=
5',(z)
b*
2o-2-4zt =lra
3 L;--z)
-lo= 3x -b
- lo
=
ffi3
6. Amachine
shop rnakes
drillbits with
C(t) :
1000+ 25x - A.7x2.
a. Find the average cost function for drill bits.
Ac=c
Y
=v(x)=w-qr
b. Find the average cost of producing
'
internret.
a
+zs- ]Y
to
drill bit at a production level of l0 drill bits per day and
loa +2s+z.5-*(,o
)=
ry
7 = Ac (r o)=$,2't At & prod.ctrtrarn [autL (toi
AC(ro)=
the
c. Estimate the average
e
Et
ffioil
0ruu o3oco>t o$ ptoduc.rrS
cost per bit at a production level
* ZS - Sv
= -tooof
C
, -+o
of
11
r
ro b'ty'do3
bct is $r24.
drill bits per day and interprg!.
o>b is \
oytc
OrO .ro Palt hdc
ynoFg
Li*
+ IZ{- \o.\O=
$ tt3.qo
a,,lo);--ffifto--- : - ro-,[=-!D.lo qu
(\rro^naa
e a9 u cost]-Ait 6g t I b'Ydy
7. Find the derivative for the following tunctions and simpliff:
tut
f.,\ :
.y2
t tz.'{s*')It -tr, - -r , z {/., ^
*(T .il' (2v)
lt(r*'*qr'tn-=)]
4- 2* (',*'.D'l.
*':')-':
$'c4 = s;i)
+2* (af*rYs,( x)=
E,f +r =
S'(*) =
y'
"?'ti;.''i"'
y (z *u.,)'o [ "*FLt:rii la
0-oox$'ncomYnangocJ*t+*t'teSaaro-\\es-texPola'nt
J
j , {*) =
9'(x) = /,-(zxt+ l.:f
s,;;
+ 2(2x"+rU
=.,:.!i:{ffi4r+2)
5,Cx)_
ffi
n7
(c) Evaluate marginal cost, revenue
and
profit
at
x:2000
and interpret.
11(C=3(>
t"\R= 3oo -fr*
MP= 24o ."* *
lvtc (eooo) =
3o
4ppuur.Cr:st 9o^ prrodrrer.mg*he aoo1"J7,1
(a1|nox Co3t $on mcr.tlb I rnalC o,phen |aooo ha't'c- bccrr tttodt)
(zooo)= $ tGb'brl
3oofu
'
e-?FhDx u,.uu,"-*g Sconn' ene s'l<' 4
/nR (zcno)
+he 2ootstn
$
,vtPlzooo) = Lrto -k qzr,oo)= | 3L.67
4pp-rrox- PnoSC 6ao-*hc5al.-
.16
,r
Loof':-TV
(d) Find the exact profit of the 200th set. Use marginal profit to estimate the profit of the 200th
vz - t sc>,ooo
Y,= L-to
- PC,
P
qq)
x -h
q
e rac-k eost= V, qzocD - Y, ( r X) = $ eS t''lo
$ 28a,13
Nte [r qq') Zttro t li)
gnol
'
fs(
set.
=
(e) Sketch Cost and Revenue together and interpret.
utettu.o
tn1{,N
Cott
l5o
,-lsoo
1 ooo
Pfod*rcc oryd-
a, b MLe bneqt- €-u€n pornts
e= (boot lbgooo)
b = ( rt foo
r Sl fooo)
srrt L6ol t9 "l{9 ilTVtS {i, llnare ofnos;t
c) !{*;. (x'*,;aca 'z
P3,o
,"
;f
S,c4=(ranj fsqa-z*- "!+G-2Cft qr*,f ( r"fi
$'(*)= -to (x?rf( 6-z*: + t>c (x'r,)= 6a nd
9'c') = -Z (x'*,J G- z"J fS qx"r,) - 4 x(6 -." )]
&'(x7
=.
$1*;=
-zQt*,fto -z"J (s r" +v-z4x rsx')
^z|f *'t(r -z$(w'-z4x+
r)
131
-
&c'7
=
gt
..f,=
Tr=
(, - ."J [s (* + s;q
5(r-axf(x+s
g, _
c:
!-vx)
=
8'= -a
S3
(,rf:',-".tJ [qc, -.J e3t
(t -
+
3x)
tz(x+5)I f '- 3"J*
(t -axla
/(x) - (*' *D4 G-2i3
on
t
A
S/= Bq (sra._r' ) - z-14
l-5
-9'=
(c)
,!-tt T+ (x+s)
(r + 5)s
(1- 3x)a
at f@)
9'(*)
=
(*+sf
(:f
Sf (J-t?:f z.'*6?
. ,a .fS
3x)
( ,-
rruerct
P't<
(d\
f(x) (*'-4)'
f ,(x)=
aY - (*1v)=- e"
x'-1 )
en(x1 4)[t1 4 - 4;J
(
$tx;
=
(*-,{)'/
3'0")=
(4">$4)
t(x--'/Y-?).
(x-- 4)"