APPLICATIONS OF DERIVATIVES IN BUSINESS 1. Let y = C(x), C(x), where where y repree!t repree!t the "#t t# t# $%!&'%"t&re $%!&'%"t&re x te$. Th&, C(x) e"r*e % the "#t '&!"t#!. Ex%$p+e The t#t%+ "#t ! th#&%! #' Pe# t# $%!&'%"t&re x e+e"tr" -e!er%t#r -e! *y
C(x) = -x3 + 15x2 +1000 /. Aer%-e Aer%-e "#t "#t per te$, AC(x), AC(x), eter$!e eter$!e *y !!- the the t#t%+ "#t *y the !&$*er #' te$.
AC(x) = C(x)/ x I! the -e! -e! ex%$p+e where C(x) = 0x 2 13x/ 21444, AC(x) = -x2 + 15x + 1000/x . 5%r-! 5%r-!%+ %+ C#t C#t (5C) (5C) -e the r%te r%te #' "h%!"h%!-e e ! the "#t "#t #' pr#& pr#&"! "!- #!e $#re te$ %'ter %'ter x te$ h%e %+re%y %+re%y *ee! pr#&"e. pr#&"e. It the %ppr#x$%te %ppr#x$%te "h%!-e ! the "#t "#t re&+t!re&+t!- 'r#$ #!e %t#!%+ %t#!%+ &!t #' #&tp&t. #&tp&t. It the 6rt er%te #' the #' the "#t '&!"t#!.
MC = C’(x) Ex%$p+e 1 The "#t t# pr#&"e pr#&"e x &!t #' %! te$ -e! *y *y C(x) = 7x / 8 9x 2 :. %. F! the %er%-e %er%-e "#t "#t per &!t t# pr# pr#&"e &"e 14 &!t &!t,, /4 &!t &!t AC(x) = C(x);x = 7x 8 9 2 :;x C(14) = 7(14) 8 9 2 :;14 = :<.: C(/4) = 7(/4) 8 9 2 :;/4 = 1<.9 *. F! the the 1t er%te #' the %er%-e "#t '&!"t#! AC>(x) = 7 8 :;x/ = (7x/ 8 :);x/ Ex%$p+e / ?e! the "#t "#t '&!"t#! '&!"t#! C(x) = /x / 2 3x 2 1:, 6! the $!$&$ %er%-e "#t. %. S#+e S#+e '#r '#r the the %er%%er%-e e "#t, "#t, AC(x). AC(x). AC(x) = /x 2 3 2 1:;x *. S#+e S#+e '#r AC>(x). AC>(x). AC>(x) AC>(x) = / 8 1:;x/ ". E@&%t E@&%te e AC>(x AC>(x)) t# 4> 4> %! #+ #+e e '#r x. x. / / 8 1:;x =4
(/x/ 8 1:);x / =4 / /(x 8 7) =4 x= x = 0 . O!+y x = !"e t p#te %! !"e C(x) = 9 p#te, x = % $!$%. e. S&*tt&te x = t# AC(x) = /x 2 3 2 1:;x AC() = /() 2 3 2 1:; = 1 There'#re, the $!$&$ %er%-e "#t P17
Ex%$p+e S&pp#e th%t the t#t%+ "#t (in hundreds of Pesos) t# pr#&"e x (in thousand units) #' %! te$ -e! *y the '&!"t#! C(x) = x2 + 100x + 500. F! the $%r-!%+ "#t '#r the '#++#w!- %+&e #' x
a! x = 5 or (5000 units) Cost for x = 5 or (5000 units) is C(x) = x 2 + 100x + 500 C(5) = (5)2 + (100 x 5) + 500 = P1100 or P110"000 (P1100 x 100) A#era$e %ost &er ite' for x = 5 or (5000 units) is AC(x) = x + 100 + 500/x AC(5) = (x5) + 100 + (500/5) = P220 or P22"000 (P220 x 100) Mar$ina %ost MC = C’(x) = x + 100 C’(5) = ( x 5) + 100 = P10 or P1"000 (P10 x 100) There'#re, %'ter 3444 &!t #' %! te$ h%e *ee! pr#&"e, the "#t t# pr#&"e 1444 $#re &!t w++ *e %ppr#x$%te+y 194 (! h&!re #' pe#) #r P19,444.
*! x = 30 or (30000 units) Cost for x = 30 or (30000 units) is C(x) = x2 + 100x + 500 C(30)= (30)2 + (100x30) + 500 = P7100 or P710"000 (P7100 x 100) A#era$e %ost &er ite' for x = 30 or (30000 units) is AC(x) = x + 100 + 500/x AC(30) = (x30) + 100 + (500/30)
= P131!7 or P13"17 (P131!7 x 100) Mar$ina %ost MC = C’(x) = x + 100 C’(30) = ( x 30) + 100 = P30 or P3"000 (P30 x 100) ,herefore" after 30000 units of an ite' ha#e *een &rodu%ed" the %ost to &rodu%e 1000 'ore units i *e a&&roxi'ate. P30 (in hundreds of &esos) or P3"000! 9. he! AC>(x) = 4. The "rt"%+ %+&e (%+&e #' x) $%y ether *e "#!ere % the $!$&$ !&$*er #' &!t re@&re t# %"hee the $!$&$ "#t #r the $%x$&$ !&$*er #' &!t re@&re t# %"hee the $%x$&$ "#t. I' C(x) is ne$ati#e" x is 'axi'u' #aue! f C(x) is &ositi#e" x is a 'ini'u' #aue! Ex%$p+e 9 I! the pre#& ex%$p+e, whe! AC>(x) = 4, x "#$p&te %
AC’(x) = 500/x 2 9 8 344;x/ = 4 (9x/ 8 344);x/ = 4 9x/ 8 344 = 4 there'#re 4 = 9(x / 8 1/3) where x =
√ 125 or 11
is a&&roxi'ate. 11 or (11000 units) S!"e C(x) = is &ositi#e" x = 11 is the 'ini'u' #aue . The $!$&$ "#t eter$!e *y &*tt&t!- the %+&e #' x t# AC(x).
AC(x) = x + 100 + 500/x AC(11) = ( x 11) + 100 + (500/11) = P1!5 or P1"5 (P1!5 x 10) 3. Ree!&e (R) 8 F#r %!y -e! e$%! '&!"t#! y = '(x), the t#t%+ Ree!&e '&!"t#!, 4, the pr#&"t #' x (the !&$*er #' &!t e$%!e) %! . (the pr"e per &!t e$%!e).
4=x.
or
4 = x f(x)
<. 5%r-!%+ Ree!&e '&!"t#! (5R) 8 r%te #' "h%!-e (er%te) #' the t#t%+ Pe# %+e re"ee wth the t#t%+ !&$*er #' &!t #+. It %+# the %ppr#x$%te "h%!-e ! ree!&e th%t re&+t 'r#$ e++!- #!e %t#!%+ &!t. d f ( x ) 2 '(x) 5R = R>= x dx
Ex%$p+e The ree!&e '#r e++!- x &!t #' %! te$ -e! *y R(x) = :44x 2 14x/. F! the 5%r-!%+ Ree!&e whe! %. x = 14, * = /4, " = / R(x) = :44x 2 14x/ R>(x) = :44 2 /4x %. R>(14) = :44 2 /4(14) = 1444 *. R>(/4) = :44 2 /4(/4) = 1/44 ". R>(/) = :44 2 /4(/) = 1994 Ex%$p+e The e$%! '&!"t#! #' % "#$$#ty -e! *y the '&!"t#! y = /44 8 3x where x the @&%!tty e$%! %! y the &!t pr"e. Deter$!e the pr"e %! @&%!tty '#r wh"h ree!&e $%x$&$. De$%! Ree!&e
y = /44 8 3x 4=x. = x (/44 8 3x) = 200x 5x2 R> = /44 8 14x
T# -et the "rt"%+ %+&e;, e@&%te R> t# 4 %! #+e '#r x /44 8 14 x = 4 x = 20 S&*tt&te x t# the e$%! '&!"t#! " . = /44 8 3(/4) = 100
,o deter'ine if x = 20 is a 'ini'a or 'axi'a" $et the 2 nd deri#ati#e of 4! f 4’ is &ositi#e" x = 20 is a 'ini'a! f 4’ is ne$ati#e" x = 20 is a 'axi'a! 4 = -10 and sin%e it is ne$ati#e" x = 20 is a 'axi'a! ,herefore a 'axi'u' re#enue is ex&e%ted for a sae of 20 units at P100 ea%h! Ex%$p+e The pr"e per *%r #' % "h#"#+%te "%!y -e! *y the '&!"t#! y= x 6 5
−
250
#w $%!y *%r $&t *e #+ t# $%x$Ge ree!&eH h%t the $%x$&$ ree!&eH Ree!&e
4=x. =x
6
x
5
250
( −
)
= = <;3 8 /x;/34 = (44 8 /x);/34
T# -et the "rt"%+ %+&e;, e@&%te R> t# 4 %! #+e '#r x (44 8 /x);/34 = 4 44 8 /x = 4 x= 134 S&*tt&te x = 134 t# the ree!&e '&!"t#! " 4 = <(134);3 8 134 /;/34 = 0 ,o deter'ine if x = 150 is a 'ini'a or 'axi'a" $et the 2 nd deri#ati#e of 4! f 4’ is &ositi#e" x = 20 is a 'ini'a! f 4’ is ne$ati#e" x = 20 is a 'axi'a!
4 = 2/150 and sin%e it is &ositi#e" x = 150 is a 'ini'a! ,herefore a 'axi'u' re#enue of P0 is ex&e%ted for a sae of 150 *ars!
. The Pr#6t F&!"t#! the ere!"e #' the ree!&e '&!"t#!, R(x) %! the "#t '&!"t#!, C(x). P(x) = R(x) 8 C(x) 5%x$Ge pr#6t 'a. *e %"hee whe! the 1 t er%te #' the Pr#6t '&!"t#! e@&%te t# 4. Ex%$p+e
A "ert%! te$ "%! *e pr#&"e %t % &!t "#t #' P14. The e$%! '&!"t#! '#r the pr#&"t D = 74 0 .4/x, where D the pr"e ! pe#, %! x the !&$*er #' &!t. %. #w $%!y &!t $&t *e pr#&"e t# $%x$Ge Pr#6tH *. h%t the pr"e th%t -e $%x$&$ pr#6tH ". h%t the $%x$&$ pr#6tH
Cost 6un%tion C(x) = 10x 4e#enue fun%tion 4(x) = x() = x(74 .4/x) = 0x !02x 2 Pro8t fun%tion P(x) = 4(x) C(x) = (0x !02x 2) 10x = 0x !02x 2 Maxi'u' &ro8t 'a. *e a%hie#ed hen 1 st deri#ati#e of P(x) = 0 P(x) = 0x !02x 2 P’(x) = 0 !0x At P’(x) = 0" 0 !0x = 0
x = 2000 ,est for P(x) to %he%9 if x = 2000 is 'axi'u' or 'ini'u'! P’(x) = 0 !0x P(x) = !0! :in%e P(x) is ne$ati#e x = 2000 is 'axi'u'! ,he &ri%e that $i#es 'axi'u' &ro8t is deter'ined *. su*stitutin$ x to the e'and fun%tion = 0 - !02x! = 0 - !02(2000) = P50 ,he 'axi'u' &ro8t is deter'ined *. su*stitutin$ x to the Pro8t fun%tion P(x) = 0x !02x2 P(2000) = 0(2000) - !02(2000)2 = 10"000 - ! 02("000"000) = 10"000 0"000 = P0"000
