DIFFERENTIAL CALCULUS REFRESHER SET 1. A water water tank has has the form form of a frustum frustum of a cone cone with with iameter iameters s of ! m at the "ottom "ottom # $ m at the to% an a hei&ht of ! m. 'ith a constant rate of inf(ow of water it was o"ser)e that when water was !* cm ee% # water was risin& at the rate of 1+mm,min. Fin the rate of f(ow in L,mina. 1.+ ". 1./+ c. !.1+ . !.!+ !. A "o0 is %roecte %roecte )ertica(( )ertica((0 0 u%war u%war such that is hei&ht hei&ht from the the &roun &roun in feet is h 2 +*t 3 14.1t5! where t 2 time in f(i&ht in secons. After how man0 secons wi(( the )e(ocit0 "e 1! f%sa. 1.16 ". *.66 c. *.+ . 1.14 $. Fin Fin the the eri eri)at )ati) i)e e of $5$ $5$7. 7. a. 3(n $7 ". (n $7
c. $(n$ $5$7 . 8$(n$ $5$7
9. Fin the eri) eri)ati) ati)e e of ! cos :! ; 75$< a. 4 sin :! ; 75$< ". 84 sin :! ; 75$<
c. 8475! sin :! ; 75$< . 475! sin :! ; 75$<
+. A ferris ferris whee( has has a raius of 1* m. Its center center is 1! m a"o)e the the &roun. &roun. 'hen a %assen&e %assen&er r is 1 m a"o)e the &roun he is mo)in& )ertica((0 at the rate of 1.61 m,s. 'hat is the s%ee of the rotation of the whee( in r%ma. + ". c. 9 . ! 4. Fin the (imit of sin ϕ, ϕ as a%%roaches a%%roaches *=. a. > ". 1 c. *
. 81
. 'hat is the s(o%e s(o%e of of the cur)e cur)e at at 0 2 e597 e597 at the %oint %oint when 7 2 *a. 9 ". * c. 1 . 89 6. 'hat is is the cur)atu cur)ature re of cur)e cur)e 05! 2 147 147 at the %oint %oint :9#6<:9#6
. 8*.*49
/. The sum of of the two num"ers is is S. 'hat is the ma7imum ma7imum )a(ue )a(ue of the sum of their their cu"esa. $S,9 ". S59,$ c.S5$,9 . 9S,$ 1*. Determine the the s(o%e of the cur)e 0 2 :18!7<5 :18!7<5!,7 !,7 at :1#1<. a. 8! ".! c. 8$
. $
SITUATI?N 1@ Let 0 2 :1 ; $7< . 11. Determine the the )a(ue of 7 on the &ra%h where the norma( norma( (ine is %ara((e( %ara((e( to the (ine 97 ; $0 2 1 a. 81 ". $ c. * . 1 1!. Determine the the )a(ue of 0 on the &ra%h where the norma( norma( (ine is %ara(( %ara((e( e( to the (ine 97 ; $0 2 1 a. ! ". 81,$ c. 8! . 81 1$. Fin the a"so(ute a"so(ute e7trema e7trema on 0 2 e5sin 7 on on B8 # a. e ". 3e c.
. 3
19. Determine the s(o%e of 0 2 7 ; 1,7 at 7 2 !. a. *.4+ ". *.61 c. *.+
. *.9*
1+. The fo((owin& fo((owin& are transcenenta( transcenenta( functions functions ECET@ a. E7%onentia( ". H0%er"o(ic c. Tri&onometric
. A(&e"raic
14. If f:7#0< 2 :7 3 0<,:7 ; 0< # fin Gf,G7 at :!#1< :!#1< usin& %artia( ifferentiat ifferentiation. ion. a. + ". 8! c. * . $ SITUATI?N !@ Com%ute the )a(ue of 7 usin& Newtons A%%ro7imation. 1. 75$ 3 !75! 3 !7 3 2 * a. 9.9/ ". 1.*99
c. $./61
. $.!46
16. +sin +sin 7 2 97 a. *.4+9
c. 1.1$1
. 1.$!!
". 1.46
1/. Fin the euation of the norma( norma( (ine to the &ra%h 0 2 7 (n 7 that is %er%enicu(ar to the (ine 7 3 0 ; 2 *. a. 7 ; 0 2 1 ". 7 3 0 2 1 c. 7 ; 0 2 81 . 7 3 0 2 81
SITUATI?N $@ A suare of sie 7 cm has area A:7< cm5!. !*. Fin the rate of chan&e of A as 7 chan&es from 9 to 9.4. a. 4.9 ". 6.4 c. .+
. 4.*
!1. Fin the rate of chan&e of A as 7 chan&es from 9 to 9.*+. a. 6.$! ". 6.!+ c. 6.1*
. 6.*+
!!. Fin the instantaneous rate of chan&e when 7 2 9. a. 1! ". 1* c. 6
. 4
!$. A car"oar %oster containin& $! cm5! of %rinte re&ion is to ha)e a mar&in of ! cm at the to% an "ottom an 1.$$$ cm at the sies. Fin the (en&th of the sma((est %iece of car"oar that can "e use to make the %oster. a. 1+. cm ". 1!.9 c. 1*./ . 6.6 !9. The %erimeter of a trian&(e is 1+./4 cm. Fin the ma7imum area of the trian&(e in s cm. a. 1.1 ". 1!.$ c. 1!.4 . 19.6 !+. Fin the %oint of inf(ection of the cur)e 0 2 75$ 3 475! 3 7 ; 1!. a. :!#4< ". :8!#4< c. :!#84< . :8!#84< !4. A c0(inrica( steam "oi(er is to ha)e a )o(ume of 1$9* cu ft. The cost of the meta( sheets to make "oi(er shou( "e a minimum. 'hat is the reuire "ase iameter in feeta. 1$.49 ". 16.94 c. 1!.$1 . 11./+ SITUATI?N 9@ A c(ose c0(inrica( tank has a ca%acit0 of +4.+4 cu m. !. Determine the reuire raius of the tank. a. 6. ". .1!
c. 9.+1
. 4.+9
!6. Determine the reuire hei&ht of the tank. a. /.*! ". 9./
c. 11.49
. 1*.6
!/. Determine the surface area at a minimum. a. $/4.1! ". 96.49
c. !1/./
. $6$.9*
$*. If f:7< 2 7:!57< # then f:7< is eua( to@ a. !57 :(n ! ; 1< ". !57 :! (n ! ; 7<
c. !57 :7 (n ! ; 1< . !57 :!7 (n 7 ; 1<
$1. A %oint mo)es in the %(ane accorin& to euations 7 2 t5$ ; !t an 0 2 !t5$ 3 4t. Fin 0,7 at t 2 *. a. 4 ". 84 c. $ . 8$ $!. The time rate of chan&e of the raius is constant for a &i)en s%here. The time rate of chan&e of the )o(ume is@ a. increasin& ". ecreasin& c. increasin& when r J 1 an ecreasin& when r K 1 . ecreasin& when r J 1 an increasin& when r K 1 $$. 'hat is the first eri)ati)e with res%ect to 7 of the function &:7< 2 9/ a. * ". 9,/ c. 9 . 1! SITUATI?N +@ The sum of the two nonne&ati)e num"ers is $4. $9. Fin the ifference of their suare roots is to "e as (ar&e as %ossi"(e. a. 16 ". * c. $4 . !+ $+. Fin the sum of their suare roots is to "e as (ar&e as %ossi"(e. a. 16 ". * c. $4
. !+
SITUATI?N 4@ Su%%ose that the cost of %roucin& 7 washin& machines is c:7< 2 !*** ; 1**7 3 *.175!. $4. Fin the a)era&e cost %er washin& machine of %roucin& the first 1** washin& machines. a. 1!+ ". 11* c. 1+* . 1+ $. Fin the mar&ina( cost when 1** washin& machines are %rouce. a. 1** ". /* c. 6*
. *
$6. Fin the cost of %roucin& the 1*1st washin& machine. a. 61.!* ". 6*.4* c. /./*
. +.4*
$/. Fin the function f:7< whose eri)ati)es is sin 7 an whose &ra%h %asses throu&h the %oint :*#!<. a. f:7< 2 cos 7 ; $ c. f:7< 2 cos 7 3 $ c. f:7< 2 8cos 7 3 $ . f:7< 2 8cos 7 ; $ 9*. 'hich of the fo((owin& is a so(ution of the euation 759 ; 7 3 $ 2 * usin& Newtons a%%ro7imation. a. 1.64+/ ". 81.64+/ c. 1.49+! . 81.49+ SITUATI?N @ i)en the cur)e 0 2 75$ 3 97 ;1# 91. Fin the euation of the (ine %er%enicu(ar to that cur)e at :!#1< a. 60 2 7 ; 1* ". 60 2 87 ; 1* c. 60 2 7 3 1* . 60 2 87 8 1* 9!. At what %oint on the cur)e ha)e the sma((est s(o%ea. :*#1< ". :1#*< c. :89#*<
. :*#9<
9$. Fin the euation of the tan&ent at the %oints where the s(o%e of the cur)e is 6. a. 0 2 867 ;1 ". 0 2 867 81 c. 0 2 67 ;1 . 0 2 67 81 99. Fin the eri)ati)e of 0 2 (o&Ma 97. a. 0 2 (o& ae , 7 ". 0 2 cos e,7
c. 0 2 sin e,7
. tan e,7
9+. A s%herica( "a((oon is inf(ate with &as at the rate of !* cu m %er min. How fast is the raius of the "a((oon chan&in& at the instant if the raius is ! cma. *.4$$ ". *.$/6 c. *.+1 . *.$19 94. Fin the 0 coorinate of the center of cur)ature of the %ara"o(a at :!#!<. a. ". 8 c. 84 . 4 9. Fin 0,7 usin& im%(icit ifferentiation at :*#8!< if 7 $ 3 70 ; 0 ! 2 9. a. *.44 ". *.6 c. 8*.$$$ . *.+ 96. An RT train 4 m a"o)e the &roun crosses OuePon A)enue at a s%ee of / m,s# at the instant that a car a%%roachin& at a s%ee of 9 m,s is 1! m u% the street. Fin the rate of the RT train an the car are se%aratin& one secon (ater. a. /.6+ m,s ". 9.1 c. $.49 . 9.16 9/. 'hat is the a((owa"(e error in measurin& the e&e of the cu"e that is intene to ho( 6 cu m if the error of the com%ute )o(ume is not to e7cee *.*$ cu ma. *.**! ". *.**$ c. *.**!+ . *.**1 +*. A man 1.6 m ta(( wa(ks awa0 from a (am% %ost 9 m hi&h at a s%ee of 1.+ m,s. How fast in m,s oes his shaow (en&then a. 1.1 ". 1.!$ c. 1.49 . 1.*