[PDF] Application of Derivatives

16223_2DGT_AOD.pdf

239Application of Derivatives2398or9 5 0ner13 an6 oer

SyllbusS Geometrical Meaning of Derivative l

yangent an= (ormal l x//ro+imation an= -rror) l wolleg) yleorem an= Xagrangeg) Mean -al,e yleorem l Ma+ima an= Minima .Xocal an=

Glo0al2 l doint of 3nflection

3f =eal4 a//lication of =erivative in vario,)

=i)ci/line)4 e5g5 in engineering4 )cience4 )ocial )cience an= man8 otler fiel=)5 qe mill learn lom tle =erivative can 0e ,)e= to =etermine rate of clange of 7,antitie)4 to fin= tle e7,ation) of tangent an= normal to a c,rve at a /oint4 to fin= tle ma+ima an= minima /oint) for a f,nction5 qe mill al)o ,)e =erivative to fin= interval) on mlicl a f,nction i) increa)ing or =ecrea)ing an= to fin= a//ro+imate val,e of certain 7,antitie)5 (TopTalPbml)GTmtPtn)od)hTlPymaPyT

Xet ,) con)i=er a f,nction 8 L f.+2 an= + an= 8

are real n,m0er)5 3f tle gra/l of 8 i) /lotte= again)t +5 yle =erivative mea),re) tle )lo/e of tli) gra/l at eacl /oint5 Xet a /oint 1.+ 6 l4 f.+ 6 l22 i) ver8 near to /oint dP+4f.+2G on 8 L f.+25 yle val,e g l i) an a//ro+imation to tle )lo/e of tle tangent mlicl me re7,ire5 xl)o4 it can 0e mritten a) clange in 8 8or mclange in + + 

3f me move 1 clo)er an= clo)er to d4 tle line d1

mill get clo)er an= clo)er to tle tangent at d an= )o4 tle )lo/e of d1 get) clo)er to tle )lo/e tlat me re7,ire5 3f me let 1 go all tle ma8 to to,cl d .i5e5 l L I24 me mo,l= lave tle e+act )lo/e of tle tangent (om4 g f.+ l2 f.+2 l l   ko4 al)o tle )lo/e d1 mill 0e given 08

Application of Derivatives

evision oints f.+ l2 f.+2ml   ',t me re7,ire tle )lo/e at d4 )o let l r I4 tlen in effect4 1 mill a//roacl d an= g l mill a//roacl tle re7,ire= )lo/e4 d,tting tli) togetler4 tle )lo/e of tle tangent at d i) + I =8 f.+ l2 f.+2lim=+ lr 

3t give) tle in)tantaneo,) rate of clange of 8

mitl re)/ect to +5

MmtnTta

yle tangent to a c,rve at a /oint d on it i) =efine= a) tle limiting /o)ition of tle )ecant d1 a) tle /oint 1 a//roacle) tle /oint d4 /rovi=e= tlat ),cl a limiting /o)ition e+i)t)5

MgT)hytmpPb)fDDlombg)ao)MmtnTtby

yle tangent to tle c,rve at d i) a line tlro,gl d mlo)e )lo/e i) tle limit of tle )ecant a) 1 r d from eitler )i=e4 ),cl tlat tle /oint) d .fi+e=2 an= 1 .moving2 coinci=e5

SloDT)od)MmtnTta

Xet 8 L f.+2 0e a contin,o,) c,rve an= let

d.+

R4 8R2 0e tle /oint on it5

ylen4

R R.+ 4 8 2

=8 =+: 9 )8 ( i) tle )lo/e of tangent to tle c,rve 8 L f.+2 at tle /oint d5 vN

2Application of Derivatives

i5e5 d=8 tan=+:  9 )8 ( L klo/e of tangent at d55 mlere4  i) tle angle mlicl tle tangent at d.+

R4 8R2 ma±e) mitl tle /o)itive =irection of

<>a+i) a) )lomn in tle a0ove fig,re5 .2mpDlT)v

3f tle e7,ation of tle tangent to tle c,rve

8 Y L a+A 6 0 at tle /oint .Y4 A2 i) 8 L B+ C D5 ylen4 a5 a L Y4 0 L | 05 a L |4 0 L Y c5 a L Y4 0 L C | =5 a L C Y4 0 L |

Sol).c2 yle e7,ation of tle c,rve i) 8Y L a+A 6 0

 Y Y .Y4 A2=8 =8 Aa+ =8 Y8 Aa+ Ya=+ =+ Y8 =+:  B  B 9 )8 (

yle e7,ation of tangent at .Y4A2 i)8 C A L Ya.+ C Y2 or Ya+ C 8 6 A C Ba L I',t tle e7,ation of tangent at .Y4 A2 i)8 L B+ C DNom/aring tle)e tmo e7,ation)4 me getYa L B an= A C Ba L C D

B a L Y

x) .Y4 A2 lie) on tle c,rve 8

Y L a+A 6 0

? L ja 6 0

B? L R9 6 0 B 0 L C |

3olpml

yle normal to tle c,rve at an8 /oint don it i) tle )traiglt line mlicl /a))e) tlro,gl d an= i) /er/en=ic,lar to tle tangent to tle c,rve at d5

SloDT)od)3olpml

qe ±nom tlat4 normal to tle c,rve at d.+

R4 8R2 i)

a line /er/en=ic,lar to tangent at d.+

R4 8R2 an=

/a))ing tlro,gl d5 klo/e of tle normal at d R klo/e of tle tangent at d

Bklo/e of normal at

R R

R RR R

.+ 4 8 2 .+ 4 8 2R =8 d.+ 4 8 2=8=+ =+:     9 ): 8 (9 )8 ( .2mpDlT)N yle e7,ation of tle normal to tle c,rve

8 L + .Y C +2 at tle /oint .Y4 I2 i)

a5 + C Y8 L Y 05 + C Y8 6 Y L I c5 Y+ 6 8 L B =5 Y+ 6 8 C B L I

Sol .a2 yle e7,ation of c,rve i)

8 L +.Y C +2 or 8 L Y+ C +

Y

B=8Y Y+=+ 

B .Y4 I2=8

Y B Y=+:    9 )8 (

yle e7,ation of tle normal at .Y4 I2 i)

R8 I .+ Y2. Y2

  

BR8 .+ Y2Y 

BY+ C + L C Y B + C Y8 L Y

4pDolamta)5oPtas)6TlmaT7)ao)MmtnTta

l3f tle tangent at d i) /arallel to <>a+i)4 tlen

7 L I5

B

R R.+ 4 8 2=8

tan I I=+:  B 9 )8 ( l3f tle tangent at d i) /er/en=ic,lar to <>a+i) or /arallel to z>a+i)4 tlen an= cot IY    B

R R.+ 4 8 2R =8

I Itan =+:  B 9 )8 (

l3f e7,ation of tle c,rve i) in /arametric form i5e5 + L f.t2 an= 8 L g.t25 ylen4 =8 =8W=t g%.t2 =+ =+W=t f %.t2  .a2 -7,ation of tangent i) g%.t28 g.t2 M+ f.t2'f %.t2   .02 -7,ation of normal i) f %.t28 g.t2 M+ f.t2'g%.t2    l3f tle tangent at an8 /oint on tle c,rve i) e7,all8 incline= to 0otl tle a+e)4 tlen =8R=+ o l3f tle tangent at an8 /oint ma±e) an e7,al interce/t on tle coor=inate a+e)5 ylen4 =8R=+ 

2Application of Derivatives

8Ttnag)od)MmtnTta9)Sub0amtnTta9)3olpml)mt7

Sub0tolpml

Xet tle tangent an= tle normal at an8 /oint

/.+4 82 of tle c,rve 8 L f.+2 meet tle <>a+i) at y an= G re)/ectivel85 Dram tle or=inate dM5 ylen tle lengtl) yM4 MG are calle= tle ),0>tangent an= ),0>normal re)/ectivel85 yle lengtl) dy4 dG are referre= to a) tle lengtl) of tle tangent an= tle normal re)/ectivel85

Nlearl84 MdG

P  

xl)o4 =8tan=+ 

From tle fig,re4 me lave

i5 Xengtl of tle tangent Yyd Md co)ec 8 .R cot 2      Y=+8 R=87 ':  6 9 )6 8 (5  ii5 Xengtl of ),0>tangent =+yM Md cot 8=8    iii5 Xengtl of normal YGd Md )ec 8 .R tan 2      Y=88 R=+7 ':  6 9 )8 (6 5  iv5 Xengtl of ),0>normal =8MG Md tan 8=+    .2mpDlT) yle lengtl of tle normal to tle c,rve + L a. 6 )in 24 8 L a.l C co) 2 at  L WY i) a5 Ya 05 a Y c5 aWY =5aW Y Sol .02 qe lave4 + L a. 6 )in 24 8 L a.R C co) 2 B=+ =8a.R co) 24 a)in= =      =8 a)intan=+ a.R co) 2 Y    

Xengtl of tle normal L

Y=88 R=+

: 9 )8 (

Ya.R co) 2 R tan

Y    

YYa)in )ec Yatan )inY Y Y Y

   l  Xengtl of tle normal at i) Ya tan )in YaY B B      hTlPymaPyT)ms)agT)6maT)od)RgmtnT yle =erivative =8 =+ re/re)ent) tle rate of clange of varia0le 8 a+ mitl re)/ect to +5 ko4 tle rate of clange of an8 /l8)ical 7,antiti8 at an8 time i) o0taine= 08 =ifferentiating tle /l8)ical 7,antit8 mitl re)/ect to time5 e5g5 Xet ) 0e tle =i)tance mea),re= from a fi+e= /oint after time t4 tlen =) =t re/re)ent) tle rate of clange of =i)tance .)2 mitl re)/ect to time.t25 i5e5 =))/ee==t

3f tmo varia0le) are var8ing mitl re)/ect to

anotler varia0le t 4 i5e58 L f .t2 + L g .t25ylen4 rate of clange of 8 mitl re)/ect to + i) =8 =8 =+ =8 =8 =t=t4 /rovi=e= I or 5=+=+ =t =+ =t =+ =t  g  .08 clain r,le of =erivative2 yl,)4 tle rate of clange of 8 mitl re)/ect to + can 0e calc,late= ,)ing tle rate of clange of 8 an= tlat of + 0otl mitl re)/ect to t5 .2mpDlT)C yle )/ee= v of a /article moving along a )traiglt line i) given 08 a 6 0v

Y L+Y4 mlere + i) tle =i)tance

from tle origin5 yle acceleration of tle /article i) a5 + 005+ a0 c5 a0+ =5 a+

22Application of Derivatives

Sol).a2 qe lave4 a 6 0vY L +Y

B=v =+ =vY0v Y+ 0v +v=t =t =t B 

Bxcceleration4 =v +

=t 0 fDDlo2PpmaPot)mt7).llols

Xet 8 L f.+25 ylen4

+ If.+ +2 f.+2lim f %.+2+r   f.+ +2 f.+2f %.+2 4+   R mlere

IRr mlen + I r

Bf.+ 6 +2 C f.+2 L f%.+25+ 6 R 5+ Bf.+ 6 +2 C f.+2 L f%.+25+ .a//ro+imatel82

B 8 L f%.+25+ .a//ro+imatel82

yl,)4 if + i) an error in +4 tlen tle corre)/on=ing error in 8 i) 85 yle)e )mall val,e) + an= 8 are calle= =ifferential)5

P, fbsoluaT).llol + i) calle= an a0)ol,te

error in +

PP, 6TlmaPyT).llol

+ +  i) calle= tle relative error5

PPP, 5TlbTtamnT).llol

+RII+ : l9 )8 ( i) calle= tle /ercentage error5 .2mpDlT)A yle a//ro+imate clange in tle vol,me of a c,0e of )i=e + m ca,)e= 08 increa)ing tle )i=e 08 A [5 a I5II9 +

A mA05 I59+A mA

c5 I5I?+

A mA=5 I5?+A mA

Sol .c2 qe ±nom tlat tle vol,me -of a c,0e of

)i=e + i) given 08

A Y=-- + A+=+ B 

Xet + 0e clange in )i=e L A[ of + L I5IA+

(om4 clange in vol,me4 =--=+ :  9 )8 ( L .A+Y2 + L .A+

Y2 .I5IA +2

Pa) + L A [ of + i) I5IA +G

L I5I?+

A mA ]ence4 tle a//ro+imate clange in tle vol,me of tle c,0e i) I5I? +

A mA56ollTls)MgTolTpXet f 0e a real>val,e= f,nction =efine= in tleclo)e= interval Pa4 0G4 ),cl tlati5 f.+2 i) contin,o,) in tle clo)e= interval

Pa4 0G5

ii5 f.+2 i) =ifferentiate in tle o/en interval .a4 02 iii5 f.a2 L f.024 tlen tlere i) )ome /oint c in tle o/en interval .a4 02 ),cl tlat4 f %.c2 L I4 (TopTalPbmlly)Hn=er tle a)),m/tion) of wolleg) tleorem4 tle gra/l of f.+2 )tart) at /oint .a4 I2 an= en=) at /oint .04 I2 a) )lomn in fig,re)5 yle concl,)ion i) tlat tlere i) atlea)t one /oint c

0etmeen a an= 04 ),cl tlat tle tangent to tle

gra/l at Mc4 f.c2' i) /arallel to tle <>a+i)5 flnTblmPb)4taTlDlTamaPot)od)6ollT's)MgTolTp 'etmeen an8 tmo root) of a /ol8nomial f.+24 tlere i) alma8) a root of it) =erivative f %.+25 .2mpDlT)L wolleg) tleorem i) not a//lica0le to tle f,nction f.+2 L V+V for C Y  +  Y 0eca,)e a5 f i) contin,o,) on PC Y4 YG

05 f i) not =eriva0le at + L I

c5 f.C Y2 L f.+2 =5 f i) not a con)tant f,nction

Sol).02 qe lave4

+4 + If.+2 ++ + I m4 3  2 .X]D at + L I2 + I + If.+2 f.I2 + Ilim lim R+ I + Ir r       an= .w]D at + L I2 + I + I f.+2 f.I2 + Ilim lim R+ I + Ir r     .X]D at + L I2 g .w]D at + L I2 ko4 f.+2 i) not =ifferentia0le at + L I5 Non)e7,entl84 wolleg) i) not a//lica0le to tle given f,nction5

8mnlmtnT's)GTmt)VmluT)MgTolTp

Xet f 0e a real f,nction4 contin,o,) on tle clo)e= interval Pa4 0G an= =ifferent:a0le in tle o/en interval .a4 025 ylen4 tlere i) atlea)t one /oint c in tle o/en interval .a4 024 ),cl tlat5 f.02 f.a2f %.c20 a 

23Application of Derivatives

(TopTalPbmlly xn8 clor= of tle c,rve 8 L f.+24 tlere i) a /oint on tle gra/l4 mlere tle tangent i) /arallel to tli) clor=5

6Tpmlus 3n tle /artic,lar ca)e4 mlere f.a2 L f.025

yle e+/re))ion f.02 f.a2 0 a   0ecome) Oero5 yl,)4 mlen f.a2 L f.02 tlen f %.c2 L I for )ome c in .a4025 yl,)4 wolleg) tleorem 0ecome) a /artic,lar ca)e of tle mean val,e tleorem5 .2mpDlT)I

Xet f.+2 L e

+4 +

RPI4 RG4 tlen a n,m0er c of tle

Xagrangeg) mean val,e tleorem i)

a5 loge .e C l2 05 loge.e 6 l2 c5 log e e =5 (one of tle)e

Sol).a2 Nlearl84 f.+2 i) contin,o,) on PI4 RG an=

=ifferentia0le on .I4R25 ylerefore4 tlere e+i)t) c

R .I4 l2 ),cl tlat

f.R2 f.I2f %.c2R I 

Bec L e C R B c L loge .e C R2

4tblTmsPtn)mt7)hTblTmsPtn)FutbaPots

Monotonicit8 Xet 8 L f.+2 0e a given f,nction

mitl D a) it) =omain Xet R

D D4 tlenS

&PH 4tblTmsPtn)FutbaPot f.+2 i) )ai= to 0e increa)ing in D

R if for ever8 +R4 +Y

R DR4

R Y R Y+ + f.+ 2 f.+ 2 B 

3t mean) tlat tlere i) a certain increa)e in tle

val,e of f.+2 mitl an increa)e in tle val,e of + .wefer to tle a=Qacent fig,re25 &PPH 3ot07TblTmsPtn)FutbaPot f.+2 i) )ai= to 0e non>=ecrea)ing in DR if for ever8 +

R4 +Y

R DR4

3t mean) tlat tle val,e ofW.+2 mo,l= never=ecrea)e mitl an increa)e in tle val,e of +.wefer to tle a=Qacent fig,re24

&PPPH hTblTmsPtn)FutbaPot f.+2 i) )ai= to 0e =ercea)ing in DR4 if for ever8 R Y R R Y R Y+ 4+ D 4 + + f.+ 2 f.+ 2R  B 

3t mean) tlat tlere i) a certain =ecrea)e in

tle val,e of f.+2 mitl an increa)e in tle val,e of + .wefer to tle a=Qacent fig,re25 &PyH 3ot0PtblTmsPtn)FutbaPot)-

3f f.+2 i) )ai= to 0e non>increa)ing in DR4 if for

ever8 +

R 4 +Y

R DR 4 +R K +Y B f.+R2  f.+Y25 3t

mean) tlat tle val,e of f.+2 mo,l= never increa)e mitl an increa)e in tle val,e of + . wefer to tle a=Qacent fig,re25 3oaT i5 f,nction i) )ai= to 0e monotonic if it i) eitler increa)ing or =ecrea)ing5 ii5 3f f.I2 L I an= f %.+2 m I + R w4 tlen f.+2 I + . 4 I2 an=  R   f.+2 I + .I4 2m  R  iii5 3f f.I2 L I an= f%.+2  I  + R w4 tlen f.+2 m I  +R .4 I2 an= f.+2 { I  + R .I4 25 iv5 3f f.+2 i) a )trictl8 increa)ing f,nction on an interval Pa4 0G4 tlen f

CR e+i)t) an= it i) al)o a

)trictl8 increa)ing f,nction v5 3f f.+2 i) )trictl8 increa)ing f,nction on an interval Va 4 0V ),cl tlat it i) contin,o,)4 tlen f

CR i) contin,o,) on Pf.a24 f.02G

.2mpDlT)w yle f,nction f.+2 L + L +

A C A+ i)

a5 increa)ing on .C 4 C lG X Pl4 2 an= =ecrea)ing on .C R4R2

05 =ecrea)ing on .C

4 C lG X Pl4 XG an= increa)ing on .C R4R2 c5 increa)ing on .I4 2 an= =ecrea)ing on .C 4 I2 =5 =ecrea)ing on .I4 2 an= increa)ing on .C 4 I2

Sol .a2 qe lave4

f.+2 L +

A C A+

2Application of Derivatives

Bf%.+2 L A+Y C A

For f.+2 to 0e increa)ing4 me m,)t lave

Y f %.+2 I A+ A Im B  m

BY+ R I + Ror + R m B   m

]ence4 f.+2 i) increa)ing on . 4 RGPR4 2   an= =ecrea)ing on .C R4R25

Gm2Ppm)mt7)GPtPpm

RotbTDa)od)8obml)Gm2Ppm)mt7)8obml)GPtPpm

Xet 8 L f.+2 0e a f,nction =efine= at + L a an=

al)o in tle vicinit8 of tle /oint + L a5 ylen4 f.+2 i) )ai= to lave a local ma+im,m at + L a4 if tle val,e of tle f,nction at + L a i) greater tlan tle val,e of tle f,nction at tle neigl0o,ring /oint) of + L a5

Matlematicall84 f.a2 K f.a C l2 an= f.a2 K

f .a 6 l24 mlere l K I5 .ver8 )mall 7,antit825 kimilarl84 f .+2 i) )ai= to lave a local minim,m at + L a4 if tle val,e of tle f,nction at + La i) le)) tlan tle val,e of tle f,nction at tle neigl0o,ring /oint) of + L a5

Matlematicall84 f.a2 { f .a C l2 an= f.a2 {

f.a 6 l24 mlere l K I5 x local ma+im,m or a local minim,m i) al)o calle= a local e+trem,m5

RlPaPbml)5oPtas

Nritical /oint) i) an im/ortant to/ic for e+amination an= a )mall /ractice can lel/ 8o, in )olving ),cl /ro0lem) ver8 7,ic±l8 a) tle level of 7,e)tion i) ea)8 to average5

3t i) a collection of /oint) for mlicl4

i5 f.+2 =oe) not e+i)t ii5 f %.+2 =oe) not e+i)t or iii5 f %.+2 L I xll tle val,e) of + o0taine= from a0ove con=ition) are )ai= to 0e critical /oint)5

3t )lo,l= 0e note= tlat critical /oint) are tle

interior /oint) of an interval5 .2mpDlT)/ yle n,m0er of critical /oint) of Y + Rf.+2+  i) a5 R 05 Y c5 A =5 (one of tle)eSol).c2 qe lave4 Y Y

Y+ R4 + R+ R+f.+2R ++

4 + R+

4 m11 31 12

Nlearl84 f.+2 i) not =ifferent: a0le at + L I an=

+ L R5 ko4 08 =efinition4 tle)e are tmo of tle critical /oint)5 For /oint) otler tlan tle)e tmo4 me lave A

A+ Y4 + R+f %.+2+ Y

4 + R+ 

4 1131 12

Nlearl84 f

%.+2 L I at + L Y5 ko4 + L R i) al)o a critical /oint)5 ]ence4 f.+2 la) tlree critical /oint)4 viO5 I4R an= Y5

GTago7)od)FPt7Ptn)Gm2Ppm)ol)GPtPpm

FPlsa)hTlPymaPyT)MTsa

x) me ±nom tlat tle f,nction attain) ma+im,m4 mlen it la) ta±en it) ma+im,m val,e an= attain) minim,m4 mlen it la) ta±en it) minim,m val,e mlicl co,l= 0e )lomn a) xt a Nritical doint + L +Ii5 qlen f.+2 attain) ma+im,m at + L a5 i5e5 from tle a0ove gra/l5 I R R I Y Y for + a4 ?I tan I or increa)ing for + a for + a4 tan I or neitler increa)ing nor =ecrea)ing for+ I4 ?I tan I or =ecrea)ing for + a

4   B   1  31

   B   2 yl,)4 me can )a84 f.+2 i) ma+im,m at )ome /oint .+ L a2

Bf.+2 i) increa)ing for + a

f.+2 i) =ecrea)ing for + a 43 2 ii5 qlen f.+2 attain) minim,m at + L a

2Application of Derivatives

i5e5 from tle a0ove gra/l5 I R R I Y Y for + a4 ?I tan I or increa)ing for + a for + a4 tan I or neitler increa)ing nor =ecrea)ing for+ a4 ?I tan I or increa)ing for + a

4   B   1  31

   B   2 yl,)4 me can )a84 f.+2 i) minim,m at )ome /oint .+ L a2

Bf.+2 i) =ecrea)ing for + a

f.+2 i) increa)ing for + a 43 2 .2mpDlT)v0 }n tle interval PI4RV tle f,nction +

YD .R C +2|D

ta±e) it) ma+im,m val,e at tle /oint a5 I 05 RWB c5 RWY =5 RWA

Sol).02 Xet f.+2 L +YA.l C +2RD

ylen4 f %.+2 L YD+YB .l C + 2|B .l C B+2 (om4 f %.+2 L I B + L I5R4 RWB

Nlearl84 f

%.+2 K I in tle left neigl0o,rloo= of RWB an= f %.+2 { I in tle riglt neigl0o,rloo= of RWB5 ko4 f %.+2 clange) it) )ign from /o)itive to negative in tle neigl0o,rloo= of RWB5 ]ence4 it attain) ma+im,m at + L R WB5 fa)m)8Tda).t7)5oPta)m)mt7)6Pnga).t7)5oPta)b

Pt)1m9)b2

Xet f.+2 0e =efine= on Pa4 0G

STbot7)hTlPymaPyT)MTsa

Fir)t me fin= tle root) of f

% .+2 L I5 k,//o)e + L a i) one of tle root) of f %.+2 L I5 (om4 fin= f %%.+2 at + L a5

R5 3f f

%%.a2 L negativeA tlen f.+2 i) ma+im,m at + L a5

Y5 3f f

%%.a2 L /o)itiveA tlen f.+2 i) minim,m at + L a5

A5 3f f

%%.a2 L OeroA tlen me fin= f%%% .+2 at + L a5 3f f %%%.a2 g I4 tlen f.+2 la) neitler ma+im,m nor minim,m .inflection /oint2 at + L a5 ',t4 if f %%.a2 L I tlen fin= f iv .a2 3f f iv L /o)itive4 tlen f.+2 i) minim,m at + L a5 3f f iv.a2 L negative4 tlen f.+2 i) ma+im,m at + L a5 an= )o on4 /roce)) i) re/eate= till /oint i) =i)c,))e=5.2mpDlT)vv3f fM+2 L Y+

A C YR+Y 6 A9+ C AI4 tlen for

f.+2 mlicl one of tle folloming i) correct B a5 f.+2 la) minim,m at + L R

05 f.+2 la) ma+im,m at + L 9

c5 f.+2 la) ma+im,m at + L R =5 f.+2 la) no ma+im,m or minim,m

Sol).c2 qe lave4 f.+2 L Y+A C YR+Y 6 A9+ C AI

Bf%.+2 L 9+Y C BY+ 6 A9 an= f%%.+2 L RY+ C BY

xt /oint) of local ma+im,m or minim,m4 me m,)t lave f %.+2 L I B 9.+Y C|+ 6 92 L I B + L l4 9

Nlearl84 f

%%.R2 L RY C BY L C AI { I an= f.92 L |Y C BY K I ko4 f.+2 la) local ma+im,m at + L l an= local minim,m at + L 95 tag)hTlPymaPyT)MTsa

3t i) notling 0,t tle general ver)ion of tle )econ=

=erivative te)t4 it )a8) tlat if4 n f %.a2 f %%.a2 f %%%.a2 555555f .a2 I    an= n R f .a2 Ig .all =erivative) of tle f,nction ,/ to or=er n vani)le) an= .n 6 l2tl or=er =erivative =oe) not vani)l at + L a4 tlen f.+2 mo,l= lave a local ma+im,m or local minim,m at + L a4 if n i) o== nat,ral n,m0er an= tlat + L a mo,l= 0e a /oint of local ma+ima4 if f n 6 R .a2 { I an= mo,l=

0e a /oint of local minima4 if f

n 6 R .a2 K I5 ]omever if n i) even4 tlen f la) neitler a ma+ima nor a minima at + La5 3t i) clear tlat tle la)t tmo te)t) are 0a)icall8 tle Matlematical re/re)entation of tle fir)t =erivative te)t5 ',t tlat )lo,l= not =imini)l tle im/ortance of tle)e te)t)5 'eca,)e at tlat time) it 0ecome) ver8 =iffic,lt to =eci=e mletler f %.+2 clange) itg) )ign or not mlile /a))ing tlro,gl /oint + L a an= tle remaining te)t) ma8 come lan=8 in tle)e ±in= of )it,ation)5

3f a f,nction i) )trictl8 increa)ing in Pa4 0G4 tlen

f.a2 i) local minim,m f.02 i) local ma+im,m 4 3 2

3f a f,nction i) )trictl8 =ecrea)ing in Pa4 0G4 tlen

f.a2 i) local ma+im,m f.02 i) local minim,m 4 3 2 fbsoluaT)&(lobmlH)Gm2Ppup3GPtPpup

RotbTDa)od)(lobml)Gm2Ppup3GPtPpup

Xet 8 L f.+2 0e a given f,nction mitl =omain D5

Xet a4 0 D5S Glo0al ma+im,mWminim,m of f.+2 in Pa4 0G i) 0a)icall8 tle greate)tWlea)t val,e of f.+2 in Pa4 0G

2Application of Derivatives

Glo0al ma+im,m an= minim,m in Pa40G mo,l=

alma8) occ,r at critical /oint) of f.+2 mitlin Pa4 0G or at tle en= /oint) of tle interval4 if f i) contin,o,) in Pa4 0G5 (lobml)Gm2Ppup3GPtPpup)Pt)1m9)b2

3n or=er to fin= tle glo0al ma+im,m an= minim,m

of a contin,o,) f,nction f.+2 inPa4 0G5 Fin= o,t all

tle critical /oint) of f.+2 in .a4 025 Xet cR4 cY4 5554cn0e tle =ifferent critical /oint)5 Fin= tle val,e of

tle f,nction at tle)e critical /oint)5 Xet f.c

R24f.cY24

55554 f.c

n2 0e tle val,e) of tle f,nction at critical /oint)5 ka84 M

R ma+Mf.a2f.cR24f.cY245554f.cn24f.02' an=

M

Y L minMf.a24 f.cY245554 f.cn24 f.02'

ylen4 M

R i) tle greate)t val,e of f.+2 in Pa4 0G

an= M

Y i) tle lea)t val,e of f.+2 in Pa4 0G5

(lobml)Gm2Ppup3GPtPpup)Pt)&m9)bH Metlo= for o0taining tle greate)t an= lea)t val,e) of f.+2in .a4 02 i) almo)t )ame a) tle metlo= ,)e= for o0taining tle greate)t an= lea)t val,e) in

Pa4 0G lomever mitl a ca,tion5

Xet 8 L f.+2 0e a contin,o,) f,nction an=

c R4 cY4 cA4 5554cn 0e tle =ifferent critical /oint) of tle f,nction in .a4 02 Xet M

R L ma+ Mf.cR25 f.cY24 f.cA245555 f.cn'' an=

MY L min Mf.c

R24 f.cY24 f.cA24 55555 f.cn2'

(om4 if + a I .or + 0 I2 limr  r  f.+2 K MR or MY4 f.+2 mo,l= not lave glo0al ma+im,m .or glo0al minim,m2 in .a4 025 yli) mean) tlat if tle limiting val,e) at tle en= /oint) are greater tlan M

R or le)) tlan MY4 tlen

f.+2 mo,l= not lave glo0al ma+im,mWminim,m in .a4 025 }n tle otler lan= if

RY+ a I + a I

.an= + 0 I2 .an= + 0 I2

M lim f.+2 an= M lim f.+2r  r 

r  r  4 tlen M

R an= MY mo,l= re)/ectivel8 0e tle glo0al

ma+im,m an= glo0al minim,m of f.+2 in .a4 02 .2mpDlT)vN

Xet f.+2 L Y+

A C ?+Y 6 RY+ 6 94 tlen a0)ol,te

ma+ima of f.+2 in PI4 YG an= .R4 A2 are re)/ectivel8 a5 I4 Y 05 R4 Y c5 Y4 Y =5 (one of tle)e

Sol).02 f.+2 L Y+A C ?+Y 6 RY+ 6 9

Y f %.+2 9+ Rj+ RY 9.+ R2 .+ Y2     

3n PI4 YG4 critical /oint of f.+2 in PI4YG i) + LR5

yl,)4 + L I i) tle /oint of a0)ol,te minim,m an= + L R i) tle /oint of a0)ol,te ma+im,m of f.+2 in PI4 YG53n .R4 A24 critical /oint of f.+2 in .R4 A2 i) + L Y5  + R + A f.Y2 RI4 lim f.+2 RRan= lim f.+2 RDr r   yl,)4 + L Y i) tle /oint of a0)ol,te minim,m in .R4 A2 an= a0)ol,te ma+im,m in .R4 A2 =oe) not e+i)t5

5oPta)od)4tdlTbaPot

Non)i=er f,nction f.+2 L +A4 at + L I4 f

%.+2LI5 xl)o4 f %%.+2 L I at + L I5 k,cl /oint i) calle= /oint of inflection4 mlere Yn= =erivative) i) Oero5

Non)i=er anotler f,nction f.+2 L )in +

f %%.+2 L C )in+5 (om4 f %%.+2 L I mlen + L n4 tlen tle)e /oint) are calle= /oint) of inflection5 fa)5oPta)od)4tdlTbaPot i5 3t i) not nece))ar8 tlat R)t =erivative i) Oero5 ii5 Yn= =erivative m,)t 0e Oero or Yn= =erivative clange) )ign in tle neigl0o,rloo= of /oint of inflection iii5 Gra/l of c,rve clange) it) concavit85 iv5 3f f %%.+2 K I gra/l i) concave tomar=) negative C>a+i) an= if f %%.+2 { I4 gra/l i) concave tomar=) /o)itive .2mpDlT)v yle /oint of inflection for tle c,rve 8 L +

DWY i)

a5 .R4 R2 05 .I4 I2 c5 .R4 I2 =5 .I4 R2

Sol .02 Given4 8 L +DWY

Y

AWY RWY

Y=8 D = 8 RD+ 4 +=+ Y =+ B 

xt + L I4 Y

Y=8 = 8

I4 I=+ =+ 

an= A A= 8 =+ i) not =efine=4 mlen + L I4 8 L I .I4I2 i) a /oint of inflection5

247Application of Derivatives

Exercise 1

(Topical Problems)

Geometrical and Physical Meaning of

Derivatives and Tangent and Normal

1.The abscissa of the point on the curvey = a (ex/a + e-x/a) where the tangent is parallel to

the X-axis, is a. 0 b. a c. 2a d. - 2a

2.The point on the curve y = x3 at which the tangent

to the curve is parallel to the X-axis, is a. 2, 2 b. 3, 3 c. 4, 4 d. 0, 0

3.The length of the subtangent at 2, 2 to the curvex5 = 2y4 is

a. 5 2b.8 5 c.2 5d.5 8

4.The equation of the normal to the curve y4 - ax3

at (a, a) is a. x + 2y = 3a b. 3x - 4y + a = 0 c. 4x + 3y = 7a d. 4x - 3y = 0

5.Let

2e if x 1g(x)log(x 1), if x 1

43  2. The equation of the normal to y = g(x) at the point (3, log 2), is a. y - 2x = 6 + log 2 b. y + 2x = 6 + log 2 c. y + 2x = 6 - log 2 d. y + 2x = - 6 + log 2 e. y - 2x = - 6 + log 2

6.The equation of the tangent to the curve

24y xx , that is parallel to the X-axis, is

a. y = 0 b. y = 1 c. y = 2 d. y = 3

7.The normal to a curve at P(x, y) meets theX-axis at G. If the distance of G form the origin istwice the abscissa of P, then the curve is aa. ellipse b. parabolac. circle d. hyperbola

8.The length of the subtangent to the curvex2 + xy + y2 = 7at (1, -3)

a. 3 b. 5 c. 15 d. 3 5

9.The length of the normal to the curvex = a( + sin ), y = a(1 - cos ) at

2  is a. 2a b.a 2 c.2 2d.2a

10.The angle between the tangents drawn from the

point (1, 4) to the parabola y

2 = 4x is

a. 6 b.6  c.3 d.2 

11.If the normal to the curve y = f(x) at the point(3, 4) makes an angle

3 4  with the positive

X-axis, then f

'(3)is equal to a. - 1 b. - 3/4 c. 4/3 d. 1

12.The intercepts on X-axis made by tangents tothe curve,

x

0y t dt, x R R., which are parallel

to the line y = 2x, are equal to a. ± 1 b. ± 2 c. ± 3 d. ± 4

13.If the line ax + by + c = 0 is a tangent to thexy = 4,thena. a < 0, b = 0 b. a

 0, b > 0 c. a < 0, b < 0 d. a  0, b < 0

14.The coordinates of the point on the curve

y = x

2 - 3x + 2, where the tangent is perpendicular

to the straight line y = x are a. (0, 2) b. (1, 0) c. (- 1, 6) d. (2, - 2)

15.If the curves

2 2 2x y

1a 12  and y = 8x intersect at

right angle, then the value of a

2 is equal to

a. 16 b. 12 c. 8 d. 4 e. 2

16.The angle between the curves y = ax and y = bx is

equal to a.

1a btan1 ab

: 9 )8 ( b.1a btan1 ab : 9 )8 ( c.1logb logatan1 logalogb : 9 )8 (

248Application of Derivatives

d.1loga logbtan1 logalogb : 9 )8 ( e.1loga logbtan1 logalogb : 9 )8 (

17.The equation of the tangent to the curve

x 4y 4e  at the point where the curve crosses Y-axis is equal to a. 3x + 4y = 16 b. 4x + = 4 c. x + y = 4 d. 4x - 3y = - 12 e. x - y = - 4

18.The equation of the tangent to the curvex2 - 2xy + y2 + 2x + y - 6 = 0 at (2, 2) is

a. 2x + y - 6 = 0 b. 2y + x - 6 = 0 c. x + 3y - 8 = 0 d. 3x + y - 8 = 0 e. x + y - 4 = 0

19.Angle between y2 = x and x2 = y at the origin is

a.

132tan4

: 9 )8 (b.14tan3 : 9 )8 ( c.2 d.4 

20.The length of tangent, subtangent, normal andsubnormal for the curve y = x2 + x - 1 at 1, 1 are

A, B, C and D respectively, then their increasing

order is a. B, D, A, C b. B, A, C, D c. A, B, C, D d. B, A, D, C

21.The point on the curve y2 = x, the tangent at which

makes an angle 45

0 with X-axis is

a.

1 1,4 2

: 9 )8 (b.1 1,2 4 : 9 )8 ( c.1 1,2 2 : 9 )8 (d.1 1,2 2 : 9 )8 (

22.The length of the subtangent to the curvex2 y2 = a4 at (- a, a) is

a. a

2b. 2a

c. a d. a 3

23.The locus of all the points on the curve

2xy 4a x asin2

: :  9 )9 )8 (8 ( at which the tangent is parallel to X-axis isa. a straight line b. a circlec. a parabola d. an ellipse

24.The slope of the normal at the point with abscissax = - 2 of the graph of the function f(x) = |x2 - |x| |

is a. 1

6b.1

3

c.1 6d.1 3

25.The equation of the normal at the point (am2, am3)

for the curve ay

2 = x3, is

a. 2x + 3my - 3am

3 - 2am2 = 0

b. 2x + 3my - 3am

4 - 2am2 = 0

c. 2x + 3m

2 y - 3am3 - 2am3 = 0

d. None of the above

26.In the curve xm + n = am - n y2n, the mth power of

the subtangent varies as the kth power of subnormal, then k is a. m b. n c. 1/n d. 1/m

27.At which point the tangent to the curvex2 + y2 = 25 is parallel to the line 3x - 4y = 7 ?

a. (3, 4), (- 3, - 4) b. (3, - 4), (- 3, 4) c. (4, 3) (- 4, - 3) d. (- 4, 3) (4, - 3)

28.If the curve y = ax and y = bx intersect at angle ,

then tana is equal to a. a b 1 ab  b.loga logb

1 logalogb

  c. a b 1 ab  d.loga logb

1 logalogb

 

29.Line joining the points (0, 3) and (5, - 2) is atangent to the curve

axy1 x, then a. a 1 3 ob.aR c.a 1 3  od.a 2 2 3  o

30.The equation of the tangent to the curve

2y 9 2x  at the point where the ordinate and

the abscissa are equal, is a.

2x y 3 3 0  b. 2x y 3 0  

c.2x y 3 0  d. None of these

31.If the normal to the curve y2 = 5x - 1 at the point

(1 - 2) is of the form ax - 5y + b = 0, then a and b are a. 4, -14 b. 4, 14 c. - 4, 14 d. 4, 2

249Application of Derivatives

32.The sum of intercepts on coordinate axes made

by tangent to the curve x y a, is  a. a b. 2a c.

2 ad. None of these

33.The tangent at (1, 7) to the curve x2 = y - 6

touches the circle x

2 + y2 + 16x + 12y + c = 0 at

a. (6, 7) b. (- 6, 7) c. (6, - 7) d. (- 6, - 7)

34.Coordinates of a point of the curve y = x log x atwhich the normal is parallel to the line2x - 2y = 3 area. (0, 0) b. (e, e)c. e2, 2e2) d. (e-2, - 2e-2)

Derivative as a Rate of Change,

Approximation and Errors

35.The diagonal of a square is changing at the rateof 0.5 cm s-1. Then, the rate of change of area,

when the area is 400 cm

2, is equal to

a.

220 2cm /sb.210 2cm /s

c.21cm /s10 2d.

210cm /s2

d.

25 2 cm /s

36.A particle is moving in a straight line. At time t,the distance between the particle from its starting

point is given by x = t - 6t

2 + t3. Its acceleration

will be zero at a. t = 1 unit time b. t = 2 units time c. t = 3 units time d. t = 4 units time

37.A ladder 10 m long rests against a vertical wallwith the lower end on the horizontal ground. Thelower end of the ladder is pulled along the groundaway from the wall at the rate of 3 m/s. Theheight of the upper end while it is descending atthe rate of 4 cm/s, isa.

4 3 mb.5 3 m

c.5 2 md. 8 m e. 6 m

38.There is an error of ± 0.04 cm in the measurementof the diameter of a sphere. When the radius is10 cm, the percentage error in the volume of thesphere isa. ± 1.2 b. ± 1.0c. ± 0.8 d. ± 0.6

39.The distance covered by a particle in t second isgiven by x = 3 + 8t - 4t2. After 1 s its velocity will

bea. 0 unit b. 3 unitsc. 4 units d. 7 units

40.A stone is thrown vertically upwards from thetop of a tower 64 m high according to the law ofmotion given by s = 48t - 16t2. The greatest height

attained by the stone above ground is a. 36 m b. 32 m c. 100 m d. 64 m

41.A lizard, at an initial distance of 21 cm behind aninsect, moves from rest with an acceleration of2 cm/s2 and pursues the insect which is crawling

uniformly along a straight line at a speed of

20 cm/s. Then, the lizard will catch the insect after

a. 24 s b. 21 s c. 1 s d. 20 s

42.The distance travelled by a motor car in f secondsafter the brakes are applied is s feet, wheres = 22t - 12t2. The distance travelled by the car

before it stops, is a. 10.08 ft b. 10 ft c. 11 ft d. 11.5 ft

43.Let y be the number of people in a village at timet. Assume that the rate of change of the populationis proportional to the number of people in thevillage at any time and further assume that the

population never increases in time. Then, the population of the village at any fixed time t is given by a. y = e kt + c, for some constant c  0 and k m 0 b. y = ce kt, for some constant c m 0 and k  0 c. y = e ct + k, for some constant c  0 and k m 0 d. y = k e ct, for some constant c m 0 and k  0

44.The circumference of a circle is measured as56 cm with an error 0.02 cm. The percentageerror in its area isa. 1/7 b. 1/28c. 1/14 d. 1/56

45.A spherical balloon is expanding. If the radius isincreasing at the rate of 2 cm/min, the rate atwhich the volume increases in cubic centimeters

per minute when the radius is 5 cm, is a. 10 b. 100  c. 200 d. 50 

46.If the surface area of a sphere of radius r isincreasing uniformly at the rate 8 cm2/s, then the

rate of change of its volume is a. constant b. proportional to r c. proportional to r2d. proportional to r

47.A spherical balloon is being inflated at the rate of35 cc/min. The rate of increase in the surfacearea (in cm2/min) of the balloon when its diameter

is 14 cm, is

250Application of Derivatives

a. 10 b.10 c. 100 d.10 10

48.If there is 2 % error in measuring the radius ofsphere, then ... will be the percentage error inthe surface areaa. 3 % b. 1 %c. 4 % d. 2 %

49.A man of 2 m height walks at a uniform speed of6 km/h away from a lamp post of 6 m height.The rate at which the length of his shadowincrease isa. 2 km/h b. 1 km/hc. 3 km/h d. 6 km/h

50.A particle moves along the curve y = x2 + 2x.

Then, the point on the curve such that x and

y-coordinates of the particle change with the same rate is a. 1, 3 b.

1 5,2 2

: 9 )8 ( c.1 3,2 4 :  9 )8 (d. (- 1, - 1)

51.Gas is being pumped into a spherical ballon at therate of 30 ft3/min. Then, the rate at which the

radius increases when it reaches the value 15 ft, is a.

1ft/min30b.

1ft/min15

c.

1ft/min20d.1ft/min15

52.The distance travelled s in metres by a particle int second is given by, s = t3 + 2t2 + f. The speed of

the particle after 15 will be a. 8 cm/s b. 6 cm/s c. 2 cm/s d. None of these

53.Moving along the X-axis there are two points withx = 10 + 6t, x = 3 + t2. The speed with which they

are reaching from each other at the time of encounter is (x is in cm and t is in second) a. 16 cm/s b. 20 cm/s c. 8 cm/s d. 12 cm/s

54.An object is moving in the clockwise directionaround the unit circle x2 + y2 = 1. As it passes

through the point

1 3,2 2

: 9 )9 )8 (, its y-coordinate is decreasing at the rate of 3 unit per second. The rate at which the x-coordinate changes at this point is (in unit per second)a. 2 b. 3 3 c.3d.2 3

55.A particle is moving along the curve x = at2 + bt + c.

If ac = b

2, then particle would be moving with

uniform a. rotation b. velocity c. acceleration d. retardation

56.The position of a point in time t is given byx = a + bt - ct2, y = at + bt2. Its acceleration at

time t is a. b - c b. b + c c. 2b - 2c d.

2 22 b c

57.The distance s metres covered by a boy in tsecond, is given by s = 3t2 - 8t + 5. The body will

stop after a. 1 s b. 3s4 c.4s3d. 4 s

58.A ladder 20 ft long has one end on the groundand the other end in contact with a vertical wall.The lower end slips along the ground. If the lowerend of the ladder is 16 ft away from the wall,upper end is moving  time as fast as the lower

end, then  is a. 1 3b.2 3 c.4 3d.5 3

59.The approximate value of square root of 25.2 isa. 5.01 d. 5.02c. 5.03 d. 5.04

60.The approximate value of

1

3(0.007) is

a. 21

120b.23

120
c.29

120d.31

120

61.A spherical balloon is pumped at the rate of10 inch3/min, the rate of increase of its radius if

its radius is 15 inch is a.

1inch/min30b.

1inch/min60

c.

1inch/min90d.

1inch/min120

251Application of Derivatives

62.x and y are the sides of two squares such thaty = x. x2. The rate of change of area of the second

square with respect to that of the first square is a.2x

2 + 3x + 1 b. 2x2 + 2x - 1

c. 2x

2 - 3x + 1 d. 3x2 + 2x + 1

63.The speed v of a particle moving along a straightline is given by a + bv2 = x2, where x is its distance

from the origin. The acceleration of the particle is a. bx b. x a c.x bd.x ab

64.The rate of change of the surface area of a sphereof radius r, when the radius is increasing at therate of 2 cm/s is proportional toa.

1 rb.21 r c. c d. r2

Rolle's Theorem and Lagrange's Mean Value

Theorem, Increasing and Decreasing Function

65.The function f(x) = x(x + 3)

1x2e : 9 )8 ( satisfies all the conditions of Rolle's theorem in (- 3, 0). The value of c is a. 0 b - 1 c. - 2 d. - 3

66.Function f is defined by f(x) = 2 +

2

3(x 1) in

[0, 2]. Which of the following is not correct ? a. f is not derivable in (0, 2) b. f is continuous in [0, 2] c. f(0) = f(2) d. Rolle's theorem is true in [0, 2]

67.The real number k for which the equation,2x3 + 3x + k = 0 has two distinct real roots in

[0,1] a. lies between 1 and 2 b. lies between 2 and 3 c. lies between - 1 and 0 d. Does not exist

68.The function f(x) = tan-1 (sin x + cos x) is an

increasing function in a. ,4 2  : 9 )8 (b.,2 4  : 9 )8 ( c.0,2 : 9 )8 (d.,2 2  : 9 )8 (

69.A value of c for which the conclusion of meanvalue theorem holds for the function f(x) = loge x

on the interval [1 3] is a. 2 log

3 e b. - loge 3

c. log

3 e d. loge 3

70.How many real solutions does the equationx7 + 14x5 + 16x3 + 30x - 560 = 0 have ?

a. 5 b. 7 c. 1 d. 3

71.A function is matched below against an interval,where it is supposed to be increasing. Which ofthe following pair is incorrectly matched?

Function Interval

a. x

3 + 6x2 + 6

( , 4)   b. 3x2 - 2x + 11,3 : ' 98  c. 2x3 - 3x2 - 12x + 6[2, ) d. x3 - 3x2 + 3x + 3( , )  

72.The function f(x) = log(1 + x) 2x

2 x is

increasing on a. ( 1, ) b.( , 0)  c.( , )  d. None of these 73.
2x 2xe 1 f(x) ise 1 : 9 )8 ( a. an increasing function b. a decreasing function c. an even function d. None of these

74.The set of all points for which f(x) = x2 e-x strictly

increases is a. (0, 2) b. (2, ) c. (- 2, 0) d. ( , ) 

75.If f(x) = x3 + bx2 + cx + d and 0 < b2 < c, then in

( , )  a. f(x) is strictly increasing function b. f(x) has a local maxima c. f(x) is strictly decreasing function d. f(x)is bounded

76.The function f(x) = cot-1 x + x increases in the

interval a. (1, )b.( 1, )  c.( , ) d.(0, )

77.If 2

2 2x 1 t xf(x) e . dt, then f(x) increases in a. (- 2, 2) b. no value of x c. (0, )d.( , 0)

252Application of Derivatives

78.The function log( x)f(x) islog(e x)

 a. increasing on (0, ) b. decreasing on (0, ) c. increasing on 0,e : 9 )8 (, decreasing on ,e: 9 )8 ( d. decreasing on 0,e : 9 )8 (, increasing on ,e: 9 )8 (

79.If f(x) = xex(1 - x), then f(x) is

a. increasing in [- 1/2,1] b. decreasing in R c. increasing in R d. decreasing in [- 1/2,1]

80.In which of the following functions, Rolle'stheorem is applicable?a. f (x) = |x| in - 2

 x  2 b. f (x) = tan x in 0  x   c. f(x) = 1 + (x - 2)

2/3 in 1

 x  3 d. f(x) = x (x - 2 )

2 in 0

 x  2

81.The interval of increase of the function

x2f(x) x e tan7 :   9 )8 ( is a. (0, )b.( , 0) c.(1, )d.( , 1)  

82.The function f(x) = (9 - x2)2 increases in

a. ( 3, 0) (3, ) X b.( , 3) (3, )  X  c.( , 3) (0, 3)  Xd. (- 3, 3) d. (3, )

83.If f(x) = sin x/ex in [0, ], then f(x)

a. satisfies Rolle's theorem and c4 , so that f ' 44 : 9 )8 ( b. does not satisfy Rolle's theorem but f ' 04 : 9 )8 ( c. satisfies Rolle's theorem and f ' 04 : 9 )8 ( d. satisfies Lagrange's Mean Value theorem but f ' 04 : g9 )8 (84.Select the correct statement from a., b., c., d..The function f(x) = xe1 - x a. strictly increases in the interval 1, 22 : 9 )8 ( b. increases in the interval (0, ) c. decreases in the interval 0, 2 d. strictly decreases in the interval (1, )

85.The function f(x) = 1 - x3

a. increases everywhere b. decreases in (0, ) c. increases in (0, ) d. None of these

86.If a < 0, the function (eax + e-ax) is a decreasing

function for all values of x, where a. x < 0 b. x > 0 c. x < 1 d. x > 1

87.For a given integer k, in the interval

2 k , 2 k2 2

 7 '   6 5  the graph of sin x is a. increasing from - 1 to 1 b. decreasing from - 1 to 0 c. decreasing from 0 to 1 d. None of these

88.If g(x) = min(x, x2) where x is a real number,

then a. g(x) is an increasing function b. g(x) is a decreasing function c. g(x) is a constant function d. g(x) is a continuous function except at x = 0 e. g(x) is a continuous function except at x = 0 and x = 1

89.Let f be a function defined on [a , b] such thatf

'(x) > 0, x [a, b] R. Then, f is an increasing function on a. (a, b) b. (a, b] c. [a, b] d. [a, b)

90.If

asinx bcosxf(x)csinx dcosx  is decreasing for all x, then a. ad - bc > 0 b. ad - bc < 0 c. ab - cd > 0 d. ab - cd < 0

91.The value of a in order that

f(x) 3sinx cosx 2ax b    decreases for all real values of x, is given by a. a < 1 b. a 1m c.a 2md.a 2

253Application of Derivatives

92.Which of the following statements is/are true?a.

2x log(1 x) x , x (0, )2    R  b.

2xlog(1 x) x , x (0, )2    R 

c.sinx x tanx, x ,2 2  :    R 9 )8 ( d.sinx x tanx, x 0,2 :    R9 )8 (

93.The function f defined by f(x) = (x+ 2)e-x is

a. decreasing for all x b. decreasing on ( , 1)   and increasing in ( 1, )  c. increasing for all x d. decreasing in ( 1, )  and increasing in ( , 1) 

94.If f(x) = (ab - b2 - 2) x +

x4 4

0(cos sin ) d  .

is decreasing function of x for all x

R R and

b

R R, b being independent of x, then

a. a (0, 6)Rb.a ( 6, 6)R  c. a ( 6, 0)R d. None of these

95.The function f(x) = tan x - xa. always increases

b. always decreases c. never decreases d. sometimes increases and sometimes decreases

96.What are the values of c for which Rolle'stheorem for the function f(x) = x3 - 3x2 + 2x in

the interval 0, 2 is verified ? a. c = ± 1 b.

1c 13 o

c. c = ± 2 d. None of these

97.If the function f(x) = cos|x| - 2ax + b increasesalong the entire number scale, the range of valuesof a is given bya.

a bb.ba2 c.1a2 d.3a2 

98.The interval in which the function y = x3 + 5x2 - 1

is a. 100,3
: 9 )8 (b. (0, 10) c.

10, 03

: 9 )8 (d. (2, 9)Maxima & Minima

99.Let f : R r R be defined by

k 2x, if x 1f(x)2x 3, if x 1   43   2 If f has a local minimum at x = - 1, then a possible value of k is a. 1 b. 0 c. 1

2d. - 1

100.The point in the interval [0, 2], where

f(x) = e x sin x has maximum slope, is a. 4 b.2  c.d.3 2 

101.The function f(x) = x3 + ax2 + bx + c, a2  3b

has a. one maximum value b. one minimum value c. no extreme value d. one maximum and one minimum value

102.If

sin(x a)y , a b, theny issin(x b)  g a. minima at x = 0 b. maxima at x = 0 c. Neither minima nor maxima at x = 0 d. None of the above

103.If  is the semi vertical angle of a cone of

maximum volume and given slant height , then tan6 is given by a. 2 b. 1 c. 2d.3

104.Let P(x) = a0 + a1x2 + a2x4 +... + anx2n be a

polynomial in a real variable x with 0 < a

0 < a1 < a2 <...< an. The function P(x) has

a. Neither a maximum nor a minimum b. only one maximum c. only one minimum d. only one maximum and only one minimum

105.The equation of the tangent to the curvey = (2x - 1) e2(1 - x) at the points its maximum, is

a. y - 1 = 0 b. x - 1 = 0 c. x + y - 1 = 0 d. x - y + 1 = 0

106.The number of values of x, wheref(x) = cos x + cos

2x attains its maximum is

a. 1 b. 0 c. 2 d. infinite

254Application of Derivatives

107.Let f(x) = 1+ 2x2 + 22 x4 +...... + 210 x20. Then,

f(x) has a. more than one minimum b. exactly one minimum c. atleast one maximum d. None of the above

108.The maximum value of

2xf(x)

4 x x  on

[- 1,1] is a. 1

3b.1

4

c.1 4d.1 6

109.The largest value of 2x3 - 3x2 - 12x + 5 for

- 2  x  4 occurs at x is equal to a. - 4 b. 0 c. 1 d. 4

110.The minimum value of 2x + 3y, when xy = 6, isa. 9 b. 12c. 8 d. 6

111.The maximum value of xy when x + 2y = 8 isa. 20 b. 16c. 24 d. 8e. 4

112.The minimum value of 2 2(2x 2x 1) sin x

e is  a. 0 b. 1 c. 2 d. 3

113.The greatest value of f(x) = (x + 1)1/3 - (x - 1)1/3

on [0, 1] is a. 0 b. 1 c. 2 d. - 1

114.The function

x 2f(x)2 x  has a local minimum at a. x = - 2 b. x = 0 c. x = 1 d. x = 2

115.Suppose S and S

' are foci the ellipse

2 2x y

125 16 . If P is a variable point on the ellipse

and if  is area of the triangle PSS ', then the maximum value of A is a. 8 b. 12 c. 16 d. 20

116.If x is a real, the maximum value of

2

23x 9x 17

3x 9x 7

    a. 41 b. 1 c. 17 7d.1 4

117.The minimum value of f(x) = sin4 x + cos4 x,

0 x is2

  a. 1

2 2b.1

4 c.1 2 d.1 2

118.Let f, g and h be real valued functions defined onthe interval (0, 1) by f (x) = ex2 + e-x2,

g(x) = xe x2 + e-x2 and h(x) = x2 ex2 + e-x2. If a, b and c denote respectively, the absolute maximum of f, g and h on (0, 1), then a. a = b and c g b b. a = c and a g b c. a g b and c g b d. a = b = c

119.If x = - 1 and x = 2 are extreme points off(x) =  In |x| + x2 + x, then

a.

12,2    b.12,2  

c.16,2    d.16,2     

120.For 5x 0,2

: R9 )8 (, define x

0f(x) t sintdt.. Then, f has

a. local minimum at  and 2 b. local minimum at  and local maximum at 2 c. local maximum at  and local minimum at 2 d. local maximum at  and 2n .

255Application of Derivatives

Exercise 2

(Miscellaneous Problems)

1.The equation of the tangent to the curve

24y xx , that is parallel to the X-axis, is

a. y = 0 b. y = 1 c. y = 2 d. y = 3

2.If the distance s covered by a particle in time f is

proportional to the cube root of its velocity, then the acceleration is a. a constant b. 3sk c.31 skd.5sk e.51 sk

3.OB and OC are two roads enclosing an angle of1200. X and Y start from O at the same time. X

travels along OB with a speed of 4 km/h and Y travels along OC with a speed of 3 km/h. The rate at which the shortest distance between X and Y is increasing after 1 h is a.37 km/hb. 37 km/h c. 13 km/h d.

13km/h

4.A line is drawn through the point (1, 2) to meetthe coordinate axes at P and O such that it formsa OPQ, where, O is the origin, if the area of the

OPO is least, then the slope of the line PO is a. - 1/4 b. - 4 c. - 2 d. - 1/2

5.Given, P(x) = x4 + ax3 + bx2 + cx + d such that

x = 0 is the only real root of P '(x) = 0. If

P (- 1) < P(1), then in the interval [- 1, 1]

a. P (- 1) is the minimum and P(1) is the maximum of P b. P (- 1) is not minimum but P(1) is the maximum of P c. P(- 1) is the minimum and P(1) is not the maximum of P d. Neither P (- 1) is the minimum nor P(1) is the maximum of P

6.Slope of normal to the curve

2

21y xx 

at (- 1, 0) isa. 4 b. 1 4 c. - 4 d.1

4

7.Suppose the cubic x3 - px + q has three distinct

real roots, where p > 0 and q > 0. Then, which one of the following holds ? a. The cubic has maxima at both p pand3 3 b. The cubic has minima at , p

3 and maxima

at p

3

c. The cubic has minima at p

3 and maxima

at p 3 d. The cubic has minima at both p

3 and p

3

8.For

5x 0,2

: R9 )8 (, define x

0f(x) t. sin t dt.

Then, f has

a. local minimum at  and 2 b. local minimum at  and local maximum at 2 c. local maximum at  and local minimum at 2 d. local maximum at  and 2

9.The radius of a cylinder is increasing at the rateof 3 m/s and its altitude is decreasing at the rateof 4 m/s. The rate of change of volume whenradius is 4m and altitude is 6 m, isa. 80 cu m/s b. 144 cu m/s

c. 80 cu m/s d. 64 cu m/s

10.A missile is fired from the ground level rises xmetres vertically upwards in t second, where

225x 100t t2 . The maximum height reached

is a. 200 m b. 125 m c. 160 m d. 190 m e. 300 m

11.If the radius of a circle be increasing at a uniformrate of 2 cm/s. The rate of increasing of area ofcircle, at the instant when the radius is 20 cm, isa. 70 cm2/s b. 70cm2/s

c. 80 cm2/s d. 80 cm2/s

256Application of Derivatives

12.The normal to the curve x = a (cos  +  sin ),

y = a (sin  -  cos ) at any point  is such that a. it is at a constant distance from the origin b. it passes through a, a2 : 9 )8 ( c. it makes angle 2  with the X-axis d. it passes through the origin

13.ST and SN are the lengths of the subtangent andthe subnormal at the point

2  on the curve x = a ( + sin ), y = a (1 - cos ), a g 1, then a. ST - SN b. ST = 2SN c. ST

2 = aSN3d. ST3 = aSN

14.If a and b are positive numbers such that a > b,then the minimum value ofa sec  - b tan 

02 :  9 )8 ( a.2 2 1 a bb.2 2 1 a b c.

2 2a bd.2 2a b

e. a2 - b2

15.The real number x when added to its inverse givesthe minimum value of the sum at x equals toa. 2 b. 1c. - 1 d. - 2

16.Let f be a real-valued function defined on theinterval

(0, ), by f(x) = ln x + x

01 sint dt..

Then, which of the following statement (s) is (are) true? a. f ''(x) exists for all x (0, )R  b. f'(x) exists for all x (0, )R  and f ' is continuous on (0, ), but not differentiate on (0, ) c. there exists  > 1 such that |f'(x) |x| f(x)| for all x ( , )R   d. Both (b) and (c) are correct

17.The value of c in (0, 2) satisfying the Mean Valuetheorem for the function f (x) = x (x - 1)2,

x [0, 2]R is equal to a. 3 4b.4 3 c.1 3d.2 3 e.5 3

18.For what values of x, the functionf(x) = x4 - 4x3 + 4x2 + 40 is monotonic

decreasing? a. 0 < x < 1 b. 1 < x < 2 c. 2 < x < 3 d. 4 < x < 5

19.Tangent is drawn to ellipse

2 2x y 127  at (3 3cos , sin ) [where, (0, /2)]  R . Then, the value of  such that sum of intercepts on axes made by this tangent is minimum, is a./3 b./6 c./8 d./4

20.Let k and K be the minimum and the maximumvalues of the function

0.6

0.6(1 x)f(x)1 x

 in [0,1] respectively, then the ordered pair (k, K) is equal to a. (2 -0.4, 1) b. (2-0.4, 20.6) c. (2 -0.6, 1) d. (1, 20.6)

21.If for a function f(x), f

'(a) = 0, f''(a) = 0, f'''(a) > 0, then at x = a, f(x) is a. minimum b. maximum c. not an extreme point d. extreme point e. None of these

22.For the curve xy = c2 the subnormal at any point

varies as a. x

3b. x2

c. y 3d. 

23.If x - 2y = 4, the minimum value of xy isa. - 2 b. 0c. 0 d. - 3

24.The maximum value of

logx x is a. e b. 2e c. 1 ed.2 e

25.If sum of two numbers is 6, the minimum valueof the sum of their reciprocals isa.

6 5b.3 4 c.2 3d.1 2

26.If f(x) = 2x3 - 21x2 + 36x - 30, then which one of

the following is correct ?

257Application of Derivatives

a. f(x) has minimum at x = 1 b. f(x) has maximum at x = 6 c. f(x) has maximum at x = 1 d. f(x) has no maxima of minima

27.The function x5 - 5x4 + 5x3 - 1 is

a. Neither maximum nor minimum at x = 0 b. maximum at x = 0 c. maximum at x = 1 and minimum at x = 3 d. minimum at x = 0

28.If there is an error of k % in measuring the edgeof a cube, then the per cent error in estimating itsvolume isa. k b. 3kc.

k

3d. None of these

29.A point of the parabola y2 = 18x at which the

ordinate increases at twice the rate of the abscissa is a. (2, 4) b. (2, - 4) c.

9 9,8 2

: 9 )8 (d.9 9,8 2 : 9 )8 (

30.The tangent to the curve y = ex drawn at the

point (c, e c) interects the line joining the points (c - 1, e c - 1) and (c + , ec + 1) a. on the left of x = c b. on the right of x = c c. at no point d. at all points

31.Angle between the tangents to the curvey = x2 - 5x + 6 at the points (2, 0) and (3, 0), is

a. 2 b.6  c.4 d.3 

32.If 4x2 + py2 = 45 and x2 - 4y2 = 5 cut orthogonally,

then the value of p isd a. 9 b. 1 3 c. 3 d. 18

33.If (a, a2) falls inside the angle made by the lines

xy2, x > 0 and y = 3x, x > 0, then a belongs to a. (3, )b.1, 32 : 9 )8 ( c.13,2 : 9 )8 (d.10,2 : 9 )8 (

34.A triangular park is enclosed on two sides by afence and on the third side by a straight river

bank. The two sides having fence are of same length x. The maximum area enclosed by the park is a. 3x

8b.21x2

c.x2d.23x2

35.Area of the greatest rectangle that can beinscribed in the ellipse

2 2

2 2x y

1isa b 

a. a bb.ab c. ab d. 2ab

36.The function f defind by f(x) = 4x4 - 2x +1 is

increasing for a. x < 1 b. x > 0 c.

1x2d.1x2

37.The second degree polynomial f(x), satisfyingf(0) = 0, f(1) = 1, f

' (x) > 0 for all x R (0, 1), is a. f(x)  b. f(x) = ax + (1 - a)x2, a (0, ) R  c. f(x) = ax + (1 - a)x2, a (0, 2)R d. No such polynomial

38.The greatest value of f(x) = (x +1)1/3 - (x - 1)1/3

on, 0, 1 is a. 1 b. 2 c. 3 d. 1 3

39.A stone is dropped into a quiet lake and wavesmove in circles at the speed of 5 cm/s. At thatinstant, when the radius of circular wave is 8 cm,how far is the enclosed area increasing?a. 6 cm2/s b. 8 cm2/s

c.

28cm /s3d. 80 cm2/s

40.If the line ax + by + c = 0 is a normal to the curvexy = 1 thena. a > 0, b > 0 b. a > 0, b < 0c. a < 0, b < 0 d. Data is insufficient

41.If the function f(x) = 2x3 - 9ax2 +12a2 x +1,

where a > 0, attains its maximum and minimum at p and q respectively such that p

2 = q, then a

equals to

258Application of Derivatives

a. 3 b. 2 c.1 d. 1 2

42.For the curve yn = an-1 x if the subnormal at any

point is a constant, then n is equal to a. 1 b. 2 c. - 2 d. - 1

43.If a2 x4 + b2 y4 = c, then maximum value of xy is

a. 2c abb. 3c ab c. 3c 2abd. 3c 2ab

44.The length of the subtangent to the curvex2 + xy + y2 = 7at (1 - 3) is

a. 3 b. 5 c. 3

5d. 15

e. 4

45.The normal to the curve x2 + 2xy - 3y2 = 0 at

(1, 1) a. does not meet the curve again b. meets the curve again in the second quadrant c. meets the curve again in the third quadrant d. meets the curve again in the fourth quadrant

46.The point on the curve y = 2x2 - 6x - 4 at which

the tangent is parallel to the X-axis, is a.

3 13,2 2

: 9 )8 (b.5 17,2 2 :  9 )8 ( c.3 17,2 2 : 9 )8 (d. (0, - 4) e.

3 17,2 2

: 9 )8 (

47.If  is the angle between the curves xy = - 2 and

x

2 + 4y = 0, then tan  is equal to

a. 1 b. - 1 c. 2 d. 3

48.The tangent and the normal drawn to the curvey = x2 - x + 4 at P(1, 4) cut the X-axis at A and

B respectively. If the length of the subtangent

drawn to the curve at P is equal to the length of the subnormal, then the area of the triangle PAB (in sq units) is a. 4 b. 32 c. 8 d. 16

49.If the curves x2 = 9A (9 - y) and x2 = A(y + 1)

intersect orthogonally, then the value of A isa. 3 b. 4c. 5 d. 7e. 9

50.The perimeter of a sector is a constant. If itsarea is to be maximum, the sectorial angle isa.

rad6 b.rad4  c. 4 rad d. 2 rad

51.If x = t2 and y = 2t, then equation of the normal at

t = 1 is a. x + y - 3 = 0 b. x + y - 1 = 0 c. x + y + 1 = 0 d. x + y + 3 = 0

52.The equation of the tangent to the curvey = (1 + x)y + sin-1 (sin2 x) at x = 0 is

a. x - y + 1 = 0 b. x + y + 1 = 0 c. 2x - y + 1 = 0 d. x + 2y + 2 = 0 e. 2x + y - 1 = 0

53.Let f be a real-valued function defined on theinterval (- 1, 1) such that

xx 4

2e f(x) 2 t 1  . dt, for all x R (- 1, 1)

and let f -1 be the inverse function of t. Then, (f -1) '(2) is equal to a. 1 b. 1 3 c.1 2d.1 e

54.The radius of a cylinder is increasing at the rateof 5 cm/min, so that its volume is constant. Whenits radius is 5 cm and height is 3 cm, then the rateof decreasing of its height isa. 6 cm/min b. 3 cm/minc. 4 cm/min d. 5 cm/mine. 2 cm/min

55.If f(x) is differentiable and strictly increasingfunction, then the value of

2 x 0f(x ) f(x)lim isf(x) f(0)r  a. 1 b. 0 c. - 1 d. 2

56.The tangent at (1, 7) to the curve x2 = y - 6

touches the circle x

2 + y 2 + 16x + 12y + c = 0 at

a. (6, 7) b. (- 6, 7) c. (6, - 7) d. (- 6, - 7)

57.Let the function

g:( , ) ,2 2  :   r 9 )8 ( be given by g(u) = 2tan -1(eu) - 2 . Then, g is

259Application of Derivatives

a. even and is strictly increasing in (0, ) b. odd and is strictly decreasing in ( , )  c. odd and is strictly increasing in ( , )  d. Neither even nor odd, but is strictly increasing in ( , ) 

58.The normal to the curve x = a (cos  +  sin ),

y = a (sin  -  cos ) at any point  is such that a. it makes a constant angle with the X-axis b. it passes through the origin c. it is at a constant distance from the origin d. None of the above

59.For the function

1f(x) x , x [1, 3]x  R, the

value of c for Mean Value theorem is a. 1 b. 3 c. 2 d. None of these

60.The value of c in Rolle's theorem for the functionf(x) = x3 - 3x in the interval 0,

3 is a. 1 b. - 1 c. 3 2d.1 3

61.The minimum value of 2x + 3y, when xy = 6, isa. 12 b. 9c. 8 d. 6

62.The function f(x) = 2x3 - 15x2 + 36x + 4 is

maximum at a. x = 2 b. x = 4 c. x = 0 d. x = 3

63.The chord joining the points where x = p andx = q on the curve y = ax2 + bx + c is parallel to

the tangent at the point on the curve whose abscissa is a. p q 2 b.p q 2  c.pq

2d. None of these

64.The function f(x) = x

x , (x R)R attains a maximum value at x which is a. 2 b. 3 c. 1 ed. 1

65.In (0,1) Lagrange's mean value theorem is NOTapplicable toa.

2

1 1x x2 2f(x)1 1x x2 2

4 113: 1

 m9 )18 (2 b. sinxx 0f(x)x 1 x 0

4g131

2 c. f(x) = x |x| d. f(x) = |x|

66.Maximum slope of the curve

y = - x

3 + 3x2 + 9x - 27 is

a. 0 b. 12 c. 16 d. 32

67.The local minimum value of the function f

' given by f(x) = 3 + |x|, x RR is a. - 1 b. 3 c. 1 d. 0

68.The fuel charges for running a train are

proportional to the square of the speed generated in mil/h and costs Rs. 48 per h at 16 mile/h. The most economical speed if the fixed charges i.e. salaries etc amount to Rs. 300 per h a. 10 mile/h b. 20 mile/h c. 30 mile/h d. 40 mile/h

69.The function f(x) = sin4 x + cos4 x increases, if

a. 0 x8  b.3x4 8    c.

3 5x8 8

  d.5 3x8 4   

70.For all

x (0,1)R a. ex < 1 + x b. loge (1 + x) < x c. sin x > x d. log e x > x - 1

71.A physical quantity A is related to four observablea, b, c and d as follows,

2 3a bAc d, the percentage

errors of measurement in a, b, c and d are

1 %, 3 %, 2 % and 2 % respectively. The

maximum error in the quantity A is equal to a. 5 % b. 7 % c. 12 % d. 14 %

72.If f

'(x) = |x| - {x}, where {x} denotes the fractional part of x, then f(x) is decreasing in a. 1, 02 : 9 )8 (b.1, 22 : 9 )8 ( c.1, 22 : '98 d.1,2 : 9 )8 (

260Application of Derivatives

73.If 2bax c, x 0x m  , where a > 0, b > 0 then

a. 27ab 2 m 4c3b. 27ab3  4c3 c. ab 2 m c3d. ab3  c3

74.If f(x) = xa In x and f(0) = 0, then the value of 

for which Rolle's theorem can be applied in [0,1] is a. - 2 b. - 1 c. 0 d. 1 2

75.The value of a, so that the sum of the squares ofthe roots of the equation x2 - (a - 2) x - a + 1 = 0

assume the least value is a. 2 b. 1 c. 3 d. 0

76.If P = (1, 1), Q = (3, 2) and R is a point on X-axis,then the value of PR + RO will be minimum ata.

5, 03 : 9 )8 (b.1, 03 : 9 )8 ( c. (3, 0) d. (1, 0)

77.The time T of oscillation of a simple pendulum oflength l is given by

T 2g l. The percentage

error in T corresponding to an error of 2% in the value of l is a. 2 % b. 1 % c. 3 % d. 1.2 %78.Oil is leaking at the rate of 16 cm3 /s from a vertically kept cylindrical drum containing oil. If the radius of the drum is 7 cm and its height is

60 cm. Then, the rate at which the level of the oil

is changing when oil level is 18 cm, is a. 16 49
 b.16 48
  c.16

49d.16

47
 

79.The points on the curve y = x3 - x2 - x + 3,

where the tangents are parallel to the X-axis, are a.

1 88, and (1, 2)3 27

: 9 )8 ( b.

1 86, and (1, 2)3 27: 9 )8 (

c.

1 86, and ( 1, 2)3 27

:  9 )8 ( d.

1 88, and ( 1, 2)3 27

: 9 )8 (

80.The maximum and minimum values off(x) = sec x + log cos2 x, 0 < x < 2n are

respectively a. (1, - 1) and 2 (1 - log 2), 2 (1 + log 2) b. (1, - 1) and {2 (1 - log 2), 2 (1 - log 2)} c. (1, - 1) and (2, - 3) d. None of the above

261Application of Derivatives

MHT-CET Corner

1.All the points on the curve y2 = 4a |x + a sinx |a |,

where the tangent is parallel to the axis of X are lies on a. circle b. parabola c. straight line d. None of these

2.The length of normal at any point to the curve,

xy cosh isc : 9 )8 ( a. fixed b. 2 2y c c. 2y cd.2y c

3.The height of right circular cylinder of maximumvolume inscribed in a sphere of diameter 2a isa.

2 3ab.3a

c.2a 3d.a 3

4.For all real x, the minimum value of

2

21 x x

1 x x     is a. 0 b. 1/3 c. 1 d. 3

5.If x + y = k is normal to y2 = 12x, then k is equal

to a. 3 b. 9 c. - 9 d. - 3

6.A particle moves along a straight line accordingto the law s = 16 - 2t + 3t3 , where s metres is the

distance of the particle from a fixed point at the end of t second. The acceleration of the particle at the end of 2s is 2011 a. 3.6 m/s

2b. 36 m/s2

c. 36 km/s

2d. 360 m/s2

7.The equation of tangent to the curve y2 = ax2 + b

at point (2, 3) is y = 4x - 5, then the values of a and b are a. 3, - 5 b. 6, - 5 c. 6, 15 d. 6, - 15

8.The equation of tangent to the curve given byx = 3 cos , y = 3 sin  at

4  is a. x y 2 b.3x y 3 2  c. x y 3 2 d.x 3y 3 2 

9.The equation of motion of a particle moving alonga straight line is s = 2t3 - 9t2 +12t, where the

units of s and t are centimetre and second. The acceleration of the particle will be zero after a.

3s2b.2s3

c.1s2d. 1s

10.The equation of the tangent to the curve y = 4xex

at 41,e
: 9 )8 ( is a. y = - 1 b.

4ye 

c. x = - 1 d.4ye 

11.The abscissa of the points, where the tangent tocurve y = x3 - 3x2 - 9x + 5 is parallel to X-axis,

are a. x = 0 and 0 b. x = 1 and -1 c. x = 1 and - 3 d. x = - 1 and 3

12.The point of the curve y2 = 2(x - 3) at which the

normal is parallel to the line y - 2x +1 = 0 is a. (5, 2) b. 1, 22 :  9 )8 ( c. (5, - 2) d.3, 22 : 9 )8 (

13.The maximum area of the rectangle that can beinscribed in a circle of radius r, isa. r2b. r2

c. r2/4 d. 2r2

14.If the function f(x) = 2x3 - 9ax2 + 12a2x +1

attains its maximum and minimum at p and q respectively such that p

2 = q, then a equals to

a. 0 b. 1 c. 2 d. None of these

15.If f(x) = kx - sin x is monotonically increasing,thena. k > l b. k > - lc. k < 1 d. k < - l

16.If a particle moves such that the displacement is

proportional to the square of the velocity acquired, then its acceleration is a. proportional to s

2b. proportional to

21
s c. proportional to 1 sd. a constant

262Application of Derivatives

17.The f(x) = tan-1 (sin x + cos x), x > 0 is always

an increasing funciton on the interval a. (0, ) b. 0,2 :

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