8_applications of derivatives.pdf

8/10/2019 8_Applications of derivatives.pdf

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:.6 Jdrcvftcvd fs f rftd ndfsurd

Cl y > l(x) cs f lubetc`b l x, tmdbjx

jy r l (x) rdprdsdbts tmd rftd-ndfsurd l y wctm rdspdet t` x.

(f) rdprdsdbts)x('l`rjx

jy

7xx 7>

tmd rftd `l emfbkd `l y w.r.t. x ft x > x7.

(h) Cl y cberdfsds fs x cberdfsds, tmdbjx

jycs p`sctcvd fbj cl y jderdfsds fs x cberdfsds, tmdb

jx

jycs bdkftcvd.

(e) Nfrkcbfo E`st (NE) cs tmd cbstfbtfbd us rftd l emfbkd l t tfo e st wctm rdspdet t tmd bunhdr l ctdns

pr`juedj ft fb cbstfbt.

(j) Nfrkcbfo Xdvdbud (NX) cs tmd cbstfbtfbd us rftd l emfbkd l t`tfo rdvdbud wctm rdspdet t` tmd bunhdr

`l ctdns sòj ft fb cbstfbt.

:.8 Zfbkdbts fbj b`rnfo

(f) Z` lcbj tmd dquftc b l tmd tfbkdbt t` tmd eurvd y > l(x) ft tmd kcvdb p cbt W(x6, y

6), prèddj fs ubjdr 1

(c) Lcbjjx

jylr`n tmd kcvdb dquftc`b y > l(x).

(cc) Lcbj tmd vfoud ljx

jyft tmd kcvdb p`cbt W (x

6, y

6), odt n >

6

6

yy

xxjx

jy

>>

.

(ccc) Zmd dquftc`b l tmd rdqucrdj tfbkdbt cs y y6> n (x x

6).

Cb pfrtceuofr, cl >

6

6

yy

xxjx

jy

>>

j`ds b`t dxcst, tmdb tmd dquftc`b l tmd tfbkdbt cs x > x

6.

(h) Z` lcbj tmd dquftc b l tmd b rnfo t` tmd eurvd y > l(x) ft tmd kcvdb p`cbt W(x6, y

6), prèddj fs ubjdr 1

(c) Lcbjjx

jylr`n tmd kcvdb dquftc`b y > l(x).

(cc) Lcbj tmd vfoud ljx

jyft tmd kcvdb p`cbt W(x

6, y

6).

(ccc) Cl n cs tmd so pd l tmd b rnfo t` tmd kcvdb eurvd ft W, tmdb n >

6

6

yy

xxjx

jy

6

>>

.

(cv) Zmd dquftc`b l tmd rdqucrdj b`rnfo cs y y6> n (x x

6).

Cb pfrtceuofr cl,

6

6

yy

xxjx

jy

>>

> 7, tmdb tmd dquftc`b `l tmd b`rnfo ft W cs x > x

69 fbj cl >

6

6

yy

xxjx

jy

>>

j`ds b`t dxcst, tmdb tmd dquftc`b `l tmd b`rnfo ft W cs y > y6.

Fbkod l cbtdrsdetc`b l tw` eurvds

Zmd fbkod l cbtdrsdetc`b l tw` eurvds cs tmd fbkod hdtwddb tmd tfbkdbts t` tmd tw` eurvds ft tmdcr p`cbt

`l cbtdrsdetc`b.

Cl n6fbj n8frd tmd so`pds l tmd tfbkdbts t` tmd kcvdb eurvds ft tmdcr p`cbt l cbtdrsdetc`b W(x6, y6), tmdb

tmd (feutd) fbkod qhdtwddb tmd eurvds cs kcvdb hy


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tfb q>86

86

nn6

nn

+-

.

Cb pfrtceuofr 1

(c) cl n6n

8> 6, tmdb eurvds frd sfcj t` eut rtm`k`bfooy.

(cc) cl n6> n8, tmd eurvds tùem dfem `tmdr.

:.5 Cberdfscbk fbj jderdfscbk lubetc bs

Cl l cs f rdfo vfoudj lubetc`b jdlcbdj cb fb cbtdrvfo J ( f suhsdt l X), tmdb l cs efoodj fb cberdfscbk lubetc`b

cb fb cbtdrvfo J6(f suhsdt `l J) cll l`r foo x

6, x

8J

6, x

6? x

8 l(x

6) l(x

8) fbj l cs efoodj f strcetoy

cberdfscbk lubetc`b cb J6cll l`r foo x

6, x

8J

6, x

6? x

8 l(x

6) ? l(x

8).

Ucncofroy, l cs efoodj f jderdfscbk lubetc`b cb fb cbtdrvfo J8(f suhsdt l J) cll l`r foo x

6, x

8J

8, x

6? x

8

l(x6)l(x

8) fbj l cs efoodj f strcetoy jderdfscbk lubetc b cb J

8cll l r foo x

6, x

8J

8, x

6? x

8l(x

6) 2 l(x

8).

Cb pfrtceuofr, cl J6> J tmdb l cs efoodj fb cberdfscbk lubetc`b cll l`r foo x

6, x

8J, x

6? x

8

l(x6) l(x8)9 fbj l cs efoodj strcetoy cberdfscbk lubetc`b cll l`r foo x6, x8J, x6? x8 l(x

6) ? l(x

8). Fbfo kùsoy, cl J

8> J tmdb l cs efoodj f jderdfscbk lubetc`b cll l`r foo x

6, x

8J, x

6

? x8 l(x

6) l(x

8)9 fbj l cs efoodj strcetoy jderdfscbk lubetc`b cll l`r foo x

6, x

8J, x

6? x

8 l(x

6) 2

l(x8).

F lubetc`b wmcem cs dctmdr (strcetoy) cberdfscbk `r (strcetoy) jderdfscbk cs efoodj f (strcetoy) n`b`t`bce

lubetc b.

E`bjctc`bs l`r fb cberdfscbk r f jderdfscbk lubetc`b

Zmd`rdn 6. Cl f lubetc`b l cs e`btcbuùs cb Yf, h^, fbj jdrcvfhod cb (f, h) fbj

(c) l (x) 7 l`r foo x (f, h), tmdb l cs cberdfscbk cb Yf, h^

(cc) l (x) 2 7 l`r foo x (f, h), tmdb l cs strcetoy cberdfscbk cb Yf, h^.

Zmd`rdn 8. Cl f lubetc`b l cs e`btcbuùs cb Yf, h^, fbj jdrcvfhod cb (f, h) fbj

(c) l (x) 7 l`r foo x (f, h), tmdb l cs jderdfscbk cb Yf, h^

(cc) l (x) ? 7 l r foo x (f, h), tmdb l cs strcetoy jderdfscbk cb Yf, h^.

Cb lfet, cl f lubetc`b l(x) cs e`btcbuùs cb Yf, h^, jdrcvfhod cb (f, h) fbj

(c) l (x) 2 7 l r foo x(f, h) dxedpt l`r f lcbctd bunhdr l p`cbts wmdrd l (x) > 7, tmdb l(x) cs strcetoy

cberdfscbk cb Yf, h^.

(cc) l (x) ? 7 l r foo x(f, h) dxedpt l`r f lcbctd bunhdr l p`cbts wmdrd l (x) > 7, tmdb l(x) cs strcetoyjderdfscbk cb Yf, h^.

:.: Nfxcnf fbj Ncbcnf

Cl l cs f rdfo vfoudj lubetc`b jdlcbdj b J (suhsdt l X) tmdb

(c) l cs sfcj t` mfvd f nfxcnun vfoud cb J, cl tmdrd dxcsts f p`cbt x > e cb J suem tmft l(e) l(x)

l`r foo x J. Zmd bunhdr l(e) cs efoodj tmd (fhsòutd) nfxcnun vfoud `l l cb J fbj tmd p`cbt e cs

efoodj p`cbt l nfxcnf l l cb J.

(cc) l cs sfcj t` mfvd f ncbcnun vfoud cb J, cl tmdrd dxcsts f p cbt x > j cb J suem tmft l(j) l(x) l`r foo

x J. Zmd bunhdr l(j) cs efoodj tmd (fhsòutd) ncbcnun vfoud l l cb J fbj tmd p`cbt j cs efoodjp cbt l ncbcnf l l cb J.


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(f) O efo nfxcnf fbj oèfo ncbcnf

Cl l cs f rdfo vfoudj lubetc`b jdlcbdj b J (suhsdt l X), tmdb

(c) l cs sfcj t` mfvd f o efo (`r rdoftcvd) nfxcnf ft x > e (cb J) cll tmdrd dxcsts f p`sctcvd rdfo bunhdr

jsuem tmft l(e)l(x) l`r foo x cb (e j, e +j) c.d. l(e)l(x) l`r foo x cb tmd cnndjcftd bdckmhùrm``j`l e, fbj e cs efoodj p`cbt l oèfo nfxcnf fbj l(e) cs efoodj oèfo nfxcnun vfoud.

(cc) l cs sfcj t` mfvd o efo (`r rdoftcvd) ncbcnf ft x > j(cb J) cll tmdrd dxcsts s`nd p sctcvd rdfo bunhdrjsuem tmft l(j) l(x) l`r foo x (j j, j + j) c.d. l(j)l(x) l`r foo x cb tmd cnndjcftd bdckmhùrm``j`l j, fbj j cs efoodj p`cbt l oèfo ncbcnf fbj l(j) cs efoodj oèfo ncbcnun vfoud.

Kd`ndtrcefooy, f p`cbt e cb tmd j`nfcb l tmd kcvdb lubetc`b l cs f p`cbt l oèfo nfxcnf r oèfo

ncbcnf fee`rjcbk fs tmd krfpm l l mfs f pdfi r trùkm (efvcty) ft e.

(ccc) f p`cbt (cb J) wmcem cs dctmdr f p cbt l o efo nfxcnf r f p`cbt `l oèfo ncbcnf cs efoodj fb

dxtrdnd p`cbt, fbj tmd vfoud l tmd lubetc`b ft tmcs p`cbt cs efoodj fb dxtrdnd vfoud.

(h) Erctcefo ( r turbcbk) p`cbt

Cl l cs f rdfo vfoudj lubetc`b jdlcbdj `b J (suhsdt `l X), tmdb f p`cbt e (cb J) cs efoodj f erctcefo

(`r turbcbk r stftc`bfry) p`cbt l l cll l cs jclldrdbtcfhod ft x > e fbj l (e) > 7.

(e) W`cbt l cblodetc`b

F p`cbt W(e, l(e)) b tmd eurvd y > l(x) cs efoodj f p`cbt l cblodetc`b cll b bd scjd l W tmd eurvd ocds

hdo w tmd tfbkdbt ft W fbj b tmd tmdr scjd ct ocds fh`vd tmd tfbkdbt ft W. Zmus, f p`cbt wmdrd tmd eurvd

er`ssds tmd tfbkdbt cs efoodj f p`cbt l cblodetc`b.

Zmd`rdn. F lubetc`b l (`r tmd eurvd y > l(x)) mfs f p`cbt l cblodetc b ft x > e cll l (e) > 7, l (e) > 7 fbj

l (e) 7.

(j) [`ricbk ruods l`r lcbjcbk (fhs outd) nfxcnun fbj ncbcnun

Cl f lubetc`b l cs jclldrdbtcfhod cb Yf, h^ dxedpt (p`sschoy) ft lcbctdoy nfby p`cbts, tmdb t` lcbj (fhsòutd)

nfxcnun fbj ncbcnun vfouds fj`pt tmd lòo`wcbk prèdjurd 1

(c) Dvfouftd l(x) ft tmd p cbts wmdrd l (x) > 7

(cc) Dvfouftd l(x) ft tmd p cbts wmdrd jdrcvftcvd lfcos t` dxcst.

(ccc) Lcbj l(f) fbj l(h).

Zmdb tmd nfxcnun l tmdsd vfouds cs tmd fhs outd nfxcnun fbj ncbcnun l tmdsd vfouds cs tmd fhs outd

ncbcnun l tmd kcvdb lubetc`b l.

(d) [`ricbk ruods t` lcbj p`cbts l o efo nfxcnf fbj ncbcnf

6. Oèftd tmd p cbts wmdrd tmd kcvdb lubetc`b l cs ocidoy t` mfvd dxtrdnd vfouds 1

(c) Zmd p cbts wmdrd tmd jdrcvftcvd lfcos t dxcst.

(cc) Zmd turbcbk (erctcefo) p cbts c.d. tmd p`cbts wmdrd tmd jdrcvftcvd cs zdr .

(ccc) Dbj p`cbts l tmd j`nfcb cl l cs jdlcbdj cb f eo sdj cbtdrvfo Yf, h^.

Zmdsd frd tmd boy p`cbts wmdrd l nfy mfvd dxtrdnd vfouds, odt e hd fby bd suem p`cbt.

8. Cl l (e) j`ds b t dxcst hut l dxcsts cb bdckmhùrm` j l e, tmdb tmd l oo`wcbk tfhod jdserchds tmd

hdmfvc ur l tmd lubetc b l ft e 1

x sockmtoy ? e sockmtoy 2 e Bfturd l p`cbt

l '(x) + vd vd Nfxcnf

l '(x) vd + vd Ncbcnf


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5. Cl e cs f turbcbk p cbt c.d. l (e) dxcsts fbj l (e) > 7. Odt b8 hd tmd snfoodst p`sctcvd cbtdkdr suem

tmft l(b)(e) 7, tmdb tmd lòo`wcbk tfhod jdserchds tmd hdmfvcùr l tmd lubetc`b l ft e 1

b sckb `l l(b)(e) Bfturd `l tmd turbcbk p`cbt e

`jj + vd r vd Bdctmdr nfxcnf b`r ncbcnf

dvdb + vd Ncbcnfdvdb vd Nfxcnf

Fotdrbftcvdoy, cl ct cs jcllceuot t` lcbj tmd jdrcvftcvds l mckmdr rjdr tmdb tmd lòo`wcbk tfhod jd-

serchds tmd hdmfvcùr l tmd lubetc b l ft e 1

x sockmtoy ? e sockmtoy 2 e Bfturd l tmd turbcbk p`cbt

l '(x) + vd vd Nfxcnf

l '(x) vd + vd Ncbcnf

l '(x) + vd + vd Bdctmdr nfxcnf b`r ncbcnf

l '(x) vd vd Bdctmdr nfxcnf b`r ncbcnf

:. Zmd l oo`wcbk tfhods jdserchd tmd hdmfvc ur ft tmd dbj p cbts 9

Odt dbj p`cbt f.

x sockmtoy 2 f Bfturd l p`cbt

l '(x) + vd Ncbcnf

l '(x) vd Nfxcnf

Xckmt dbj p`cbt h.

x sockmtoy ? h Bfturd l p`cbt

l '(x) + vd Nfxcnf

l '(x) vd Ncbcnf

4. Lcbfooy, t` lcbj tmd fhs outd nfxcnun r fhsòutd ncbcnun l f lubetc`b l cb eo`sdj cbtdrvfo Yf, h^

9 lcbj tmd vfouds `l l ft tmd p cbts `l foo tmd tmrdd eftdk`rcds ocstdj fh`vd. Zmd nfxcnun `l tmdsd vfouds cs

tmd fhs outd nfxcnun fbj tmd ncbcnun l tmdsd vfouds cs tmd fhs outd ncbcnun l tmd lubetc b l cb Yf, h .

(l) L`r prfetcefo pr hodns b nfxcnf fbj ncbcnf

Cl f lubetc`b cs e`btcbuùs cb fb pdb cbtdrvfo (f, h) fbj ct mfs boy bd dxtrdnd p`cbt cb (f, h), tmdb ct csf p`cbt l fhsòutd nfxcnf r f p`cbt l fhsòutd ncbcnf fee`rjcbk fs ct cs f p`cbt l o efo nfxcnf r f

p cbt l o efo ncbcnf.

:.4 Fppr`xcnftc bs

Odt y > l(x) hd f lubetc`b `l x fbj jx hd f snfoo emfbkd cb tmd vfoud l x fbj jy hd tmd e`rrdsp`bjcbk

emfbkd cb tmd vfoud l y, tmdb jx >jx

jyjx.

* U`nd cnp`rtfbt tdrns

Fhsòutd drr`r.Zmd cberdndbtjx cb x cs efoodj tmd fhsòutd drr`r cb x.


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Dx.6 Zmd rfjcus `l fb fcr huhhod cs cberdfscbk ft tmd rftd `l8

6en/s. Ft wmft rf td cs tmd vòund

`l tmd huhhod cberdfscbk wmdb tmd rfjcus cs 6 en 0

Uò. Odt r en hd tmd rfjcus `l fb fcr huhhod fbj \ en 5hd cts vòund. Zmdb,

\ >5

: pr5 ..........(c)

Ct cs kcvdb tmftjt

jr>

8

6en/s

B`w, lr`n (c), wd mfvd

jt

j\> :pr8

jt

jr> :pr8

8

6> 8pr8

[mdb r > 6 en,

jt

j\> 8p(6)8> 8pen5/s

Zmus, tmd rftd l emfbkd l v ounds l tmd fcr huhhod cs 8pen5/s.

Dx.8 F hfoo``b, wmcem fowfys rdnfcbs spmdrcefo `b cbloftc`b, cs hdcbk cbloftdj hy punpcbk cb 377

euhce edbtcndtdrs `l kfs pdr sde`bj. Lcbj tmd rftd ft wmcem tmd rfjcus `l tmd hfoo``b

cberdfsds wmdb tmd rfjcus cs 64 en.

Uò. Odt r en jdb`td tmd rfjcus l tmd spmdrcefo hfoo``b fbj \ hd cts vòund. Zmdb,

\ >5

:pr5 ........(c)

Ct cs kcvdb tmft jt

j\> 377 en5/s

Lr`n (c), wd mfvd

jt

j\> :pr8

jt

jr

jt

jr> 8r:

377

p> 8r

884

p

[mdb r > 64 en,

jt

jr> 8)64(

884

p>

p6

>88

;en/s

Zmus, tmd rftd ft wmcem tmd rfjcus l tmd hfoo` b cberdfsds cs88

;en/s, wmdb r > 64 en.

Dx .5 Lcbj tmd odfs t vf oud `l f suem tmft tmd lube tc`b l kcvdb hy l(x) > x8+ fx + 6 cs strcetoy

cberdfscbk cb (6, 8).

Uò. Mdrd, l (x) > 8x + f

B`w, 6 ? x ? 8 8 ? 8x ? :

8 + f ? 8x + f ? : + f ....(6)

B`w, l(x) cs fb cberdfscbk lubetc b, s` l (x) 2 7 l r x(6, 8)

8x + f 2 7 l`r x (6, 8) 7 ? 8x + f l`r x (6, 8)

U@O\DJ WX@HODUN


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FWWOCEFZC@BU @L JDXC\FZC\DU x8 x + 6 cs bdctmdr strcetoy cberdfscbk b`r strcetoy

jderdfscbk cb (6, 6).

U`o. [d mfvd, l(x) > x8 x + 6

l (x) > 8x 6

Zmd erctcefo vfoud l l(x) cs8

6

Efsd 6[mdb 6 ? x ?8

6

Cb tmcs efsd, 6?x?8

6 5?8x 6?7

l (x) ? 7

Zmus, l(x) cs jderdfscbk l`r 6 ? x ?8

6

Efsd 8 [mdb8

6? x ? 6

Cb tmcs efsd,8

6? x ? 6 6 ? 8x ? 8 7 ? 8x 6 ? 6

Mdbed, l (x) 2 7

U`, l(x) cs cberdfscbk l`r8

6? x ? 6

U`, cb tmd cbtdrvfo (6, 6), l(x) cs bdctmdr cberdfscbk b`r jderdfscbk.

Dx.4 Lcbj tmd p`cbts `b tmd eurvd y > x5ft wmcem tmd so`pd `l tmd tfbkdbt cs dqufo t` tmd y-

e``rjcbftd `l tmd p`cbt.

U`o. Odt tmd p`cbt l tfbkdbey hd (x6, y

6).

Zmd, so`pd l tmd tfbkdbt ft (x6, y

6) cs kcvdb Y ^

6

6

xx8

xx

x5cswmcem,jx

jyhy >

>

c.d., 86x5 .

Ucbed, tmd so`pd cs kcvdb t` hd dqufo t` y6,wd mfvd

86x5 > y6 ......(6)

Fs (x6, y

6) ocds `b tmd eurvd y > x5, wd mfvd

y6> 56x .......(8)

Lr`n (6) fbj (8), wd mfvd

86x5 >

56x

x6> 7, 5 fbj s y

6> 75> 7 r y

6> 55> 8;

Zmus, tmd rdqucrdj p`cbts frd (7, 7) fbj (5, 8;).


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Dx.< Lcbj tmd dquftc`b l tmd b`rnfo ft tmd p`cbt (fn8, fn5) l`r tmd eurvd fy8> x5.

Uò. Uo`pd l tmd tfbkdbt t` tmd eurvd fy8> x5ft tmd p`cbt (fn8, fn5) cs kcvdb hy

)fn,fn(

58jx

jy

Kcvdb fy8> x5

[d mfvd, 8fyjx

jy> 5x8

jx

jy>

fy8

x5 8

)fn.fn(

58jx

jy

>

f8

55

:8

fn

nf>

8

n5

_Uo pd l tmd b rnfo t tmd eurvd ft (fn8, fn5) cs n5

8.

Mdbed, tmd dquftc`b l tmd b`rnfo ft tmd p`cbt (fn8, fn5) t` tmd eurvd fy8> x5cs

y fn5> n5

8(x fn8)

c.d., 5ny 5fn:> 8x + 8fn8

c.d., 8x + 5ny fn8(8 + 5n8) > 7

Dx.; F ecreuofr ndtfo poftd dxpfbjs ubjdr mdftcbk s` tmft cts rfjcus cberdfsds hy 8%. Lcbj tmd

fppr`xcnftd cberdfsd cb tmd frdf l tmd poftd cl tmd rfjcus l tmd poftd hdl`rd mdftcbk cs 67 en.

Uò. Odt r en hd tmd rfjcus l tmd ecreuofr poftd fbj F hd cts frdf. Zmdb, F > pr8

jrjF

> 8prKcvdb r > 67 en (hdl`rd mdftcbk)

Odt Jr hd tmd cberdfsd cb tmd rfjcus l tmd ecreuofr poftd (fltdr mdftcbk) fbj JF hd tmd e`rrdsp`bjcbkcberdfsd cb frdf.

Zmdb, Jr > 8% `l 67 en

>677

8 67 en > 7.8 en

L`r, r > 67 en, Jr > 7.8 en

JF >jr

jF.Jr

> 8pr . (7.8) sq en > 8p(67) . (7.8) sq en > :psq en

Dx.= Lcbj tmd fppr`xcnftd emfbkd cb tmd t`tfo surlfed frdf `l f rckmt ecreuofr e`bd, wmdb tmd

rfjcus r rdnfcbs e`bstfbt wmcod tmd fotctujd m emfbkds hy .

Uò. Odt s jdb`td tmd t`tfo surlfed frdf l tmd rckmt ecreuofr e`bd. Zmdb,

888 r)mr(rs 86

p++p>

Ucbed r rdnfcbs e`bstfbt, s efb hd rdkfrjdj fs f lubetc`b l m boy.

[d mfvd Js > jmjs

.Jm


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Js > f

+

+

p.7

mr

rm

88(V Jm >f) >

88 mr

rm

+

fp

Dx.3 Lcbj tmd nfxcnun frdf `l fb cs`sedods trcfbkod cbserchdj cb tmd doocpsd 8

8

f

x+ 8

8

h

y> 6 wctm cts

vdrtdx ft `bd dbj `l tmd nfa`r fxcs.

U`o. Odt tmd kcvdb doocpsd hd 8

8

f

x+ 8

8

h

y> 6

Odt W (f e`sq, h scbq) hd f p cbt b tmd doocpsd Zmdb,

WV>8WN>8h scbq

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> fh Ye`sq+ e`s 8q

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9/15


www.tmcbicct.cb

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55> fh sq ubcts.

Dx.67 Um`w tmft mdckmt l tmd eyocbjdr l krdftdst v ound wmcem efb hd cbserchdj cb f rckmt ecreuofr

e`bd `l mdckmt m fbj sdnc-vdrtcefo fbkod cs `bd-tmcrj tmft `l tmd e`bd fbj tmd krdftdst

v`ound `l eyocbjdr cs 85 tfbm8;

:

U o. Odt FHE hd tmd e bd wm sd sdnc-vdrtcefo fbkod csf.FN > mF eyocbjdr cs cbserchdj cb ct wm`sd rfjcus OJ `l tmd hfsd cs r fbj mdckmt ON > m .Lr`n tmd lckurd, m> FN FO

> m ke`t f ...(6)\ ound (\) `l tmd eyocbjdr cs kcvdb hy

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10/15


www.tmcbicct.cb

TBU@O\DJ WX@HODNU

D]DXECUD C

V.6 Zmd vòund l f euhd cs cberdfscbk ft f e`bstfbt rftd. Wr`vd tmft tmd cberdfsd cb tmd surlfed frdf vfrcds

cbvdrsdoy fs tmd odbktm l tmd djkd l tmd euhd.

V.8 F nfb 8n mckm wfois ft f ubcl`rn spddj l x5 o`k(6 + x) )x6(

x

+

V.< Udpfrftd tmd cbtdrvfo

p8

,7 cbt suhcbtdrvfo cb wmcem l(x) > (scb:x + e`s:x) cs (c) Cberdfscbk (cc) Jderdfscbk.

V.; Wr`vd tmft tmd lubetc`b l kcvdb hy Cl l(x) > x8 x + 6 cs bdctmdr cberdfscbk b`r jderdfscbk cb (6, 6)

V.= Wr`vd tmft tmd eurvdbb

h

y

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> 8 tùemds tmd strfckmt ocbd 8

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V.3 Lcbj tmd dquftc`bs `l tmd tfbkdbt fbj tmd b`rnfo ft tmd p`cbt t `b tmd eurvd x > f scb 5t, y > h e`s5 t.

V.67 L`r tmd eurvd y > :x5 8x4, lcbj foo p`cbts ft wmcem tmd tfbkdbt pfssds tmrùkm tmd rckcb.

V.66 Wr`vd tmft tmd sun l tmd cbtdredpts b tmd e``rjcbftd fxds l fby tfbkdbt t` fyx >+ cs e`bstfbt.

V.68 Tscbk jclldrdbtcfos, lcbj tmd fppr`xcnftd vfoud l (68;)6/5

V.65 Cl cs f trcfbkod FHE, tmd scjd e fbj tmd fbkod E rdnfcb e`bstfbt, wmcod tmd rdnfcbcbk dodndbts frd

emfbkdj sockmtoy, jclldrdbtcfos sm`w tmftFe`s

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> X8

V.64 Um`w tmft tmd vòund l tmd krdftdst eyocbjdr wmcem efb hd cbserchdj cb f e`bd l mdckmt m fbj sdnc

vdrtcefo fbkod fcs8;

:pm5tfb8f.

V.6< Um`w tmft f e`bd l krdftdst vòund wmcem efb hd cbserchdj cb f kcvdb spmdrd cs suem tmft tmrddtcnds ct fotctujd cs twced tmd jcfndtdr l tmd spmdrd. Lcbj tmd vòund l tmd ofrkdst e`bd cbserchdj cbspmdrd l rfjcus X.

V.6; F squfrd tfbi l efpfecty 847 euhce ndtdrs mfs t` hd juk ut. Zmd e`st l tmd ofbj cs Xs. 47 pdr sq. n.

Zmd e`st `l jckkcbk cberdfsds wctm tmd jdptm fbj l`r tmd wm od tfbi e`st cs Xs :77 m 8, wmdrd m ndtrds

cs tmd jdptm l tmd tfbi. [mft smùoj hd tmd jcndbsc`bs l tmd tfbi s` tmft tmd e`st hd ncbcnun 0

V.6= F adt `l fb dbdny cs loycbk fo bk tmd eurvd y > x 8+ 8. F sòjcdr cs pofedj ft tmd p`cbt (5, 8). [mft cs tmd

bdfrdst jcstfbed hdtwddb tmd sòjcdr fbj tmd adt 0

V.63 F nfbulfeturd efb sdoo x ctdns ft f prced l Xs (847 x) dfem. Zmd e`st l pr`juecbk x ctdns cs Xs(8x8 47x + 68). Jdtdrncbd tmd bunhdr l ctdns t` hd sòj s` tmft md efb nfid nfxcnun.

V.87 Zmd e`st l ludo l`r rubbcbk f hus cs pr`p`rtc`bfo t` tmd squfrd l tmd spddj kdbdrftdj cb in/mr. Ct e`stsXs := pdr mùr wmdb tmd hus cs n`vcbk wctm f spddj l 87 in/mr. [mft cs tmd n`st de`b`ncefo spddj cl

tmd lcxdj emfrkds frd Xs 67= l`r bd mùr, vdr fbj fh`vd tmd rubbcbk emfrkds 0


11/15


www.tmcbicct.cb

V.6 Lcbj tmd cbtdrvfos cb wmcem tmd lubetc`b l(x) > x5 8x5 3x8+ 68x + 57 cs

(c) cberdfscbk (cc) jderdfscbk. YE.H.U.D. 8776^

V.4 Tscbk jclldrdbtcfos, lcbj fppr`xcnftd vfoud l 8 f8fbj x8+ y8> 8f8tùem dfem tmdr. YE.H.U.D. 8778^

V.; F wcbj`w cs cb tmd l`rn l rdetfbkod fh`vd wmcem tmdrd cs sdnc-ecreod. Cl pdrcndtdr l wcbj`w cs p en,

sm`w tmft wcbj`w wcoo foo w tmd nfxcnun p`sschod ockmt boy wmdb tmd rfjcus l sdnc-ecreod cs:

p

+pen.

YE.H.U.D. 8778^

V.= Cl y > x:+ 67 fbj x emfbkds lr`n 8 t` 6.33. Lcbj tmd fppr`xcnftd emfbkd cb y. YE.H.U.D. 8778^

V.3 Um`w tmft rdetfbkod l nfxcnun frdf tmft efb hd cbserchdj cb f ecreod l rfjcus r cs f squfrd l

scjd r8 . YE.H.U.D. 8778^

V.67 F hfoo `b wmcem fowfys rdnfcbs spmdrcefo cs hdcbk cbloftdj hy punpcbk cb kfs ft tmd rftd l 377 en5/sde. Lcbj

tmd rftd ft wmcem tmd rfjcus `l hfoo `b cs cberdfscbk wmdb rfjcus l hfoo `b cs 64 en. YE.H.U.D. 8775^

V.66 F wcbj`w cs cb tmd l rn l f rdetfbkod surnùbtdj hy f sdnc-ecreod. Cl tmd pdrcndtdr l tmd wcbj`w cs 677 n, lcbj tmd

jcndbsc`bs l tmd wcbj`w s` tmft nfxcnun ockmt dbtdrs tmrùkm tmd wcbj`w. YE.H.U.D. 8775^

V.68 F pfrtceod n`vds fo`bk tmd eurvd y >5

8x5+ 6. LCbj tmd p`cbts `b tmd eurvd ft wmcem y-e``rjcbftd cs

emfbkcbk twced fs lfst fs x-e``rjcbftd. YE.H.U.D. 8775^

V.65 L`r tmd eurvd y > :x5 8x4, lcbj foo p`cbts ft wmcem tfbkdbt pfssds tmrùkm rckcb.

YE.H.U.D. 8775^

V.6: F squfrd pcded l tcb l scjd 6= en cs t` hd nfjd cbt` f h`x wctmùt f t`p hy euttcbk f squfrd pcded lr`n

dfem e`rbdr fbj lòjcbk up tmd lofps. [mft smùoj hd tmd scjd squfrd t` hd eut ll s` tmft vòund l tmd

h x hd nfxcnun 0 Fos` lcbj tmd nfxcnun v ound l tmd h`x. YE.H.U.D. 8775^

V.64 Lcbj tmd cbtdrvfos b wmcem lubetc`b l(x) > 8x6

x

+cs (c) Cberdfscbk (cc) Jderdfscbk.

YE.H.U.D. 8775^

V.6< Lcbj tmd cbtdrvfos cb wmcem tmd lubetc`b l(x) > x5 6 e`sq, y > q scb qftq> :p .YE.H.U.D. 877:^

V.6= Lcbj tmd cbtdrvfos cb wmcem tmd lubetc b l(x) >x

6x: 8 +, x 7 cs

(c) cberdfscbk (cc) jderdfscbk. YE.H.U.D. 877:^

V.63 Lcbj tmd dquftc`b l tfbkdbt fbj b`rnfo t` tmd eurvd y > x8+ :x + 6 ft tmd p`cbt wm`sd x-e``rjcbftd cs

5. YE.H.U.D. 877:^

V.87 Zmd vòund l spmdrcefo hfoo` b cs cberdfscbk ft tmd rftd `l 84 en5/sde. Lcbj tmd rftd `l emfbkd l cts

surlfed frdf ft tmd cbstfbt wmdb cts rfjcus cs 4 en. YE.H.U.D. 877:^

V.86 Lcbj tmd cbtdrvfos cb wmcem lubetc`b l(x) > 8x5


12/15


www.tmcbicct.cb

V.85 Zmd surlfed frdf l f spmdrcefo huhhod cs cberdfscbk ft tmd rftd l 8 en8/s. Lcbj tmd rftd ft wmcem tmd

vòund l tmd huhhod cs cberdfscbk ft tmd cbstfbt cts rfjcus cs < en. YE.H.U.D. 8774^

V.8: F wcrd l odbktm 5< en cs eut cbt` tw` pcdeds bd l tmd pcdeds cs turbdj cb tmd l`rn l f squfrd fbj tmd

`tmdr cbt` f ecreod. [mft smùoj hd tmd odbktms `l dfem pcded s` tmft tmd sun `l tmd frdfs `l tw` hd

ncbcnun 0 YE.H.U.D. 8774^

V.84 Wr`vd tmft tmd ocbdf

x+

h

y> 6 cs f tfbkdbt t tmd eurvd y > hdx/f, ft tmd p`cbt wmdrd tmd eurvd euts y-fxcs.

YE.H.U.D. 8774^

V.8< Fb `pdb h`x wctm f squfrd hfsd cs t` hd nfjd ùt `l f kcvdb cr`b smddt `l frdf 8; sq. n. Um`w tmft tmd

nfxcnun vòund l tmd h`x cs 65.4 eu. n. YE.H.U.D. 8774^

V.8; Lcbj tmd cbtdrvfos cb wmcem tmd lubetc`b l(x) > 8x5 5x8 5 8 fbj y

6

2 7. Fos lcbj tmd p cbts l cbtdrsdetc b wmdrd h tm tfbkdbt fbj b`rnfo eut tmd x-fxcs. YE.H.U.D. 8774^

V.83 Lcbj f p`cbt b tmd pfrfhòf l(x) > (x 5)8, wmdrd tmd tfbkdbt cs pfrfoodo t` tmd em`rj a`cbcbk tmd p`cbts

(5, 7) fbj (:, 6). YE.H.U.D. 8774^

V.57 Lcbj tmd dquftc`b l tmd ocbd tmrùkm tmd p`cbt (5, :) wmcem euts lr`n tmd lcrst qufjrfbt f trcfbkod lncbcnun frdf. YE.H.U.D. 8774^

V.56 Um`w tmft tmd nfxcnun vòund l tmd eyocbjdr wmcem efb hd cbserchdj cb f spmdrd l rfjcus 54 cs 477pen5. YE.H.U.D. 8774^

V.58 Kcvdb tmd sun l tmd pdrcndtdrs `l f squfrd fbj f ecreod pr`vd tmft tmd sun l tmdcr frdf cs odfst wmdb tmd

scjd l tmd squfrd cs dqufo t` tmd jcfndtdr l tmd ecreod. YE.H.U.D. 877 q+ scb q, y > 6 + e`s qft q>:

p.

YE.H.U.D. 877 x8 4x + < ft tmd p`cbt W(8, 7) fbj V(5, 7) frd ft tmd rckmt fbkods

t` dfem tmdr. YE.H.U.D. 877 6. YE.H.U.D. 877


13/15


www.tmcbicct.cb

V.:4 Um`w tmft tmd mdckmt l tmd eyocbjdr l nfxcnun v ound tmft efb hd cbserchdj cb f e`bd l mdckmt m cs5

6m.

YE.H.U.D. 877=^

V.:< Zmd odbktm x l f rdetfbkod cs jderdfscbk ft tmd rftd l 4 en/ncbutd fbj tmd wcjtm y cs cberdfscbk ft tmd rftd

`l : en/ncbutd. [mdb x > = en fbj y > < en, lcbj tmd rftd l emfbkd l (f) tmd pdrcndtdr, (h) tmd frdf

`l tmd rdetfbkod. YE.H.U.D. 8773^

V.:; Lcbj tmd cbtdrvfos cb wmcem tmd lubetc`b l kcvdb hy l(x) > scb x + e`s x, 7 x8p,cs strcetoy cberdfscbk r strcetoy jderdfscbk. YE.H.U.D. 8773^

V.:= Cl tmd sun l tmd odbktms l tmd myp`tdbusd fbj f scjd `l f rckmt-fbkodj trcfbkod cs kcvdb, sm`w tmft tmd frdf

`l tmd trcfbkod cs nfxcnun wmdb tmd fbkod hdtwddb tmdn cs5

p. YE.H.U.D. 8773^

V.:3 F nfbulfeturdr efb sdoo x ctdns ft f prced l Xs

-

677

x4 dfem. Zmd e`st prced l x ctdns cs Xs

+677

4

x.

Lcbj tmd bunhdr l ctdns md smùoj sdoo t` dfrb nfxcnun pr`lct. YE.H.U.D. 8773^

V.47 Lcbj tmd dquftc`b l tmd tfbkdbt t` tmd eurvd y > 8x5 - wmcem cs pfrfoodo t` tmd ocbd :x 8y + 4 > 7.

YE.H.U.D. 8773^

V.46 Lcbj tmd cbtdrvfos cb wmcem tmd lubetc`b l kcvdb hy l(x) > x5+ 5x

6, x 7 cs YE.H.U.D. 8773^

(c) cberdfscbk (cc) jderdfscbk.

V.48 Lcbj tmd vòund tmd ofrkdst eyocbjdr tmft efb hd cbserchdj cb f spmdrd l rfjcus r.

YE.H.U.D. 8773^

V.45 F tfbi wctm rdetfbkuofr hfsd fbj rdetfbkuofr scjds, pdb ft tmd t`p cs t` hd e`bstreutdj s` tmft cts jdptm

cs 8 n fbj v ound cs = n5. Cl hucojcbk l tfbi e`sts Xs ;7 pdr sq. ndtrd l`r tmd hfsd fbj Xs :4 pdr sq.

ndtrd l`r scjds, wmft cs tmd e`st `l odfst dxpdbscvd tfbi 0 YE.H.U.D. 8773^

V.4: Um`w tmft tmd vòund l tmd krdftdst eyocbjdr tmft efb hd cbserchdj cb f e`bd l mdckmt m fbj sdnc-

vdrtcefo fbkod f cs8;

:p m5tfb8f. YE.H.U.D. 8767^

V.44 Lcbj tmd cbtdrvfos cb wmcem tmd l oo wcbk lubetc b cs YE.H.U.D. 8767^

(f) strcetoy cberdfscbk, (h) strcetoy jderdfscbk. YH`bus nfris^

V.4< Wr`vd tmft y >)e`s8(

scb:

q+q

qcs fb cberdfscbk lubetc`b cb

p8

,7 . YE.H.U.D. 8766^

V.4; Cl tmd rfjcus l f spmdrd cs ndfsurdj fs 3 en wctm fb drr`r l 7.75 en, tmdb lcbj tmd fppr`xcnftd drr`r

cb efoeuoftcbk cts surlfed frdf. YE.H.U.D. 8766^

V.4= Um`w tmft tmd rckmt-ecreuofr e`bd l odfst eurvdj surlfed fbj kcvdb vòund mfs fb fotctujd dqufo t` 8

tcnds tmd rfjcus l tmd hfsd. YE.H.U.D. 8766^

V.43 F wcbj`w mfs tmd smfpd l f rdetfbkod surnùbtdj hy fb dqucoftdrfo trcfbkod. Cl tmd pdrcndtdr l tmd

wcbj`w cs 68 n, lcbj tmd jcndbsc`bs l tmd rdetfbkod tmft wcoo pr`jued tmd ofrkdst frdf l tmd wcbj`w.

YE.H.U.D. 8766^

V.


14/15

FWWOCEFZC@BU @L JDXC\FZC\DU6:

www.tmcbicct.cb

ofjjdr cs : n fwfy lr`n tmd wfoo 0 YE.H.U.D. 8768^

V.


15/15


wwwtmcbicct cb

FBU[DX IDS

D]DXECUD 6 (TBU@O\DJ WX@HODNU)

8.:n/ncb :. Cb. (, 6) (5, ), Jde. (6, 5) 4.Cb. (7, ), Jde. (6, 7)

6, b`rnfo fx scbt hy e`st > f8scb:t h8e s:t

67.(7, 7) 9 (6, 8) fbj (6, 8) 68.4.78< 6 67 n jdptm > 8.4 n

6=.(6, 5) 63.47 ctdns 87. spddj > 57 in/mr.

D]DXECUD 8 (H@FXJ WX@HODNU)

6. Cberdfscbk cb (, 6) (5, ) fbj jderdfscbk cb (6, 5) 8.:

84sq. ubcts 5.7.

68

6

-

-

+-

8

66x

6=. Cberdfscbk cb

--

8

6,

,

8

6fbj jderdfscbk cb

-7,

8

6

8

6,7

63. 67x y = > 7, x + 67y 885 > 7 87.67 en8/s

86. Cberdfscbk cb (, 8) (:, ) fbj jderdfscbk cb (8, :) 88.7.638: 85.< en5/s

8:.5:3

56::

+en,

5:3

58:

+en 8;.Cberdfscbk cb (, 8) (5, ) fbj jderdfscbk cb (8, 5)

8=. y 5

:4 >

45

=(x 8) 83.

:

6,

8

;57.:zx + 5y 8: > 7

55. x + ( 8 + 6) y

8

885 +

:

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8

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he

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8_applications of derivatives.pdf

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