5_Continuity and Differentiability

8/10/2019 5_Continuity and Differentiability

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CLERIE[IRY AEB BIGGK_KERIAOIFIRY

3.1 OASIC CLECKXRS AEB IJXL_RAER _KS[FRS

(a) Cletieuity lg a rkaf guectile at a pliet

A guectile g is saib tl ok fkgt cletieulus lr cletieulus grlj tdk fkgt at x : c igg

(i) g(c) kxists (ii) -cxFt g(x) kxists aeb (iii) -cx

Ft g(x) : g(c).

A guectile g is saib tl ok ri`dt cletieulus lr cletieulus grlj tdk ri`dt at x : c igg

(i) g(c) kxists (ii) +cxFt g(x) kxists aeb (iii) +cx

Ft g(x) : g(c).

A guectile g is saib tl ok cletieulus at x : c igg

(i) g(c) kxists (ii)cx

Ft

g(x) kxists aeb (iii)cx

Ft

g(x) : g(c).

Dkeck, a guectile is cletieulus at x : c igg it is oltd fkgt as wkff as ri`dt cletieulus at x : c.

Pdke cxFt g(x) kxists out kitdkr g(c) blks elt kxist lr cxFt g(x) g(c), wk say tdat g

das a rkjlvaofk biscletieuity> ltdkrwisk, wk say tdat g das ele-rkjlvaofk biscletieuity.

(o) Cletieuity lg a guectile ie ae ietkrvaf

A guectile g is saib tl ok cletieulus ie ae lpke ietkrvaf (a, o) igg g is cletieulus at kvkry

pliet lg tdk ietkrvaf (a, o) > aeb g is saib tl ok cletieulus ie tdk cflskb ietkrvaf a, oZ igg g is

cletieulus ie tdk lpke ietkrvaf (a, o) aeb it is cletieulus at a grlj tdk ri`dt aeb at o grlj

tdk fkgt.

Cletieulus guectile. A guectile is saib tl ok a cletieulus guectile igg it is cletieulus at

kvkry pliet lg its bljaie. Ie particufar, ig tdk bljaie is a cflskb ietkrvaf, say ^a, oZ, tdke g

just ok cletieulus ie (a, o) aeb ri`dt cletieulus at a aeb fkgt cletieulus at o.

Rdk skt lg aff pliet wdkrk tdk guectile is cletieulus is caffkb its bljaie lg cletieuity. Rdk

bljaie lg cletieuity lg a guectile jay ok a prlpkr suoskt lg tdk bljaie lg tdk guectile.

3.? X_LXK_RIKS LG CLERIE[L[S G[ECRILES

Xrlpkrty 1.Fkt g, ` ok twl guectiles cletieulus at x : c, tdke

(i) ag is cletieulus at x : c, " a_ (ii) g + is cletieulus at x : c

(iii) g is cletieulus at x : c (iv) g` is cletieulus at x : c

(v)

`

gis cletieulus at x : c, prlvibkb `(c) 8.

Xrlpkrty ?.Fkt B1aeb B

?ok tdk bljaies lg cletieuity lg tdk guectiles g aeb rkspkctivkfy

tdke

(i) ag is cletieulus le B1glr aff a_ (ii) g + is cletieulus le B

1B

?

(iii) g ` is cletieulus le B1B

?(iv) g` is cletieulus le B

1B

?

(v)`

gis cletieulus le B

1B

?kxckpt tdlsk pliets wdkrk `(x) : 8.

Xrlpkrty 3.A plfyeljiaf guectile is cletieulus kvkrywdkrk.

Ie particufar, kvkry clestaet guectile aeb kvkry ibketity guectile is cletieulus.

Xrlpkrty 2.A ratileaf guectile is cletieulus at kvkry pliet lg its bljaie.Xrlpkrty ;.Ig g is cletieulus at c, tdke | g | is afsl cletieulus at x : c.


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Ie particufar, tdk guectile | x | is cletieulus glr kvkry x _.

Xrlpkrty 7.Fkt g ok a cletieulus lek-lek guectile bkgiekb le a, oZ witd rae`k c, bZ, tdke

tdk ievkrsk guectile g1= ^c, bZ ^a, oZ is cletieulus le ^c, bZ

Xrlpkrty 0.Ig g is cletieulus at c aeb ` is cletieulus at g(c), tdke `lg is cletieulus at c.

Xrlpkrty 4.Aff tdk oasic tri`leljktric guectiles i.k. sie x, cls x, tae x, clt x, skc x aeb

clskc x ark cletieulus.

Xrlpkrty 9.Aff oasic ievkrsk tri`leljktric guectiles i.k. sie1x, cls1x, tae1x, clt1x, skc

1x, clskc1x ark cletieulus (ie tdkir rkspkctivk bljaies).

Xrlpkrty 18. Rdklrkj. Ig a guectile is biggkrketiaofk at aey pliet, it is ekckssarify cletieulus

at tdat pliet.

Rdk clevkrsk lg tdk aolvk tdklrkj jay elt ok truk i.k. a guectile jay ok cletieulus at a

pliet out jay elt ok bkrivaofk at tdat pliet.

3.3 BK_IWARIWK LG WA_IL[S G[ECRILES

(a) Bkrivativk lg cljplsitk guectiles

Rdklrkj. Ig u : `(x) is biggkrketiaofk at x aeb y : g(u) is biggkrketiaofk at u, tdke y

is biggkrketiaofk at x aebbx

by:

bu

by.

bx

bu.

Ig ` is biggkrketiaofk at x aeb g is biggkrketiaofk at `(x), tdke tdk cljplsitk guectile d(x) :

g(`(x)) is biggkrketiaofk at x aeb d(x) : g(`(x)). (x).

Cdaie _ufk.Rdk aolvk rufk is caffkb tdk cdaie rufk lg biggkrketiatile, sieck bktkrjieie` tdk

bkrivativk lg y : g(`(x)) at x ievlfvks tdk glfflwie` cdaie lg stkps =

(i) Girst, gieb tdk bkrivativk lg tdk lutkr guectile g at (x).

(ii) Skcleb, gieb tdk bkrivativk lg tdk ieekr guectile at x.

(iii) Rdk prlbuct lg tdksk twl bkrivativks `ivks tdk rkquirkb bkrivativk lg tdk cljplsitk

guectile gl` at x .

(i)bx

by:

bt

bxbt

by

, prlvibkbbt

bx8. (ii)

bx

by:

by

bx

1, prlvibkb

by

bx8.

(iii)bx

by.

by

bx: 1 (iv)

bx

b(| x |) :

|x|

x, x 8.

(o) Bkrivativks lg ievkrsk tri leljktric guectiles

(i)bx

b(sie1x) :

?x1

1

-, x (1, 1) i.k. | x | 6 1

(ii)bx

b(cls1x) :

?x1

1

-, x (1, 1) i.k. | x | 6 1

(iii)bx

b(tae1x) :

?x1

1

+, glr aff x _

(iv)bx

b(clt1x) : ?x1

1

+, glr aff x _


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(v)bx

b(skc1x) :

1xx

1

? -, x 5 1

(vi)bx

b(clskc1x) :

1xx

1

? -, x 5 1

(c) Bkrivativks lg af koraic aeb tri leljktric guectiles

(i)bx

b(xe) : exe 1 (ii)

bx

b(xx) : xxfl` kx

(iii)bx

b(sie x) : cls x (iv)

bx

b(cls x) : sie x

(v)bx

b(tae x) : skc?x (vi)

bx

b(clt x) : clskc?x

(vii)bx

b(clskc x) : clskc x clt x.

(b) Bkrivativks lg kxpleketiaf aeb fl`aritdjic guectiles

(i)bx

b(kx) : kx, glr aff x _

(vi)bx

b(fl`

a| x |) :

afl`x

1, x 8, a 5 8, a 1.

(ii)

bx

b(ax) : axfl` a, a 5 8, a 1, x _

(v)bx

b(fl` | x |) :

x

1, x 8

(iii)bx

b(fl` x) :

x

1, x 5 8

(iv)bx

b(fl`

ax) :

afl`x

1, x 5 8, a 5 8, a 1

(k) Fl`aritdjic biggkrketiatile

Ig u,eark biggkrketiaofk guectiles lg x, tdkebx

b(ue) : ue

bx

b(efl` u).

(g) Bkrivativks lg guectiles ie parajktric glrj

Ig x aeb y ark twl variaofks sucd tdat oltd ark kxpficitfy kxprksskb ie tkrjs lg a tdirb

variaofk, say t, i.k. ig x : g(t) aeb y : `(t), tdke sucd guectiles ark caffkb parajktric

guectiles aeb

bx

by:

bt

bxbt

by

, prlvibkbbt

bx8.


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(`) Bkrivativk lg skcleb lrbkr

Ig a guectile g is biggkrketiaofk at a pliet x, tdke its bkrivativk g is caffkb tdk girst bkrivativk lr

bkrivativk lg girst lrbkr lg tdk guectile g. Ig g is gurtdkr biggkrketiaofk at tdk sajk pliet x,

tdke its bkrivativk is caffkb tdk skcleb bkrivativk lr bkrivativk lg tdk skcleb lrbkr lg g at

tdat pliet aeb is bkeltkb oy g.

Ig tdk guectile g is bkgiekb oy y : g(x), tdke its girst aeb skcleb bkrivativks ark bkeltkb oy g

(x) aeb g(x) lr oybx

byaeb

?

?

bx

yblr oy y

1aeb y

?lr oy y aeb y rkspkctivkfy..

3.2 _LFFKS RDKL_KJ AEB FA@_AE@KS JKAE WAF[K RDKL_KJ

(i) _lffks tdklrkj

Ig a guectile g is

(i) cletieulus ie tdk cflskb ietkrvaf a, oZ

(ii) bkrivaofk ie tdk lpke ietkrvaf (a, o) aeb

(iii) g(a) : g(o),

tdke tdkrk kxists atfkast lek rkaf eujokr c ie (a, o) sucd tdat g(c) : 8.

Rdus clevkrsk lg _lffks tdklrkj jay elt ok truk.

(ii) Fa rae`ks jkae vafuk tdklrkj

Ig a guectile g is

(i) cletieulus ie tdk cflskb ietkrvaf a, oZ aeb

(ii) bkrivaofk ie tdk lpke ietkrvaf (a, o),

tdke tdkrk kxists atfkast lek rkaf eujokr c ie (a, o) sucd tdat g (c) :ao

)a(g)o(g

-

-

Rdk clevkrsk lg Fa`rae`ks jkae vafuk tdklrkj jay elt ok truk.


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SLFWKB X_LOFKSJ

Kx.1 Is tdk guectile bkgiekb oy g(x) : x? sie x + ; cletieulus at x :


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aeb -8xfij

g(x) : 8dfij g(8 d) : 8d

fij d8

)d8sie(

-

-: 8d

fij d

sied

-

-: 1

Afsl, g(8) : 8 + 1 : 1

Sieck, +8xfij

g(x) : -8xfij

g(x) : g(8), g is cletieulus at x : 8.

Dkeck, tdkrk is el pliet lg biscletieuity lg g.

Kx.2 Bktkrjiek ig g bkgiekb oy

g(x) :

:

8xig,8

8xig,x

1siex

?

is a cletieulus guectile.

Slf. It is suggiciket tl kxajiek tdk cletieuity lg tdk guectile g at x : 8.

Dkrk g (8) : 8

Afsl, +8xfij

g(x) : 8dfij g(8 + d)

: 8dfij

++

d8

1sie)d8( ? : 8d

fij

d

1sied? : 8

aeb -8xfij

g(x) : 8dfij g(8 d)

: 8dfij

--

d8

1sie)d8( ? : 8d

fij

- d

1sied? : 8

1

d

1sieQ

Dkeck, +8xfij

g(x) : -8xfij

g(x) : g(8)

Sl, g is cletieulus at x : 8

Rdis ijpfiks tdat g is a cletieulus guectile at aff x _.

Kx.; Kxajiek tdk cletieuity lg g, wdkrk g is bkgiekb oy

g(x) :

:

8xig,18xig,xclsxsie

Slf. Dkrk, g(8) : 1

Afsl, +8xfij g(x): 8d

fij g(8+d): 8d

fij ^sie(8+d)cls(8+d)Z

: 8dfij ^sie d cls dZ : 1

aeb -8xfij

g(x): 8dfij g(8d): 8d

fij ^sie(8d)cls(8d)Z

: 8dfij ^sie d cls dZ ^Q sie (d) : sie dZ

: 8 1 : 1 aeb cls (d) : cls dZ

Dkeck, +8x fij g(x) : -8x fij g(x) : g(8)

Sl, g is cletieulus at x : 8> aeb dkeck cletieulus at aff x _.


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Kx.7 Gieb tdk vafuk lg n sl tdat tdk glfflwie` guectile g is cletieulus at tdk iebicatkb pliet=

(i) g(x) :

5

;xig,;x3;xig,1nx

at x : ;

(ii) g(x) :

5

?xig,3

?xig,nx?

at x : ?

Slf. (i) Sieck g is ivke tl ok cletieulus at x : ;, wk davk

+;xfij

g(x) : -;xfij

g(x) : g(;)

8dfij g(; + d) : 8d

fij g(; d) : g(;)

8dfij ^3(;+d);Z: 8d

fij n(;d)+1Z : ;n + 1

18 : ;n + 1 n :;9

(ii) Sieck g is ivke tl ok cletieulus at x : ?, wk davk

+?xfij

g(x) : -?xfij

g(x) : g(?)

8dfij g(? + d) : 8d

fij g(? d) : g(?)

8dfij (3) : 8d

fij ^n(? d)

?Z : 2n

3 : 2n n :2

3

Kx.0 Gieb tdk vafuks lg a aeb o sucd tdat tdk guectile bkgiekb oy

g(x) :

18xig,?1

18x?ig,oax?xig,;

is a cletieulus guectile.

Slf. Sieck tdk guectile g is cletieulus, it is cletieulus at x : ? as wkff as at x : 18.

Sl, +?xfij

g(x) : -?xfij

g(x) : g(?)

i.k., 8dfij g(? + d) : 8d

fij g(? d) : g(?)

i.k., ?a + o : ; (......1)

aeb +18xfij

g(x) : -18xfij

g(x) : g(18)

i.k., 8dfij g(18 + d) : 8d

fij g(18 d) : g(18)

i.k., ?1 : 18a + o (......?)

Grlj (1) aeb (?), wk gieb tdata : ? aeb o : 1


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Kx.4 Sdlw tdat tdk guectile bkgiekb oy

`(x) : x ^xZ is biscletieulus at aff ietk`raf pliets. Dkrk, xZ bkeltks tdk rkat-

kst ietk`kr fkss tdae lr kquaf tl x.

Slf. Rdk guectile g(x) : x ^xZ cae ok writtke as

g(x) :

+66-66---

1nxnig,nxnx1nig),1n(x

, wdkrk n is ae aroitrary ietk`kr..

Elw, +nxfij

g(x): 8dfij g(n + d): 8d

fij ^(n+d) nZ:8

aeb -nxfij

g(x): 8dfij g(nd): 8d

fij ^(nd) (n1)Z:1

Sieck, +nxfij

g(x) -nxfij

g(x), tdk guectile g is elt cletieulus at x : n.

Sieck n is ae aroitrary ietk`kr, wk cae kasify clecfubk tdat tdk guectile is biscletieulus at aff

ietk`raf pliets.

Kx.9 Wkrigy FJW Rdklrkj glr tdk guectile

g(x) :

5

1xwdke,x3Z.?,1^le1xwdke,?x

3

Slf. Oltd x3+ ? aeb 3x ark plfyeljiaf guectiles. Sl, g (x) is cletieulus aeb biggkrketiaofk kvkry-

wdkrk kxckpt at x : 1.

Dkrk, 31.3)x(gfij1x

::+

3?1)x(gfij 3

1x:+:

-

As

.1xatcletieulusis)x(g),1(g)x(gfij)x(gfij1x1x

:::-+

Lovilusfy, tdke g(x) is cletieulus le ^1, ?Z. A`aie tl tkst tdk biggkrketiaoifity lg g(x) at x : 1,

wk davk

F g (1) : -1xfij

1x

)1(g)x(g

-

-: -1x

fij1x

)?1()?x( 33

-

+-+

: -1xfij

1x

1x3

-

-: -1x

fij(x?+ x + 1) : 3

_ g (1): +1xfij

1x

)1(g)x(g

-

-

: +1xfij

1x

1.3x3

-

-: +1x

fij(3) : 3

As F g (1) : _ g (1), tdk guectile g (x) is biggkrketiaofk at x : 1. Dkeck, g is biggkrketiaofk ie (

1, ?).

Rdus, oltd tdk clebitiles rkquirkb glr tdk appficaoifity lg tdk FJW Rdklrkj ark satisgikb

aeb dkeck, tdkrk kxists at fkast lek c (1, ?) sucd tdat

g (c) :)1(?

)1(g)?(g

--

-- g (c) :

3

17 -:

3

;


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Elw, ie x 5 1, g (x) : 3. Sl, g (c) caeelt ok3

;ie tdis ietkrvaf.

Ie x 1. g (x) : 3x?

g (c) : 3c?

Lovilusfy, 3c?

: 3

;

ivks c?

: 9

;

lr c : 3

;

Oltd3

;aeb

3

;fik ie (1, ?). Rdus, FJW is vkrigikb glr g(x) aeb ^1, ?Z.

Kx.18 Wkrigy _lffks tdklrkj glr tdk guectile g (x) : x (x 3)?ie tdk cflskb ietkrvaf 8 x

3.

Slf. (i) Dkrk, g(x) : x (x 3)?

: x (x? 7x + 9)

: x3 7x?+ 9x

Sieck g(x) is a plfyeljiaf guectile lg x, it is cletieulus ie 8, 3Z

(ii) g (x) : 3x? 1?x + 9

kxists ueiqukfy ie tdk lpke ietkrvaf (8, 3)

(iii) g(8) : (8)3 7(8)?+ 9(8)

: 8 8 + 8 : 8

g(3) : (3)? 7(3)?+ 9(3)

: ?0 ;2 + ?0 : 8

g(8) : g(3)

Rdus, aff tdk tdrkk clebitiles ark satisgikb, Dkeck, _lffks Rdklrkj is appficaofk.

Fkt us elw slfvk g (c) : 8

i.k. 3c? 1?c + 9 : 8

3(c? 2c + 3) : 8

(c 3) (c 1) : 8

c : 3, 1

SIeck, c : 1 (8, 3), tdk _lffks Rdklrkj is vkrigikb glr tdk guectile.

g(x) : x(x 3)?ie tdk cflskb ietkrvaf ^8, 3Z.

Kx.11 Wkrigy _lffks Rdklrkj glr tdk guectile g(x) : (x a)j(x o)eie ^a, oZ > j, e okie`

plsitivk ietk`krs.

Slf. Dkrk, g(x) is a plfyeljiaf guectile lg bk`rkk (j + e). Sl, it is a cletieulus guectile ie ^a, oZ.

g (x) : (x a)j 1(x o)e 1 j (x o) + e (x a)Z kxists ueiqukfy ie (a, o). Sl, it is bkrivaofk

j (a, o).

Gurtdkr, g(a) : 8 aeb g(o) : 8. Sl, g(a) : g(o)

Rdus, aff tdk tdrkk clebitiles lg _lffks Rdklrkj ark satisgikb. Dkeck, _lffks Rdklrkj is

appficaofk.

Fkt us elw slfvk g (c) : 8

(ca)j 1(c o)e 1^j (c o) + e (ca)Z : 8

c : a lr c : o lr c :ej

eajo

+

+

Sieck c :ej

eajo

+

+(a, o), tdk _lffks Rdklrkj is vkrigikb.


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Kx.1? Sdlw tdat tdk guectile g(x) :

5

?xig,x;?xig,x1

is cletieulus at x : ?, out elt biggkrketiaofk

at x : ?.

Slf. At x : ?,

+

?x

fijg(x) :

8d

fij g(? + d) :

8d

fij ;(?+ d)Z : 3

-?xfij

g(x) : 8dfij g(? d) : 8d

fij 1+(?d)Z : 3

Afsl, g(?) : 1 + ? : 3

Sieck, +?xfij

g(x) : -?xfij

g(x) : g(?), g(x) is cletieulus at x : ?.

Ekxt, Fg (?) : 8dfij d

)?(g)d?(g

-

--

: 8dfij : d)?1()d?1( -

+--+: 1

_g (?): 8dfij d

)?(g)d?(g -+: 8d

fij : d

)?1()d?(; +-+-: 1

Sieck, Fg (?) _g (?), tdk guectile g is elt biggkrketiaofk at x : ?.

Kx.13 Sdlw tdat tdk guectile g(x) :

6

1xig,1x

1xig,x1? is cletieulus at x : 1, out elt biggkrketiaofk

tdkrkat.

Slf. Rdk guectile is cletieulus at x : 1, okcausk

+1xfij

g(x) : -1xfij

g(x) : g(1) as sdlwe okflw =

+1xfij

g(x) : 8dfij g(1 + d) : 8d

fij ^(1 + d)

? 1Z : 8dfij (d

?+ ?d) : 8 >

-1xfij

g(x) : 8dfij g(1 d) : 8d

fij ^1 (1 d)Z : 8d

fij (d) : 8

aeb g(1) : (1)? 1 : 1 1 : 8

Gurtdkr, _g (1) : 8dfij

d

)1(g)d1(g -+

: 8dfij

?^(1 d ) 1Z ^8 Z

d

+ - -: ?

Fg (1) : 8dfij

g(1 d) g(1)

d

- -: 8d

fij

?^(1 d ) 1Z ^8 Z

d

- - -

: 8dfij

- d

d: 8d

fij (1) : 1

Sieck, _g (1) Fg (1), tdk guectile is elt biggkrketiaofk at x : 1.


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Kx.12 Sdlw tdat tdk guectile g bkgiekb as

g(x) :

5

?xig,2x;

1x1ig,xx?

1x8ig,?x3?

is cletieulus at x : ?, out elt biggkrketiaofk tdkrkat.

Slf. At x:?, +?xfij

g(x): 8dfij g(?+d): 8d

fij ^;(?+d)2Z:7

-?xfij

g(x) : 8dfij g(?d): 8d

fij ?(?d)

?(?d)Z

: 8dfij ^?(2 2d + d

?) (? d)Z

: 8dfij ^7 0d + ?d

?Z : 7

aeb g(?) : ? (?)? ? : 4 ? : 7

Sieck +?xfij

g(x) : -?xfij

g(x):g(?), tdk guectile g is cletieulus at x : ?.

Ekxt, Fg (?) : 8dfij d

)?(g)d?(g

-

--

: 8dfij d

Z2)?(;^)d?()d?(? ?

-

-----

:d

7d?d07 ?

-

-+-: 0

g (?) : 8dfij d

)?(g)d?(g -+

: 8dfij d

Z2)?(;^Z2)d?(;^ ---+

: 8dfij d

Z2)?(;^Z2)d?(;^ ---+

: 8dfij

-+

d

7d;7: ;

Sieck, Fg (?) : _g (?), tdk guectile g is elt biggkrketiaofk at x : ?.


12/20


13/20


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Q.9 Gieb tdk bkrivativk lg g(x) :

:-

+-

-

1x,3

1

1x,;x0x?

1x?

at x : 1

Q.18 Gieb tdk vafuk lg a aeb o, sl tdat tdk guectile g(x) :

5+++

1xig,?ox1xig,ax3x?

is biggkrketiaofk at

kacd x _.

Q.11 Biggkrketiatk tdk glfflwie` w.r.t x =

(i) fl`xcls1

xcls1

+

-(ii) fl`

a

++ ?? axx (iii) fl` (skc x + tae x)

(iv) fl`

+

?

xcls

?

xsie (v) fl`

-

+

xsiex1

xsiex1(vi) fl`

-+

++

xax

xax

??

??

Q.1? (i) Ig y :x1

x1

+

-, prlvk tdat (1 x?)

bx

by+ y : 8

(ii) Ig y :

--+

-++

1x1x

1x1x, prlvk tdat

bx

by:

1x

1xx

?

?

-

-+

(iii) Ig y :xsiexcls

xsiexcls

-

+, sdlw tdat

bx

by: skc?

p+

2x

(iv) Ig y :xtaexskc

xtaexskc

-

+, sdlw tdat

bx

by: skc x (skc x + tae x)

Q.13 (i) Ig y : x + x

1, sdlw tdat ?x

bx

by+ y : ? x (ii) Ig y : x sie y, prlvk tdat x

bx

by:

)yclsx1(

y

-

(iii) Ig x y1+ + y x1+ : 8, prlvk tdat bx

by: ?)x1(

1

+

-, x y

Q.12 Ig y : ,........xxx +++ prlvk tdatbx

by:

)1y?(

1

-

Q.1; @ivke tdat cls?

x. cls

2

x . cls

4

x......:

x

xsie, prlvk tdat ??

1skc?

?

x+ 2?

1skc?

2

x+ ......:

clskc?x?x

1


14/20


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Q.17 Ig x : tae1 ?t1

t?

-aeb y : sie1 ?t1

t?

+, sdlw tdat

bx

by: 1.

Q.10 Biggkrketiatk =(i) sie1

-+- ?x1xx1x (ii) tae1

+ ?x1;1

x?

(iii) tae1

++ 1xx

1? + tae

1

++ 3x3x

1? + tae

1

++ 0x;x

1? + ............ tl e tkrjs.

Q.14 Biggkrketiatk =(i) tae1

- ?x1

x+ tae1

-+ ?x11

x(ii) cls1

-

;

xsie2xcls3

Q.19 Biggkrketiatk = sie1

+ ?x1

1+ tae1

-+

x

1x1 ?

.

Q.?8 Biggkrketiatk = (i) sie1

-++

?

x1x1(ii) cls1

+

--

-

1

1

xx

xx

Q.?1 Biscuss tdk cletieuity lg tdk guectile g(x) :

+6-

8xig,1x?8xig,1x?

Q.?? Ig a guectile g(x) is bkgiekb as g(x) :

:

--

2x,8

2x,2x

|2x|sdlw tdat g is kvkrywdkrk cletieulus kxckpt

at x : 2.

Q.?3 Biscuss tdk cletieuity lg tdk guectile g(x) : | x | + |x 1| ie tdk ietkrvaf ^1, ?Z

Q.?2 Sdlw tdat g(x) : | x | is elt biggkrketiaf at x : 1.

Q.?; Fkt g(x) :

6-+

8xig,x?8xig,x?

, sdlw tdat g(x) is elt bkrivaofk at x : 8.

Q.?7 Sdlw tdat tdk guectile g(x) :

:

8xig,8

8xig,x

1siex?

is biggkrketiaf at x : 8 aeb g (8) : 8.


15/20

CLERIE[RIRY AEB BIGGK_KERIAOIFIRY1;

www.tdieniit.ie

OLA_B X_LOFKS

KTK_CISK II

Q.1 Ig xy: kx y, prlvk tdatbx

by: ?

)xfl`1(

xfl`

+. C.O.S.K. ?888Z

Q.? Ig xpyq: (x + y)p + q, prlvk tdatbx

by:

x

y. ^C.O.S.K. ?888Z

Q.3 Gieb?

?

bx

ybwdke y : fl`

x

?

k

x. ^C.O.S.K. ?888Z

Q.2 Ig y : ak?x+ okx, prlvk tdat ?

?

bx

yb

bx

by ?y : 8. ^C.O.S.K. ?888Z

Q.; Ig y : A cls ex + O sie ex sdlw tdat ?

?

bx

yb+ e?y : 8. ^C.O.S.K. ?881Z

Q.7 Biscuss tdk cletieuity lg tdk guectile g(x) at x : 8 ig g(x) :

+6-

8x,1x?8x,1x?

^C.O.S.K. ?88?Z

Q.0 Sdlw tdat tdk guectile g(x) : ?x | x | is cletieulus at x : 8. ^C.O.S.K. ?88?Z

Q.4 Ig tdk guectile g(x) :

6-:5+

1x,o?ax;1x,111x,oax3

is cletieulus. at x : 1, gieb tdk vafuks lg a aeb o.

^C.O.S.K. ?88?Z

Q.9 Ig y :x?sie1

x?sie1

+

-, prlvk tdat

bx

by+ skc?

-

px

2: 8. ^C.O.S.K. ?88?Z

Q.18 Ig y : fl`

+

p

?

x

2tae , sdlw tdat

bx

by skc x : 8. ^C.O.S.K. ?88?Z

Q.11 Wkrigy Fa`rae`ks jkae vafuk tdklrkj glr tdk glfflwie` guectiles ie tdk ivke ietkrvafs.

^C.O.S.K. ?88?Z

Afsl gieb c lg tdis tdklrkj = (i) g(x):x?+x1 ie 8, 2Z (ii) g(x): 2x? - le ?, 2Z

Q.1? Ig y : kx(sie x + cls x), prlvk tdat ?

?

bx

yb ?

bx

by+ ?y : 8. ^C.O.S.K. ?88?Z

Q.13 Biggkrketiatk tdk glfflwie` w.r.t. x ^C.O.S.K. ?883Z

(i) fl`

+

-

xcls1

xcls1(ii) fl` (x + ?x1+ )


16/20


www.tdieniit.ie

Q.12 Ig x : a(t + sie t), y : a(1 cls t), giebbx

byat t :

?

p. ^C.O.S.K. ?883Z

Q.1; Biggkrketiatk tdk glfflwie` guectiles w.r.t x = ^C.O.S.K. ?882Z

(i) tae1

-

+

xsie1

xsie1. (ii) clt1

+

-

xsie1

xsie1(iii) tae1

-+

x

1x1 ?

(iv) sie1

-+

13

x11?x; ?

(v) tae1

-++

--+

??

??

x1x1

x1x1

(vi) tae1

-++

--+

x1x1

x1x1

Q.17 Xrlvk tdatbx

b

+- -

?

xsie

?

axa

?

x 1?

??: ?? xa - . ^C.O.S.K. ?882Z

Q.10 Ig y : (sie x)x+ (cls x)tae x, giebbx

by^C.O.S.K. ?882Z

Q.14 Giebbx

by, wdke x : a ?

?

t1

t1

+

-, y : ?t1

ot?

+^C.O.S.K. ?882Z

Q.19 Biggkrketiatk tae1

- ?x1

x?w.r.t. sie1

+ ?x1

x?. ^C.O.S.K. ?882Z

Q.?8 Ig g(x) :

x3?

x1

x3 +

+

+, gieb g (8). ^C.O.S.K. ?88;Z

Q.?1 Giebbx

byig = x : a

-

+?

?

t1

t1, y : ?t1

t?

-^C.O.S.K. ?88;Z

Q.?? Ig y : x fl`

+ oxax , prlvk tdat

?

?

bxyb :

x1

?

oxaa

+. ^C.O.S.K. ?88;Z

Q.?3 Ig tdk guectile g bkgiekb oy g(x) :

5-+

:

6-

8x,

2x17

x

8x,a

8x,x

x2cls1?

is cletieulus at x : 8, gieb tdk vafuk

lg a. ^C.O.S.K. ?887Z


17/20


www.tdieniit.ie

Q.?2 Ig y : x +x

1, tdke sdlw tdat ?x

bx

by+ y : ? x . ^C.O.S.K. ?887Z

Q.?; Biggkrketiatk w.r.t. x = tae1

--+

-++

xsie1xsie1

xsie1xsie1^C.O.S.K. ?887Z

Q.?7 Ig y 1x ? + : fl` ( 1x ? + x), prlvk tdat (x?+ 1)bx

by+ xy + 1 : 8. ^C.O.S.K. ?887Z

Q.?0 Ig x : a sie ?t (1 + cls ?t) aeb y : o cls ?t (1 cls ?t), sdlw tdata

o

bx

by

2t

:

p

:

.

^C.O.S.K. ?887Z

Q.?4 Ig y : clskc x + clt x, sdlw tdat sie x . ?

?

bxyb : y?. ^C.O.S.K. ?887Z

Q.?9 Wkrigy FJW > gieb c g(x) : x?+ ?x + 3 ie ^2, 7Z ^C.O.S.K. ?887Z

Q.38 Ig g(x) :

:

-

-

;x,n

;x,;x

?;x?

is cletieulus at x : ;, gieb tdk vafuk lg n. ^C.O.S.K. ?880Z

Q.31 Ig y : 3k?x+ ?k3x, prlvk tdat ?

?

bx

yb ;

bx

by+ 7y : 8. ^C.O.S.K. ?880Z

Q.3? Ig y : A kjx+ O kex, prlvk tdat?

?

bx

yb (j + e)

bx

by+ jey : 8. ^C.O.S.K. ?880Z

Q.33 Ig y : sie (fl` x), prlvk tdat x??

?

bx

yb+ x

bx

by+ y : 8. ^C.O.S.K. ?880Z

Q.32 Glr wdat vafuk lg n is tdk glfflwie` guectile cletieulus at x : ?

Documents

5_Continuity and Differentiability