differentiation all pratic sit www.euelibrary

8/3/2019 Differentiation All Pratic Sit Www.euelibrary

1/11

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2/11

G,nr

- practiccShcet #l, Function, Dornain & Rangc,':,

l:ind out thc dornain and rangc ol'tlrc fbllowing Iunctions and also sketch the graph:

"f (x) - x-42x

.r'-

.I'1.r

,I(0,0(x


3/11

V(;tJx -12. /(x)=lLJ_XIrind lirn./(.r)

[x'+lliIIl+x./'(x)

r l.l],-t[,'"f(x)

..$ $ry\Sa14. ./(x) =Find linr-t->0

irg./(x)rrd

l2x +l15. ./(x) =.j.L-) -.r

8.

ti,',', Ir, 1)', ,,rr\ x )

l. lirn --j---,u Jx+l-l

Eastcrn UnivcrsityCalculus I (MAT-101)Practicc Shect#2

Irind lim'./'(x)

I ,xl

+l ,rll.r) ciist?,

,t'> 0,.I=0,I'( 0

-l ,'r'zl\t I \4 \V,0


4/11

Practice Sheet #3ContinuitY and DifierentiabilitYGulshan Ikhtun

(a) Test the continuity of the following functions:[cosx, r20 at x=0. 6.l. f(x)=l_.orr, x


5/11

Gulsh:rlr I(ahtrrnCalculus I (MA'I':I0l)I)racticc Shcet#4 ..I ntlctcrnri natc Forrns

I llr lT, 2ly; ;i*.", 3 i,:t;(*-#J,4. lll#,5' lll,l {-liil'l}, 6 ,r-,lllCr, , pl(.'*,-)'", *. l,j;(* ,*), ,. 1if,." ,,10. lilll(-*,r)c.r.x , n. rirng+q. 12. rir, IlEr, 13. rirrrl+. 14. rirrr:I- .'-''r ltt(tatt x) i-ri; .t ' .r-,o ...-, r -r., c ,t5. ti.rIi- .L l. I6. tirnf'Ia",, \ r Xe' )' r-ro 5ip'J; '

t'l. \rcrili,thc hyltothcsis / Discuss r.lrc application ol. Rolle's Thcorom lor rhc Ibllovl,ingli i rrc I iolt s:

' Fitrcl thc linrit usirrg 1.,,1-lospital rulc:

;r) ,/'(.r) .= .r2 * 6.r--r- 8; [2, +1*-5f,f(r) = cos.r:; 1itZ,3r /Z)c) /'(.i') = Jes-'; [0,5]2. Vcril), thc hypothcsis / Discuss thc applical.ionlill Iorving Ibnctions:

a) ./(.r) =.Y3 r' x-4; t-1,2)tr) / (r) .. J.^' ',1 t' ; [0, 3 |c) / (.r) = {25 4; [0, 5l

I

ol' Mcan Valuc 'Ihcorcrn tbr tlrc

iMaclaurin arrri 'l-avlor Sericsl.I;indthc,I.aylorScricsforthetbllorvingfultctions;..:''.',..-,'.-

2. l:xlxrnd .,l' - llr.r. in thc l)o\\/cr ol' "r-- ? arrcl ),= a,,, irt tlrc powcr ol' ri- I ..j. ljind tlrc MaclaLrrin Serics lbr tho lirnctiolr c"' ilnd cos.r.4. l:xpand /- = ln(x -r- I) ancl .)u - sirr .r / cos x ilr thc potvcr of x .


6/11

Calculus I (MAT-I 0t'),;.'i;,1 ",:P ril cticc S lta,st#Sr ;,i,;',. r',.

i..'.: ",l'cchrrirlrrcs of DilTcrciltiition'i.,i.i:::,: .,..'lrincl thc dil'tcrcntial cocllicicnts (,, *)of thc following f'unitions w.r..to r'

(i) 3x' - x' ), -v 2yi - A (ii) y =1

(iv) .r=.sint t . y= talt/

, .'Irirrd tltc n th clcrivativc of tlrc fottu*ing functions:

-x)' r-;;.1''r .b-) 'y=(r,.r+il1' -a) y=ln(ax+J+) d|-ye+{r+a} e)-y=cos{ortr}' j1

3) .)': sin(ax + 6) t ' -\ isirr Dx , thcrt sltow thflt .y2 -?a.1,, + (rl2 + hz )1,- 0.sirr r', lhcn show thal yu a'4y = Q .

Lcibnitz.'s Thcoremx. tlrcn show that.(l +.rl )y,,,, +2(n it)*.y,*, * r(,r''*0r,. = 6.; i.;i'i;'::1-':''i""" '- sirr-rx, tlrcn show thar (t-r')y,,.,-(21.*i\!y,-'rl2 y,,-, =0.i: -'. tlrcn show that (l* r').p,,*, +(Z.nx+ 2x- l)4y1.,,*,1.1,1i l).1',, = 0.

'' .r)', thcn show that (l - r')y,,r -(2,,+l)* y,*, - n2ln

. .i -.i',r::-r'r | '

l) .1'= c'l[' .1' ;- ."'ril' t,= c'

il' .t,= tan-llt' t,{ I - .r"'

l:ldll Y=AIIl' .r,= (sirr =fl.

),,,b' * *'

I+x\-tl-r) (iii) xt' = y' (iv) (sin x)r"'-

(v) ./ = sit't xsin 2xsin 3:r.

Il' .r, = grrrsin-r'* , t6cn show tlrnt (l - "')),,,*r, -(ir, +l)*y,*, =Q.


7/11

e-,.t/

'Proccss of dctcrmininglhc meximum rnd minintum valuc of a function:Lct f(x) is a l'unction. .To dbtcrntinc tlrc rnaxi,rrrunr arrd minimum value of the function wc may

' foltow thc lbllowing stcPs-: ng points sotvc it for'/''(r) = 0 ' I-ct thc solution is aStcp-2'. to delcrmine the turning points solvc it f-o .. . .,.Stcp-3: if .f'(a)giys5 *'vc vatuc tlrcn thc I'urtcriott ltas a rttittitrrunl valuc fora. *' 'a'rrrf" "' .lf .f'(a)givcs -'vc valuc thcn thc lurrctiott has a maximunr valuc fora .Sten-.{: thcn.to ljn


8/11

l.

Assignment # IMaxima and minima

Last date of submission 18/02/08

Find (a) the open intervals on which / is increasing , (b) the open intervals onwhich/ is decreasing, (c) the open intervals on which / is concave up' (d) ) theopen intervals on which f is concave down and (e) the x- coordinate of allinflection Points.(i).f(x) = xz -5x+6 (ii) f(x) = 5 +llx- xl(iii)f(x)=x4 -8x2 +16 (iv)/(x) =* (v),f(x) ='J;+2"ut?(vi) f (x) = Cosx; 10,2r) (vii) f (x)= t&nri Vi'ilIocate the critical numbers and identify which critical numbers correspond tostationary Points.

(i) f (x) = x3 +3x2 -9t+l (ii)f (x) = x4 - 6xz -'3(iii)f (x) =;e Gv) f (x) = x2t3(r)/(x) = x't3 (x + 4) (vi) f (x) = cos3x'Find the relative extrema (maximai minima) using both the first and secondderivative tests.(i)/(x) =1xt -9x' +12x (ii)f (x)= I-sinx' 0 < x


9/11

Eastern UniversityFaculty of Engineering & TechnologyB.Sc. Engg. Mid Term Exam, Spring- 20llCourse Name: Calculus I (Day)Course No. MAT-l0llll7(Group 1 & 3)Total Time: 90 Minutes Total Marks: 30NB..I. Answer anv three questions.

2. The right margin indicates the marks associated with each question.,t't.a),/verify the Rolle's theorem for the function -f (x) = x' - x' - 4x+'4 in the l4l/ interval (-t t\

/rrrlvr vsr \ o, o I

b) A.furrction is defined as fbllows:,/ l2x+6 ,-21x 10/ /rrl=]8 ,o'-x


10/11

Define i) funct'ion & ii) limii with example.- . : .State Leibnitz's Theorem.

- , . a' ', IIf y = (-"-tr)' , then show that,(t - r' )f*, -(2n + l\' * ! n*,r' n, y, =0 . (uslng feibriitz's Theorem)

':i

\:': : '

4Ja)b)

"c)t3l

trl

, . ., .r. -").{iskre.S;r!.r':::is1]:,,,i L_ -.-i:.,.l*d'"t.'


11/11

,rn estthe Differe".'f:i'1#.:: f"t*Let f(x)=lire ,'.x2e at x=e/vetify the Mean Value theorem for the firnction f (x) = x - xt in the tsl( interval(-2,1).

- 4. lf y = "2sin'.r", then using Leibnitz's theorem showthat'(, - *') h., - (2, +1\ x v,*, - (o' * 4) t, = o -

OR--' ,/*{t y= s' sinx, then show that la + 4y = Q.Eastcrn Univenity

B.Sc- Engg. Quidl, SPring- 20ll 'Course Namc Celculus I(MAT 101) @ay & Group 3)

Eastern UniversitYB.Sc. Engg. Quiz#2, SPring- 2011Course Name: Calculus I(MAT 101) (Day & Group 3)

Total Time: 30 minutesAnswer all the questions..al. ll Define the continuitY?-,/'ft State the Rolle's theorem.

Total Time: 30 minuteAnswer all the questions.1. Define domain and range of a function with example'2. A function is defined as follows:

f (x)=x' ,r(0x,0

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