MTH621 Final Term Solved Subjective by MS

T lmorcru (Thu duuu rule): Suppose lbnt g i, dllforculinblc Ill .ro nod / i• cll ffer m• t,rnbl r ai q(xo), Thn1 1 ho c·o11111 osit ,. f1111etlo11 h - Jo !J, dmul'<I by h(r) = f(g(r)).

is rliffcrm1inblt1111 ro. with lt'(L-O) = J'(g(.ro))g'(xo).

Proor Siu«• f I• d1fft1't'11t inbli• nl g(ro). w,• can writ,,

/(t) - /(.9(ro)) = l/'(9(:co)) + E(t)l[t - g(ro)I,

wh,-r"

4. 1. D crivnl ivc

Lc•I l ing I= g(r) yldd• Jim E(t) = E(g(.rol) = 0. 1-->g{zo)

/(g(r)) - f(g(rn)) = 1/'(g(ro)) + E(g(r))}lg(r) - g(ro)}.

Si nco h(r) = /{g(r)) . this lmJ>lil"ll thnl ( ,1.3)

l,(r) - h(xo) = 1/'(g(xo)) - E(g(x))Jg(.r) - g(:ro) _ (-1 I) r:-r:o

r-ro

Si uc4' g ii, coot I uuous at .r.o WI! ha,·~ lim E(g(.r)) = E(g(.ro)) = 0.

L-t.AO

Th c,roforc, ( I 1) l1t1plie1< t h hl 1,(r) h(x0 ) h {ro) = Jim ---- = J1(r1(:ro))g (ro), z-ro z - ro

3ll iJ.aLed Tbeormm . 5Up[h:Jise ~ lun f ~ l'i co1ninir_101~s on t hi~ .-,1,., i-;~·~i h,1

1 ·n·ill lo. hj ru:111 di fTL,. {au-

L1 nhJu uu lhc opn11 inl CT\'liJ (,~ . £). nntl J(,1j =- J(b). Tbrn J'(c)-==-0 for so,me c JH the

• lfHm i rrt r.:,- J ( a r b~ ~