Question

Consider the generalized integrator function (2) discussed in class, defined by its proper- ties: | dr 8(x) = 1, Ve > 0, | dx 8(x) = 12+ = ſo if r* <0 11 if x* 20' dx 8(2 – c)f(x) = f(c), VER, where dc 8() is understood as a slight abuse of notation and f(x) in the last formula is a suitably well-behaved (at least bounded and continuous - and perhaps even smoother- in a neighborhood of x=c) function of x. Remember that (2) can be defined by a limiting process with respect to a sequence of functions chosen by our desire/need for smoothness vs. compactness (~ localization to some bounded interval of x) in any particular instance, e.g.: So if x <0 or x 2 8(2) = lim na n+ n if 0 < x < 1 3(x) = Lim »(13 - 1,-_) – lim {iro n +00 = lim nềrt- 100 1-1 - 2rl + -- n = limen22/2 = lim ** 100 V 27 0 2t (a) Express (ar) for a ER in terms of 8(2). What is 8(-r)? Hint: consider the change of variables y = ar. (b) Consider a bounded Cl function g(x) with a finite set of real roots: {TER: g(x)=0, i=1, ..., M}, where all M of the x are assumed to be simple roots, i.e. g'(x=r) +0. Write (g(x)) and dx (g(x))f(x) in terms of the x. Interpret the value of the integral de 8(g(2)) 9' (2) in relation to the properties of g(2). (c) Express 8(x-c) and (a(x-c)) for a € R and x, CE RN in terms of the N individual components (Li-ci), i = 1, ..., N. 8(x+h) - 0(2) (d) Define the derivative 8'(x) of 8(x) by: 8'(x) = limu h+0 h Assuming that the usual integration by parts formulaſ dg f = fg - ſ df g holds for g(x)=8'(20) and g(x)=8(.1), evaluate de 8'(:1 — c)f(x) for f(c) E C(R). J-00

Answer 1

olution: RO. f oli vai are then stayda = $conde > Sam stoly a scorder where we substitute, ya aral and une change of vouchten. 56, s(ov) = ran Špay, for ayo. Apgain, but aco chen Sie einde Sibasa) dhe se non solo by to renta where we substitute y=-x [al oned one the change of venialien. So, $(69) thg 5(2), fre aso. Thus, d (9x) = 1 a 8(x) for all ato. &(-) = F 867) = $(x) = $(47)= 8(2).. (6) Comidoring all the given concertion, we may write 8(2609) 8(96) – Tgs (2017 : 5 scam) frode = 3 Toren $(427){29th = ? Toen DF, 82=18(e), then we directly chain (špan)Yelek= 1 = there 18(90) 1 760)da M = number of simple zero of 969–o. - 00

As a menowote, then-dion 7 O w product megoldle of the 11-dimemimal desta function in each vcruiable setorately. (C) For X, CERN we have $(2-4)= 6(x,-4, 7-6,.-, X-w) = $(1,-4)5(-6)....(w=) :: 8(4-5)= FT 8(2-6). uring the property i (a), we get <f«(2-ci)= K 8 (2-2)= ikat fits (:-(:). (1) thing integration ay par to, we get Sate of E-c) feo = -c)302 ) –c) f(a)dx = -f(e), since 8(2-0)70 a antas. Shun patu s'la-c) f(2)= -f%CC). loooox 200
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