Question

(1) True or False. Explain why or why not. Also, for each of (1)-(iv), graph f and L (if it exists) on one set of axes. (i) The linear approximation to f(x) = x2 at x = 0 is L(x) = 0. (ii) Linear approximation at x = O provides a good approximation to f(x) = (xl. (iii) If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x). (iv) When linear approximation is used to estimate the value of f(x) = In x near x = e the approximations are underestimates of the true value.

Answer 1

f(x) = x² 'y=x2 x font f(2)=4², f(0) =0 F'(x) = 2x, f(0) = 0 Now, L(x) = 0 +(x-o0x0 it is true that the linear appoonimation to f(n)=x? is Lex)=0 True Ang (ii) Linear apponimation to for at let us check, Len) = f (0) + (-a) f'(o) frm=181, f(0) = 0 104) Obatth of for But f'(x) does not enift of f(x)=1*). (1) True or False Explain why of why not? Criven f function L linear appoonimation of f (1) find linear appronimation of fem) at x=0 Formula used; . 10,0) linear appronimation Len) of fered at r=a is Lex) = f(a) + (4-8) f'(a) LM) = f(0) + (1-6) FO) f(x)=121 Because at r=0 slope of f(x) = 1 n) is two. there are two slepe enist at azo of ul. lim f(n) = -1 & lim flu) =+ , so f'(x) does hat enist. v not 20-

ch Since f'(x) does not enist so linear approximatis Len) of fonalul does not provide good approximation. false And gy ya math fra) = b im Na m ་ལ་ if f(x) = matb let find linear appsonimation of f(n)=mutb at general print x = a. 166) = f(a) + B-a) F'(a) f(M)= muitb: matb xx f'(o) = flea) = m. [(y) = ma +b + (x-a) xm =matbtmn-ma LAME mub = f(n) f(u) > so if f(M)=mutb, then linear approximation to of at any point is £en) = f(n) true Ane civ) fcm) f(x) = lns at rze f'm) = Kifice) = ė fle) = lne=1 Lex) = -f(e) + (n-e)f'ce) LOM= 1 + (x-e)xt It ze : 을 Llul x

how take borint near x=e. red-Y fraafn214-8070.875468737 0.875468737 ( tore value) Leno= x= 24 0.882910659 e clearly L (2) > Fen) near n=e so linear apponimation Len) of fen) = lnx over estimate near aze het underestimate. False) An xo e Len) y=inne e [(*) > fen) e