Question

Problem 1.1. Suppose that f: R → R and that f is differentiable at z = a. 1. Show that, given an angle 6, we can choose 6(0) > 0 small enough so that for all r such that r - al < (0) we have that the graph of f(r) lies inside of the cone with angle e around the tangent line. 2. Can you find explicit formulas for 6(0) for the function f(x) = c²+cx+ ca for any arbitrary a? Hint: first solve for g(h) = f(a+h)– f[a) – La(h) where, of course La(h) = f'(a)h y= f(r) = f(a) + L(h) + sh) fla) + l(A) 26(0) f(a) y- f(a) + La(A) - ch f(a) + L(h) + eh Figure 1: Here is a picture to stimulate your thoughts and explorations. La(h) is a linear function from R to R - a line through the origin.

number 1 and 2 pls

Answer 1

0 we know lim ten) fic) afle) щ ч ) . +) +{\cs) •h - €, .. Cпh) = a*) – 4 ж) Фо: 2rn" (0) E: Ten tel: ) we get хян) - 8«++) = "h) +€ 8 мм) -8» (к) = 34) - tim ст4) - 8 th) . тел (6(1 ) hto От 14) - Чең) - - - -т-( ) ho sine Tan (2) » i iн е 4 (0) ясън) - см) »o 4 4 (h) 4.0) Жети) - 56 (4) к se) . ms 1 < 0) , (0)] + ho | k4 6 2) Яс»++) {сти) «Ч» си) » Спн) • усън) екст) 8t ev) Кл) +) + (14) с 10 )

(2) Giren fem) - cim" 724 +63 flim) = 2an the - f'ca) = 2act feath) - tha) + lach) + g(n) =) , Cath the cath) to ca teza tl3 th (sacitee)+ geh = .. cur a cla te atyt non +2h trauch taauh teh teh Cla tcha tez - hlaacite it gh) so gin) = ant point on some orbitary and his it is since Yanis constant gen) = ca is given as formula Hence explicit y = a +6₂ a tatzac, te to-gen)