MATH 210 Exam 2, Spring 2004

Math 210 Quiz #2, April 26, 2004 “Oh! the little more, and how much it is! And the little less, and what worlds away!” [ Robert Browning] 1. (20 pts.) Let fbe the function defined by x*sin(l),x #0 f= 4° SS 0, x=0. Prove that fis differentiable, and that fis continuous but not bounded on (—o, co). 2. (20 pts.) Let c, be the sequence defined by 1 1 1 + Peeedt 2Jn,n = 1,2,3,.... Prove that c, is monotone and bounded. Ch = 1+ 3. (20 pts.) Letf: R+ Rbea differentiable function and suppose that f(x)| < 0.49 for all x € R. Prove that the equation f(x) = Fasing has a unique solution in R. ~ 4. (20 pts.) Let: [a,b] + R be a given function and suppose that, for each x € [a, d], lim,x/(¢) exists. Prove that fis bounded on (a, 6]. Hint: Denote the limit of fat x by Le. ie. lim-./(‘) = Lz, and show that, for each x € (a, b] , there is a nbhd V, where fis bounded. Be sure to specify a bound in V,. 5. (20 pts.) Let/: [a,b} + R be continuous on (a, b] and differentiable on (a, 6). Suppose that f(a) = 0, and there is a constant M > 0 such that if (x)| < Mif(x)| for all x € (a,b). () Ifc € (4,6) anda < x < c, prove that e)| < MM-\x — al < MMe — al, where Me = supecsee f(x). (ii) Ife € (a,b) and M. + 0, prove that 1 < Mic—al. (iti) Prove that f(x) = 0 for every x € (a, 5). y] J 2 € J. 2,48 ft so Pi aos Sa, Chr) ~ Ca = _ L-, = +5 wn° we a n - vp Fe yw fact weh Sty < — Cn