Math 117 Section A1 Solution to Final ExamFall 2003-04Time: 2 hoursInstructor: Dr. Michael Li1. (20%) State the Following Theorems.(1) Balzano-Weierstrass TheoremA bounded sequence has a convergent subsequence.(2) Intermediate Value TheoremAssume thatfis continuous in [a, b] and thatf(a)≤M≤f(b) orf(b)≤M≤f(a).Then there existsc∈(a, b) such thatf(c) =M.(3) Lagrange Mean Value TheoremAssume that (i)f(x) is continuous in [a, b],(ii)f(x) is differentiable in (a, b).Thenthere existsc∈(a, b) such thatf(c) =f(b)-f(a)b-a.(4) Fundamental Theorem of CalculusAssume thatbaf(x)dxexists andF(x) =f(x).Thenbaf(x)dx=F(b)-F(a).2. (20%) Show that functionf(x) =x2sin1xx= 00,x= 0is differentiable atx= 0 but is notC1atx= 0.