Math 117 Final exam solution - Math 117 Section A1 Solution to Final Exam Fall 2003-04 Time 2 hours Instructor Dr Michael Li 1(20 State the Following

Math 117 Final exam solution - Math 117 Section A1 Solution...

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Math 117 Section A1 Solution to Final Exam Fall 2003-04 Time: 2 hours Instructor: Dr. Michael Li 1. (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)Mf(b) orf(b)Mf(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.

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