Actual Final Solutions - Chapter 6 Curve Fitting Two types of curve tting Least square regression Given data

# Actual Final Solutions - Chapter 6 Curve Fitting Two types...

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Solutions Final Exam — May. 14, 2014 1.(a) (10 points) State the formal definition of a Cauchy sequence of real numbers.A sequence,{an}nN, of real numbers, is Cauchy if and only if for every>0, there is aNNso that ifm, n > N, then|an-am|<.(b) (5 points) Give an example of a sequence of real numbers,{an}nN, which satisfies limn→∞|an+1-an| →0, but which isnotCauchy. You do not need to justify your answer.
Math 405, Fall 2015 Solutions to Final Exam — May. 14, 2014 Page 2 of 12 (c) (15 points) Arguing directly from the definition, show that if both{an}nNand{bn}nNareCauchy, then so is the sequence{anbn}nN.
Math 405, Fall 2015 Solutions to Final Exam — May. 14, 2014 Page 3 of 12 2.(a) (10 points) State the formal definition of a compact subset (ofR). (b) (5 points) Give an example of a non-compact setAand a continuous functionf:ARso thatthere is nox0Aso thatf(x0)f(x) for allxA– i.e.,fdoesnotachieve its maximum. Youdo not need to justify your answer.
Math 405, Fall 2015 Solutions to Final Exam — May. 14, 2014 Page 4 of 12
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