page 1 of 11 Math 1B Final Exam Friday 14 August 2009 Instructor Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Name Problem Number Score Maximum 1 2 3 20 20 10 4 5 6 7 15 10 10 15 Total 100 Please do not begin this test until 2 10 p m You may work on the exam until 4 p m Please do not leave during the last 15 minutes of the exam time You must always justify your answers show your work show it neatly and when in doubt use words and pictures to explain your reasoning Please box your final answers Calculators are not allowed Please sign the following honor code I the student whose name and signature appear on this midterm have completed the exam by myself without any help during the exam from other people or from sources other than my allowed one page hand written cheat sheet Moreover I have not provided any aid to other students in the class during the exam I understand that cheating prevents me from learning and hurts other students by creating an atmosphere of distrust I consider myself to be an honorable person and I have not cheated on this exam in any way I promise to take an active part in seeing to it that others also do not cheat Signature Name Math 1B Final Exam 14 August 2009 page 2 of 11 1 20 pts total 2 pts each For each of the following statements determine if the conclusion ALWAYS follows from the assumptions if the conclusion is SOMETIMES true given the assumptions or if the conclusion is NEVER true given the assumptions You do not need to show any work or justify your answers for these question only your answer will be graded P a 2 pts If limn an 1 then 1 an converges ALWAYS SOMETIMES b 2 pts If the series ALWAYS P 1 an and P 1 NEVER bn both converge then SOMETIMES P 1 an bn converges NEVER c 2 pts If the sequences an and bn both diverge then an bn diverges ALWAYS d 2 pts If SOMETIMES PN 1 ALWAYS e 2 pts If ALWAYS NEVER an 10 for every N and if an 0 for every n then SOMETIMES P 1 cn 2n converges absolutely then SOMETIMES P 1 an converges NEVER P 1 cn 2 n converges conditionally NEVER Name Math 1B Final Exam 14 August 2009 f 2 pts If P 1 cn 2 n converges conditionally then ALWAYS P 1 SOMETIMES cn converges absolutely NEVER g 2 pts If bn is a decreasing positive sequence then ALWAYS page 3 of 11 P SOMETIMES 1 1 n bn converges NEVER h P 2 pts If p is a real number then the Ratio Test can be used to determine whether p 1 1 n converges ALWAYS i 2 pts If 0 an bn and ALWAYS SOMETIMES P 1 an converges then SOMETIMES j 2 pts If limn bn exists then ALWAYS P 1 NEVER P 1 bn converges NEVER bn bn 1 converges to b1 SOMETIMES NEVER Name Math 1B Final Exam 14 August 2009 page 4 of 11 2 20 pts total 5 pts each Determine whether each of the following series is ABSOLUTELY CONVERGENT CONDITIONALLY CONVERGENT or DIVERGENT You must specify which test s you use for each series and why the series satisfies the conditions of the test a 5 pts X 1 n n 1 n 1 b 5 pts n X 1 n n2 n 0 n 1 Name Math 1B Final Exam 14 August 2009 c 5 pts X 1 n n 2 d 5 pts ln n X 1 n n3 n n 1 page 5 of 11 Name Math 1B Final Exam 14 August 2009 3 10 pts Find the interval of convergence of the following power series X x 4 n n 2 n ln n 2n page 6 of 11 Name Math 1B Final Exam 14 August 2009 page 7 of 11 4 15 pts Evaluate the following definite integral as a series Z 1 p 3 1 x6 dx 0 You must both Write your final answer in notation but you may leave your answer in terms of the binomial coefficients nk Write out the first four terms of the series Name Math 1B Final Exam 14 August 2009 page 8 of 11 5 a 5 pts Find power series representations centered at 0 for each of the following functions and state the intervals of convergence for each series ln 1 x ln 1 x2 ln 1 x b 5 pts Show that as power series the representations for the above functions satisfy ln 1 x2 ln 1 x ln 1 x Name Math 1B Final Exam 14 August 2009 page 9 of 11 6 a 2 pts State the power series representation for arctan x centered at 0 What is its interval of convergence b 4 pts What is tan 6 Use this value for x in the power series representation to find a series that converges to 6 Is the convergence absolute or conditional c 4 pts Use the error estimate for an Alternating Series to determine how many summands you would need in order to use this series to estimate 6 correct to three decimal places error 0 001 Name Math 1B Final Exam 14 August 2009 page 10 of 11 7 15 pts Find a power series representation centered at 0 for the solution to the following initial value problem y 00 xy 0 2y 0 y 0 1 y 0 0 1 Name Math 1B Final Exam 14 August 2009 page 11 of 11 References All the problems on this midterm are due to the instructor although they are loosely based on the material in Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart The honor code language is adapted from the Stanford Honor Code http www stanford edu dept vpsa judicialaffairs guiding honorcode htm and from the exams by Zvezda Stankova Feel free to use this page for extra scrap work
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