Calculus and Analytic Geometry
Calculus and Analytic Geometry MATH 222
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This 4 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 222 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/205291/math-222-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.
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Date Created: 09/17/15
Worksheet 10 Math 222 7 Lecture 1 7 Section 315 WES Wednesday Sept 24 2008 H A True or False if true explain why if false give 2 examples 1 g exists then both lim 1 and lim 9 exist 4111 4111 1 g exists then both lim 1 and lim 9 exist a 4111 4111 f z g exists and lim 9 exists then lim 1 exists 4111 4111 to For the following evaluate or show they diverge r 2 vs a 57 dz sec z A d 0 b tanm 7 12 d C 16 7 2mm 3 7 m0 03 For each of the following predict the values of r that make the integral converge Why Then actually gure it out For the values of r for which the integral converges to what number does it converge 00 1 d b 00 71 d W 1 ltgt Law 1 4 a Consider the integral 1 dm m0 For what values of r does the integral convergediverge If it converges to what value b Using the results from a and the last problem for what values of r does the integral CO dm m0 converge c Repeat part a carefully for the integral 1 dm zln z m0 T 5 ls there a constant T such that lnz dz 0 If so nd it and draw a graph of y lnz which m0 includes r and show geometrically what s happening If not explain why Worksheet 14 Math 222 7 Lecture 1 7 Section 315 WES Friday7 Oct 107 2008 Happy Friday 1 Complete the following de nitions A sequence is A series is The sequence an has limit L if and only The series 221 an converges to the sum L if 2 For each of the following7 0 write the series in the form Zg t an7 o calculate the nth partial sum break into cases if need he7 and 0 state whether the series converges or not If it does7 what does it converge to ai111 f 15 lillillilli b111m g 2 2 3 3 4 4 hllll 61111 2 3 4 5 2 3 d1 1166 e e1 j1e 1e 2e 3 U a 1 Find an example of a series with non zero terms that satis es CO 00 a 2 an e an 0 and 2 an converges 1 n1 CO 00 b 2 an 73 f lim an 0 and 2 an diverges n1 H00 n1 CO 00 c 2 an t for t 7 0 g lim an 7 0 and 2 an converges n1 H00 n1 CO CO d an 0 h lim an 0 and an diver es g mace 1 n1 Classify the following statements as true of false If true explain lf false give a counter example a If the sequence an converges then the sequence 02 converges b If the sequence an diverges then the sequence 02 diverges If the sequence 02 converges then the sequence an converges If the sequence 02 diverges then the sequence 02 diverges CO CO A C d AAA e If the series E an converges then the series E agn converges n1 n1 00 00 f If the series 2 an diverges then the series 2 agn diverges n1 n1 00 00 g If the series 2 agn converges then the series 2 an converges n1 n1 00 00 h If the series 2 agn diverges then the series 2 an diverges n1 n1 Calculate the following without the use of a calculator e n0 10 10 a Z 2 b 2 2 0 n2 00 1 Show that if an is any sequence of positive numbers then S an 7 diverges an n1 Prof Nagel talked Monday about the Fibonacci sequence 1 1 2 3 5 8 de ned recursively by an1 an 1771 a1 17 a2 1 He calculated that the sequence bn MMan converges to L 1 a Does this limit depend on our choice of a1 and a2 For example what is limnH00 1 if we instead chose an 4 and a2 7139 b What happens if we de ne a new Fibonacci like sequence de ne recursively by an1 2am 1771 a1 17 a2 17 Write down the rst few terms in this sequence What is limH00 b Worksheet 11 Math 222 7 Lecture 1 7 Section 315 WES Friday7 Sept 267 2008 H Fill in the blank iz p dz converges if and only p gt 1 m p dz converges if and only p lt 1 Explain the phrase A if and only if B77 to For each of the following integrals7 determine whether it converges i on the interval 07 17 ii on the interval 17 00 iii on both or iv on neither Then calculate the integral over the interval 07 oo7 or state that it diverges a dz d 7jdx g dz 1 z 1 b Fm e mdm h 2L1dm Hint for at least one of the converging integrals7 it may be useful to verify u41 u22u1u272u1 Also7 1 1 1 U2 x2u 1u2 7 x2u 1 2u22u1 2u27x2u7l71 For a real number n the Gamma function is de ned by 00 tn le zt 0 It is used widely in probability and statistics and generalizes a very important basic idea in mathe matics7 which we will investigate below a Find P1 03 PM A b Use integration by parts to show Pn 1 c What is NB N4 N6 any patterns
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