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# Calculus and Analytic Geometry MATH 222

UW

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This 6 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 Zajj Daugherty in Fall. Since its upload, it has received 8 views. For similar materials see /class/205270/math-222-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15

1 to Worksheet 23 Math 222 7 Lecture 1 7 Section 315 WES Monday Nov 3 2008 Power series everyones favorite functions are polynomials l have a secret function in a box about which I claim two things 1 it can be written as an in nite power series and 2 when I take its derivative 1 get 0 times the function back ie of c is some non zero constant a What is this super secret series Hint all power series look like 00 a0 alm 1 0Lng ang 0an n1 Take this things derivative and set it equal to the series and solve for the coef cients Look for patterns b For what z does this series converge c What s another function you know which satis es part 2 d For those x which causes this series to converge what does it converge to e is irrational Recall that a rational number is one which can be written as the ratio of integers and that an irrational number is one that can t In general proving that a number is rational is a lot easier than proving that it isn tithe former just requires you to nd the appropriate fraction whereas the latter can t be done by example you can t try all possibilities since there are in nitely many Use the series expansion of em to prove that 15 is not rational and therefore neither is e hint a Assume you can write 5 as pq where p and q are integers and are reduced there are no factors that p and q have in common So what s 15 b Plug in z 71 into the series expansion for em and use some partial sum to estimate 15 What kind of series do you get What s the error of your estimate What does that mean about the distance between qp and 3 Write this inequality down c You can pick an 71 large enough so that there is something you can multiply both sides by to get integers on both sides How large does 71 have to be How do you clear all the fractions in a useful way What can you conclude d Find a contradiction of your assumption in part a What does this mean Recall we can write decimal expansions like power series with z 110 00 b 2a110 n0 where an is an integer and for n 2 1 an is inclusively between 0 and 9 a What are the an if b Q 999999 b Take another look at the series from the last part what kind of series is this What does it converge to What have you shown 8457 32 Consider the series 3 00 nn1 2 n1 What is the radius of convergence of this series AA 793 VV Convince yourself that for certain values of y 0 Zyn1 392 1 n 1 Differentiate this identity twice c asusiuioncanyoumaeoryinpar oge onesieooo 1eour Wht bt tt kf tbt t dtlkl k series d What does our series converge to For what z p 8457 35 Quality control a Assume z lt 1 and differential the series 1m2 17x to get a series for 171 b In one throw of two dice7 what is the probability of getting a 7 call this proba bility p If you throw the dice repeatedly7 the probability that a sum of 7 will appear for the rst time at the nth throw is qn lp7 where q 1 7 p why The expected number of throws untill a sum of 7 rst appears is 2201 nqn lp why Find the sum of this series A O V As an engineer applying statistical control to an industrial operation7 you inspect items taken at random from the assembly line You classify each sampled item as either good or bad If the probability of an item being good is 197 then the probability of the item being bad is q 1 7 p Then the probability that the rst bad item is the nth one is pn lq So the average number inspected up to and including the rst bad item found is 2201 npn lq What are the bounds on p and q Evaluate the average Worksheet 24 Math 222 7 Lecture 1 7 Section 315 WES Tuesday7 Nov 57 2008 Recall that the Taylor series expansion around the point 0 is 1 2 03 F M 7 0 f 0z Explain the ideas behind Taylor series Calculate the Taylor series for the following function around center z 0 some of these will be repeats from yesterday7 but go through them on your own now anyway 3 a x sinm W 52 b 1 00890 0 x Let s convince ourselves that this series expansion business might make sense Show that some of our favorite trig identities hold for these power series a sin7m 7 sinm b cos7m cosz c cos2z sin2m 1 hint multiplying in nite polynomials requires a little care7 but is not impos sible Start by asking where the constant term comes from7 where the coefficient on x comes from7 where the coefficient on 2 comes from7 and so on d sina b sina cosb sinb cosa hint start on the right Stay organizedll Use the power series expansion of em to evaluate an wherei v71 Can you write this series in terms of the series for sinm and cosz What s the non intuitive identity you ve just proven 5 a p 845 32 Consider the series i nn 1 n1 35 a What is the radius of convergence of this series b Convince yourself that for certain values of y i yn1 242 1 i y n 1 Differentiate this identity twice c What substitution can you make for y in part b to get one side to look like our series d What does our series converge to For what z e Go backwards 7 take what you found this series to converge to and calculate its Maclaurin series Only after you ve nished the rest If you ve nished the rest of the worksheet ask for a list of numbers As we ve already heard the other day as Ruth and Zajj were cleaning up an alien appeared in the WES classroom The alien told them I m thinking of two integers both not smaller than 3 and not bigger than 97 I will tell their sum to Ruth and their product to Zajj The alien did this and then disappeared The following conversation occurred Ruth You dont know what the numbers are Zajj Well I didn t but now I do Ruth Now I do too Assuming Zajj and Ruth weren t using their psychic powers determine what the two integers were Hint Think about the sum rst 1 Can it be the sum of two primes 2 Can it be big enough that 97 could be a summand 3 Can it be big enough that 47 can be a summand 4 Can it be the sum of a power of 2 and a prime Worksheet 22 Math 222 7 Lecture 1 7 Section 315 WES Happy Halloween Friday Oct 31 2008 Power series everyones favorite functions are polynomials Unpacking the concepts this stuff might look familiar but it s good to go through 1 For what z does 231 azn l converge converge conditionally converge absolutely What kind of series is this 2 For those z for which 2301 azn l converges what does it converge to 3 For each of the following z s i estimate 231 zn l by calculating the rst few partial sums ii what is 231 mn l iii how far out in the sequence of partial sums do you have to go to get within 1100 of the actual value a 35110 c 3512 e 35910 b 357110 d 35712 f 357910 1 Conjecture the sequence of partial sums of 221 mn converge faster when 4 Turn to page 799 in your book and read up on testing for convergence For what z to the following series converge absolutely converge conditionally diverge Z n 4 c Zzninm n1 a M8 E18 E 3 H H 5 a T I have a secret function in a box about which I claim two things 1 it can me written as an in nite power series and 2 when I take its derivative I get the function back ie dz a What is this super secret series Hint all power series look like 00 a0 alm 0Lng ang Zanm n1 Take this thing s derivative and set it equal to the series and solve for the coef cients Look for patterns b For what z does this series converge c What s another function you know which satis es part 2 d For those x which causes this series to converge what does it converge to Recall that a rational number is one which can be written as the ratio of integers and that an irrational number is one that can t In general proving that a number is rational is a lot easier than proving that it isn tithe former just requires you to nd the appropriate fraction whereas the latter can t be done by example you can t try all possibilities since there are in nitely manyll e is irrational Use the series expansion of em to prove that 15 is not rational and therefore neither is e hint a Assume you can write 5 as pq where p and q are integers and are reduced there are no factors that p and q have in common So what s 15 b Plug in z 71 into the series expansion for em and use some partial sum to estimate 15 What kind of series do you get What s the error of your estimate What does that mean about the distance between qp and 3 Write this inequality down A O V You can pick an 71 large enough so that there is something you can multiply both sides by to get integers on both sides How large does 71 have to be How do you clear all the fractions in a useful way What can you conclude d Find a contradiction of your assumption in part a What does this mean The other day as Ruth and Zajj were cleaning up an alien appeared in the WES classroom The alien told them I m thinking of two integers both bigger than 3 and smaller than 97 I will tell their sum to Ruth and their product to Zajj77 The alien did this and then disappeared The following conversation occurred Ruth You dont know what the numbers are Zajj Well I didn t but now I do Ruth Now I do too Assuming Zajj and Ruth weren t using their psychic powers determine what the two integers were

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