MAT 137Y: Calculus! Problem Set 9. Due in tutorial on March 21-22 Instructions: Print this cover page, ll it out entirely, and STAPLE it to the front of your problem set solutions. (You do not need to print the questions.) Doing this correctly is worth 1 mark. Submit your problem set ONLY in the tutorial in which you are enrolled. Before you attempt this problem set do all the practice problems from sections 12.1, 12.2, 12.3. PLEASE NOTE that so far over 26 students have been penalized for academic misconduct and now have a record with OSAI. Do not be the next one. Re-read \"Important notes on collaboration\" on the cover page for Problem Set 1. Last name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tutorial code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TA name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Please, double-check your tutorial code on blackboard, and double-check your TA name on the course website. Remember that if there is a discrepancy between Blackboard and ROSI/ACORN, then your correct tutorial is the one on Blackboard, not on ROSI/ACORN. See http://uoft.me/137tutorials 1. Assume we know the following about a sequence: lim an = 0. With only this inn formation, we are going to show you various series. For each one of them decide whether the series must be convergent, the series must be divergent, or it could be either. If you think the series must be convergent, prove it. If you think the series must be divergent, prove it. If you think it could be either, give one example that shows the series may be convergent and one example that shows the series may be divergent. (a) The series an . n=0 a2 . n (b) The series n=0 |an |. (c) The series n=0 (d) The series n=1 an . n 2. Read the Limit Comparison Test for series (Theorem 12.3.7 on the book) and its proof. (a) Assume that in the statement of the LCT we accept the case L = 0. Show with an example that the theorem is no longer true. (b) While the conclusion of the theorem is no longer true, one of the two directions in the \"i\" statement is true. Which one is it? Prove it. 3. Innite decimal expansions. We can interpret any nite decimal expansion as a nite sum. For example: 2.13096 = 2 + 3 0 9 6 1 + 2+ 3+ 4+ 5 10 10 10 10 10 Similarly, we can interpret any innite decimal expansion as an innite series. For example: = 3.141592... = 3 + 1 4 1 5 9 2 + 2 + 3 + 4 + 5 + 6 + ... 10 10 10 10 10 10 Interpret the following numbers as a series, then add up the series to calculate its exact value as a rational number. (a) 0.11111 . . . (b) 0.9999 . . . (c) 0.25252525 . . . (d) 0.3121212 . . . 4. In Problem Set 5 you used the Mean Value Theorem to learn how to approximate functions via polynomials and estimate the error. In this problem you are going to take a dierent approach to the same problem. For this problem, assume that all functions have all their derivatives at all points of their respective domains, and that they are all dened at least on an interval around 0. (a) Denition: Assume that lim f (x) = lim g(x) = 0. We say that f (x) approaches x0 x0 0 faster than g(x), as x 0, when f (x) = 0. x0 g(x) lim Consider the functions h2 (x) = x2 , h3 (x) = x3 , h4 (x) = x4 . According to the denition, as x 0, which one approaches 0 the fastest of the three? Which one approaches 0 the slowest? Draw their graphs on the same axes. Your graph should agree with the explanation you just gave. (b) Let f be a function. Let's say we want to approximate f using a dierent, simpler function P (probably a polynomial) near x = 0. We dene the error of this approximation as E(x) = f (x) P (x). If the approximation is going to be any good, we want that lim E(x) = 0. Actually, we would like something else: x0 we would like that E(x) approaches 0 \"fast\" as x 0. Denition: We say that P is a good approximation of order n for f near x = 0 when E(x) approaches 0 faster than hn (x) = xn . x2 is a good approximation of order 3 for f (x) = cos x Prove that P (x) = 1 2 near x = 0. (c) Let f and P be arbitrary functions. Complete and prove the following theorem: IF [ some condition about the derivatives of f and P at 0] THEN P is a good approximation of order n for f near x = 0. Hint: Write down the limit that must equal 0, then use L'Hpital's Rule reo peatedly to generate a list of conditions about the derivatives of f and P . It is vital that you check that you are allowed to use L'Hpital's Rule at every step. o (d) Let f be a function. Obtain a formula for a polynomial of degree as small as possible which is a good approximation of order n for f near x = 0. Hint: Reread Questions 3 and 6 from Problem Set 5