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# CalculusScience & Engineering MATH 20B

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This 4 page Class Notes was uploaded by Claire Harris I on Thursday October 22, 2015. The Class Notes belongs to MATH 20B at University of California - San Diego taught by Staff in Fall. Since its upload, it has received 44 views. For similar materials see /class/226774/math-20b-university-of-california-san-diego in Mathematics (M) at University of California - San Diego.

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Date Created: 10/22/15

71509 Math 20B Lecture Example Solutions for Instructors Section 102 Summing an in nite seriesT Example 1 Imagine that you want to go from a point A toward a second point B two miles away Imagine that rst you go one mile Figure 1 Then you go a half mile further Figure 2 Next you go half that distance Figure 3 and so forth so that at each stage you go half as far as you did in the previous stage a How far have you gone after one two three ve and eight stages b Predict the limit of your distance from A as the number of stages tends to 00 A B A B A B l A l l A l 0 1 2 s 0 1 2 s 0 1 2 3 FIGURE 1 FIGURE 2 FIGURE 3 Answer a The distances are in the table below b The distance seems to be approaching 2 miles Stages Total distance Decimal approximation 1 1 1 172 1 1 5 173 1 g 175 175 1234 19375 18 1 2345 19921875 ltgt6 ltgt7 SOLUTION FOR INSTRUCTORS a The distances are in the table above i b The distance seems to be approaching 2 miles ILecture notes to accompany Section 102 of Calculus Early Transcendentals by Rogawski All or most of the examples in these notes are examples or exercises from Al Shenk7s calculus manuscript Math 2013 Lecture Example Solutions 71509 Section 102 p 2 Example 2 Now suppose you go one mile from A toward B in the rst stage 1 Z mile back toward A in the second stage if mile away from A in the third 1 a mile toward A in the fourth stage etc so that at each stage you go onefourth the distance and in the opposite direction from the previous stage a How far have you gone after one two three ve and eight stages b Predict the limit of your distance from A as the number of stages tends to 00 stage if Answer a The distances are in the table below b The distance seems to be approaching 08 miles Stages Total distance Decimal approximation 1 1 1 1 12 17 Z 075 173 17 i 2 08125 175 17 i i2 7 13 14 079688 1 1 2 1 3 1 4 1 5 18 17 Z 1 61 1 7 1 07999987 Hz 7 1 SOLUTION FOR INSTRUCTORS a The distances are in the table above b The distance seems to be approaching 08 miles Theorem 1 Finite geometric series For any constant Iquot and any nonnegative integer N N N1 ifr1 Zr 1rr2r3mrN 1 7 IN1 n0 if I39 y Example 3 Give a concise formula for Z 099quot and nd its approximate decimal n2 value 511 510 1 i 099 Answer E 099 0992 97427602 1 i 099 712 SOLUTION FOR INSTRUCTORS 511 509 1 0 99510 Z 099n 099 099n 0992 Em 97427602 n2 n0 Section 102 p 3 Math 20B Lecture Example Solutions 71509 400 Example 4 Evaluate 71quot n0 400 Answer E 71quot 1 710 SOLUTION FOR INSTRUCTORS 400 V 1 1401 400 V ZeiJ gt i o 0rZeiJ1ei1e1ei11 n0 n0 Example 5 A rich uncle agrees to put 1 in a trust fund for you on your rst birthday 2 on your second birthday 4 on your third birthday and to put in twice the amount each year as the year before until you are eighteen How much will be in the fund after your eighteenth birthday 218 262144 Answer After your eighteenth birthday the fund will contain 263143 SOLUTION FOR INSTRUCTORS 1 7 218 After your eighteenth birthday the fund Will contain 1 2 22 23 217 218 e 1 262144 71 263143 dollars Example 6 Suppose that when a ball is dropped from height ft above a oor it bounces up to a height of h feet What is the total distance the ball travels when it hits the ground for the tenth time if it is dropped from a height of 10 feet Answer When the ball hits the ground for the tenth time it has traveled 10 601 7 i 65495 feet SOLUTION FOR INSTRUCTORS The ball falls 10 feet When it hits the ground the rst timer 0 Then it bounces up 10 feet and falls 10 feet so When it hits the ground a second time it has traveled a total of 10 210 feet 0 On the next bounce it goes up and down 2210 feet and the total distance it has gone When it hits the ground the third time is 10 210 2210 feet 0 When it hits the ground for the tenth time it has traveled 10 2ltgt10gt 2lt2gt2lt10gt 2ltgt9lt10gt10 20llt gt 2 if 17 9 1020 1 m 8 1015 10601e gm 65495 feet 4 eo Theorem 2 In nite geometric series For lI39l lt 1 the geometric series Z 139quot converges and n0 has the value The series diverges if M 2 l Math 2013 Lecture Example Solutions 71509 Section 1027 p 4 Example 7 Check the predictions a in Example 1 and b in Example 2 Answer a In Example 1 the limit of the distance is 2 o The prediction was correct b In Example 2 the 4 limit of the distance is g 0 The prediction was correct SOLUTION FOR INSTRUCTORS 0 a In Example 1 the limit of the distance is Z 1 l 2i 0 n0 2 2 The prediction was correct 0 b In Example 2 the limit of the distance is Z 7 gr 0 n0 4 4 The prediction was correct 0 Example 8 Give the exact value of the in nite geometic series Z 7quot n3 Answer Z 7 n 7 713 SOLUTION FOR INSTRUCTORS gt0 3 4 5 3 2 Z 7 J 7 7 7 1 ltp gt 7 n3 3 3 Do 3 n 3 3 1 Z 21 01 01 01 1 3 i m 0 7 Z 1 Example 9 Express the in nite repeating decimal 0002222 as a fraction Answer 0002222 SOLUTION FOR INSTRUCTORS 2 2 2 2 2 1 1 1 010022227777c7lt1i77 103 104 105 106 103 10 102 103 gt0 1 1 1 2 1 3 1 1 n 1 1 1 1 1 m i1 i l WZ E mltjgt mli lm n0 10 A test for divergence x Theorem 3 If the numbers an x 2 an diverges nn0 nine do not tend to zero as n tends to 00 then the in nite series

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