Assignment 2 Complete this assignment after you have finished Unit 5, and submit your work to your tutor for grading. Remember to attach a tutor-marked exercise form (paper or electronic). Be certain to keep a copy of your work, in case the original is lost in the mail. This exercise is worth 15 per cent of your final grade. Total points: 180 (plus 12 bonus points) 1. Determine the values of r, if any, such that y = e rx 9 pts is a solution of the differential equations given below. a. y 00 + 6y 0 + 9y = 0 b. y 00 - 5y 0 + 6y = 0 c. y 000 - 3y 0 + 2y = 0 12 pts 2. Use Euler’s method with the given step size (?x or ?t) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. a. dy dx = v3 y, y(0) = 1, 0 = x = 4, ?x = 0.5 b. cos y, y(0) = 1, 0 = t = 2, ?t = 0.5 6 pts 3. A tank with a 1000-gal capacity initially contains 500 gal of water that is polluted with 50 lb of particulate matter. At time t = 0, pure water is added at a rate of 20 gal/min and the mixed solution is drained off at a rate of 10 gal/min. How much particulate matter is in the tank when it reaches the point of overflowing? 9 pts 4. Polonium-210 is a radioactive element with a half-life of 140 days. Assume that 20 milligrams of the element are placed in a lead container and that y(t) is the number of milligrams present t days later. a. Find a formula for y(t). b. How many milligrams will be present after 12 weeks? c. How long will it take for 75% of the original sample to decay? 12 pts 5. Determine whether the following sequences are (eventually) decreasing, (eventually) increasing, or neither. Explain. a. 4 - (-1)n n b. (n!)2 (2n)! c. n 22 n n! d. 1 · 3 · 5 . . .(2n - 1) (2n) n Introduction to Calculus II 24 pts 6. Evaluate the limit (if it exists) of each of the following sequences. Indicate the results (definition, theorems, etc.) you use to support your conclusion. a. an = n - 3 n n b. an = (n!)2 (2n)! c. an = n 22 n n! d. 1 3 5 , - 1 3 6 , 1 3 7 , - 1 3 8 , . . . . e. an = v n2 + 3n - n f. an = (-1)n 2n 3 n3 + 1 6 pts 7. A bored student enters the number 0.5 in her calculator, and then repeatedly computes the square of the number in the display. Taking a0 = 0.5, find a formula for the general term of the sequence {an} of the numbers that appear in the display, and find the limit of the sequence {an} 9 pts 8. Give an example of two sequences {an} and {bn} such that a. {an} and {bn} are divergent, but {an + bn} is convergent. b. {an} is convergent, {bn} is divergent, and {anbn} is divergent. c. {an} is divergent, and {|an|} is convergent. 12 pts 9. Find the sum of each of the convergent series given below. a. X8 k=1 1 2 k - 1 2 k+1 b. X8 k=2 1 k 2 - 1 c. X8 n=5 e p n-1 d. X8 i=1 5 3i 7 1-i 24 pts 10. Determine whether each of the series below is divergent, absolutely convergent (hence convergent) or conditionally convergent. Indicate the test, result or results you use to support your conclusion. a. X8 k=1 v k v k + 3 b. X8 k=1 (-1)k p k(k + 1) Mathematics 266 / c. X8 k=1 3k 2 - 1 k 4 d. X8 k=1 tan-1 k k 2 e. X8 n=1 (-1)n+1 3 2n-1 k 2 + 1 f. X8 k=1 5 k + k k! + 3 11. Use an appropriate Taylor polynomial of degree 2 to approximate tan 61o 4 pts . 8 pts 12. Give the Taylor polynomial of order n about x = x0 a. sin(px); x0 = 1 2 b. ln x; x0 = e 15 pts 13. Find the radius of convergence and the interval of convergence for each of the series listed below. a. X8 k=1 5 k k 2 x k b. X8 n=1 (-1)nx 2n (2n)! c. X8 k=1 (ln k)(x - 1)k k 8 pts 14. Find the first four terms of the Malaurin series for each of the functions given below. a. e x sin x b. ln(1 + x) 1 - x 8 pts 15. Use Maclaurin series to find a. limx?0 tan-1 x - x x 3 b. Z e x - 1 x dx 14 pts 16. a. Use the relationship Z 1 v 1 + x dx = sinh-1 x + C to find the first four nonzero terms in the Maclaurin series for sinh-1 x. b. Express the series in sigma notation. c. What is the radius of convergence? Introduction to Calculus II Bonus Question 12 pts 1. a. Prove tan-1 x + tan-1 y = tan-1 x + y 1 - xy where - p 2 < tan-1="" x="" +="" tan-1="" y="">< p 2 . hint: use an identity for tan(x + y). b. use part (a) to show that tan-1 1 2 + tan-1 1 3 = p 4 . c. use the first four terms of the maclaurin series of tan-1 x and part (b) to approximate the value of p. mathematics / p="" 2="" .="" hint:="" use="" an="" identity="" for="" tan(x="" +="" y).="" b.="" use="" part="" (a)="" to="" show="" that="" tan-1="" ="" 1="" 2="" ="" +="" tan-1="" ="" 1="" 3="" ="p" 4="" .="" c.="" use="" the="" first="" four="" terms="" of="" the="" maclaurin="" series="" of="" tan-1="" x="" and="" part="" (b)="" to="" approximate="" the="" value="" of="" p.="" mathematics="">