Math 116 Practice for Exam 3

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1 Math 6 Practice for Eam 3 Generated December, 6 Name: SOLUTIONS Instructor: Section Number:. This eam has 6 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.. Do not separate the pages of the eam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the eam. 3. Please read the instructions for each individual eercise carefully. One of the skills being tested on this eam is your ability to interpret questions, so instructors will not answer questions about eam problems during the eam. 4. Show an appropriate amount of work (including appropriate eplanation) for each eercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate. 5. You may use any calculator ecept a TI-9 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 5 note card. 6. If you use graphs or tables to obtain an answer, be certain to include an eplanation and sketch of the graph, and to write out the entries of the table that you use. 7. You must use the methods learned in this course to solve all problems. Semester Eam Problem Name Points Score Fall Winter Fall 4 3 chicken coop 8 Fall Winter Bessel 9 Winter 6 3 Hanoi Tower 5 Total 45 Recommended time (based on points): 55 minutes

2 Math 6 / Final (December, 4) page 8. [7 points] Consider the power series (+) n 3 n n. a. [ points] At which -value is the interval of convergence of this power series centered? Solution: This power series is centered on =. (+) n b. [5 points] The radius of convergence for the power series 3 n is 3. Find the n interval of convergence for this power series. Thoroughly justify your answer. Solution: Since the radius of convergence for this power series is 3 and it is centered on =, the interval of convergence contains the open interval ( 3, +3) = ( 5,). Now we only need to check the endpoints = 5 and =. (+) n For = : 3 n = diverges by the p-test with p = n n (this is the harmonic series). ( 5+) n ( ) n For = 5: 3 n = which converges by the n n alternating series test. Therefore, the interval of convergence for this power series is [ 5, ). 9. [5 points] Find the radius of convergence for the power series Solution: Let the n-th term be denoted by a n a n+ ((n+))! (n+) (n!) = a n ((n+)!) n = ()(n+) ((n+) (n!) n Therefore, we can use the ratio test: a n+ = ()(n+) n a n n ((n+) = 4. So this series converges for with 4 <, or rather with < 4 radius of convergence is /. which implies that the Fall, 4 Math 6 Eam 3 Problem 9 Solution

3 Math 6 / Final (April 3, 5) page 5 5. [ points] a. [5 points] Determine the radius of convergence of the power series ( ) n ( 5) 4n n 5 (6) n. Solution: The above series will converge for values such that by the ratio test. We have n n and so the desired inequality holds if Thus, the radius of convergence is. 5 4n+4 (n+) 5 (6) n+ 5 4n+4 (n+) 5 (6) n+ < 5 4n n 5 (6) n = 5 4n n 5 (6) n <. This is equivalent to 5 <. The radius of convergence is. () n b. [5 points] The power series n 4 has radius of convergence. Determine the + interval of convergence for this power series. Solution: For =, we get the series n 4. We have that + n The it comparison test then tells us that the series if the series n 4 + converges. For =, we get the series converges absolutely. n 4 + n 3 n 4 + n 4 +n 3 = n n 4 + =. converges if and only n 3 converges. Since is a p-series with p > it converges, and so n3 The interval of convergence for ( ) n () n 4. By the work shown above, this series + () n n 4 + is then The interval of convergence is [, ]. Winter, 5 Math 6 Eam 3 Problem 5 Solution

4 Math 6 / Final (December, 4) page 3. [8 points] Franklin, your friendly new neighbor, is building a large chicken sanctuary. You decide to help Franklin build a special chicken coop with volume (in cubic km) given by the integral cos( )d. This integral is difficult to evaluate precisely, so you decide to use the methods you ve learned this semester to help out Franklin. Your friend and president-elect, Kazilla, stops by to give you a hand. She suggests finding the 4th degree Taylor polynomial, P 4 (), for the function cos( ) near =. a. [4 points] Find P 4 (). Solution: We can use the Taylor series epansion for cos( ) so Therefore cos(y) = cos( ) = ( ) n y n ( ) n ( ) n = y! + + ( )n y n + = ( )! P 4 = ( 4! ) = 4! + + ( )n ( ) n + b. [4points]SubstituteP 4 ()for cos( )intheintegralandcomputetheresultingintegral by hand, showing all of your work. Solution: P 4 ()d = = = = 4 = Fall, 4 Math 6 Eam 3 Problem (chicken coop) Solution

5 Math 6 / Final (December 7, 5) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 4. [8 points] Let f() = 3 +. a. [5 points] Find the first 3 nonzero terms of the Taylor series for f centered at =. Solution: Using the Taylor series for (+y) /3 centered at y =, ( 3 )( ) +y = + 3 y y +...! = + y 3 y Substituting y =, 3 + = b. [3 points] For what values of does the Taylor series converge? Solution: The binomial series for 3 +y converges when < y <. Substituting y =, this converges when < <, or < <. 5. [3 points] Determine the eact value of the infinite series + 3! 5! + + ( )n+ (n+)! +... No decimal approimations are allowed. You do not need to show your work. Solution: + 3! 5! + = ( ) n+ (n+)! = ( ) n ( ) n+ (n+)! = sin( ). Fall, 5 Math 6 Eam 3 Problem 4 Solution

6 Math 6 / Final (April 3, 5) page 4 4. [9 points] We can define the Bessel function of order one by its Taylor series about =, ( ) n n+ J () = n!(n+)! n+. a. [3 points] Compute J (5) (). Write your answer in eact form and do not try evaluate using a calculator. Solution: The 5th derivative of J at = corresponds to 5! times the 7th coefficient in the Taylor series above. This gives us that J (5) (5)! () = (7)!(8)! 5. J (5) () = (5)! (7)!(8)! 5 b. [4 points] Find P 5 (), the Taylor polynomial of degree 5 that approimates J () near =. Solution: P 5 () = P 5 () = c. [ points] Use the Taylor polynomial from the previous part to compute J () 3. J () Solution: 3 = = 6 J () 3 = 6 Winter, 5 Math 6 Eam 3 Problem 4 (Bessel) Solution

7 Math 6 / Final (April, 6) DO NOT WRITE YOUR NAME ON THIS PAGE page. [5 points] The Hanoi tower is made by rotating the region depicted below around the y-ais. The region is made up of infinitely many adjacent rectangles. The nth rectangle has width and height a n = where n =,,,3,... The rectangle touching the y-ais corresponds to n =. Note that the y-ais is not to scale. n!(n+) y a a a a 3 a 4 a Compute the volume of the Hanoi Tower. Give an eact answer. Solution: To compute the volume of the Hanoi Tower, we focus on each rectangle separately. The volume of the object made by the revolution of the nth rectangle is given by [π(n+) πn ] a n = π(n+) n!(n+) = π n! The total volume is given by adding the volume of all those objects for n =,,,3,... π n! = π n! = πe Winter, 6 Math 6 Eam 3 Problem (Hanoi Tower) Solution