1 Math 6 Fial Exam December 9, 06 UMID: EXAM SOLUTIONS Iitials: Istructor: Sectio:. Do ot ope this exam util you are told to do so.. Do ot write your ame aywhere o this exam. 3. This exam has 3 pages icludig this cover. There are problems. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur to a problem o which you are stuck. 4. Do ot separate the pages of this exam. If they do become separated, write your UMID o every page ad poit this out to your istructor whe you had i the exam. 5. Please read the istructios for each idividual problem carefully. Oe of the skills beig tested o this exam is your ability to iterpret mathematical questios, so istructors will ot aswer questios about exam problems durig the exam. 6. Show a appropriate amout of work (icludig appropriate explaatio) for each problem so that graders ca see ot oly your aswer, but also how you obtaied it. Iclude uits i your aswer where that is appropriate. 7. You may use a TI-84, TI-89, TI-Nspire or other approved calculator. However, you must show work for ay calculatio which we have leared how to do i this course. You are also allowed two sides of a 3 5 ote card. 8. If you use graphs or tables to fid a aswer, be sure to iclude a explaatio ad sketch of the graph, ad to write out the etries of the table that you use. 9. Tur off all cell phoes, pagers, ad smartwatches, ad remove all headphoes. Problem Poits Score Total 00
2 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page. [4 poits] Suppose that the power series c (x 3) coverges at x = 6 ad diverges at x =. What ca you say about the behavior of the power series at the followig values of x? For each part, circle the correct aswer. Ambiguous resposes will be marked icorrect. a. [ poit] At x = 3, the power series... CONVERGES DIVERGES CANNOT DETERMINE b. [ poit] At x = 0, the power series... CONVERGES DIVERGES CANNOT DETERMINE c. [ poit] At x = 8, the power series... CONVERGES DIVERGES CANNOT DETERMINE d. [ poit] At x =, the power series... CONVERGES DIVERGES CANNOT DETERMINE. [5 poits] Determie the radius of covergece of the power series ()! (!) x. Justify your work carefully ad write your fial aswer i the space provided. Limit sytax will be eforced. For = 0,,..., let a = ()!. We have (!) a + a = (( + ))! (( + )!) (!) ( + )( + ) = ()! ( + ) 4 as. Hece the radius of covergece is 4 =. Radius of covergece =
3 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 3 3. [8 poits] For =,, 3,... cosider the sequece a give by a = (+)/ if is odd, a = if is eve. / 3 a. [ poits] Write out the first 5 terms of the sequece a. The first five terms are, 3, 4, 9, 8. b. [ poits] The series a is alteratig. I a setece or two, explai why the Alteratig Series Test caot be used to determie whether a coverges or diverges. The coditio a + < a does ot hold for all. (It does ot eve hold evetually.) c. [4 poits] The series a coverges. Show that it coverges, either by usig theorems about series, or by computig its exact value. Oe possible aswer is that the series is equal to the differece of two coverget geometric series: k= 3 k k= k = 3 3 =. Aother aswer uses the Compariso Test; for =,,..., let b =, ad otice that a b evetually. Sice b coverges by the p-test (p = ), a coverges by compariso. Hece the origial series coverges.
4 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 4 4. [5 poits] The followig series diverges: = + l(). Use theorems about ifiite series to show that the series diverges. Give full justificatio, showig all your work ad idicatig ay theorems or tests that you use. for all. Sice compariso. Oe solutio uses the Compariso Test. Notice that Alteratively, let a = Sice + l() + = diverges by the p-test (p = ), the origial series diverges by + l() ad b = a lim =. b for all, ad otice that = b coverges by the p-test (p = ), the origial series diverges by the Limit Compariso Test. 5. [5 poits] Let α > 0 be a costat. Compute the first 3 terms of the Taylor series of x f(x) = about x = 0. Write the appropriate coefficiets i the spaces provided. + αx 0 + x + α x +
5 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 5 6. [0 poits] After receivig a termiatio otice, The Iter has begu to read up o the global job market. A dubious pop-ecoomics book he is readig claims that the rate at which iters are hired or termiated i a large compay is purely a fuctio of the umber of iters at the compay. Specifically, it states that dh dt = g (H), where H(t) gives the umber of iters at a compay, i thousads, after t days, ad g(h) is a differetiable fuctio. A graph of g(h) (ot g (H)) is give i the book: g(h) H - a. [ poits] What are the uits of g (H)? The uits are thousads of iters per day. b. [3 poits] Are there ay stable equilibrium solutios of the differetial equatio? If so, what are they? Yes; the stable equilibrium solutios are H = ad H = 4.
6 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 6 6. (cotiued). Recall that the umber of iters i thousads H(t) satisfies dh dt = g (H), where a graph of g(h) (ot g (H)) is give below: g(h) H - c. [ poits] If a compay starts with 3,500 iters, what will happe to the umber of iters i the log ru? The umber of iters will approach 4,000 asymptotically from below. d. [ poit] Estimate the umber of iters at which the umber of iters is decreasig the fastest. The umber of iters is decreasig the fastest whe there are 4, 500 iters. e. [ poits] Suppose that a compay begis with 5,500 iters. If you used Euler s method to estimate how may iters there will be 5 days from ow, would you expect a uderestimate or a overestimate? Justify your aswer briefly. The correspodig solutio of the differetial equatio is cocave up, so we expect Euler s method to yield a uderestimate.
7 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 7 7. [7 poits] The Legedre equatio is a differetial equatio that arises i the quatum mechaical study of the hydroge atom. I oe of its forms, the Legedre equatio is ( x )y xy + y = 0. For this problem, let y be a solutio to the Legedre equatio satisfyig y( ) = ad y ( ) = 3. Assume that the Taylor series for y(x) about x = coverges to y(x) for all < x < 3. a. [4 poits] I the blak below, write dow P (x), the degree Taylor polyomial of y(x) ear x =. Your aswer should ot cotai the fuctio y(x) or ay of its derivatives. P (x) = ( + 3 x ) ( 4 x ) b. [3 poits] Compute the limit y(x) lim 3x x / (x. ) The limit is 4.
8 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 8 8. [ poits] I this problem, we cosider the parametric curve give by x = f(t) y = g(t) for all t, where f ad g are twice-differetiable fuctios. Some values of f ad g ad their derivatives are give i the tables below. t f(t) g(t) t f (t) g (t) a. [ poit] I the space provided, write a itegral that gives the arc legth of the parametric curve from t = to t = 5. 5 (f (t)) + (g (t)) dt Arc legth = b. [3 poits] Use a midpoit sum with as may subdivisios as possible to estimate your itegral from part a. Write out all the terms i your sum, ad do ot simplify. The midpoit sum is ( 0 + ( ) ). c. [3 poits] Fid the Cartesia equatio for the taget lie to the parametric curve i the xy-plae at t =. I poit-slope form, the taget lie is give by y 5 = (x + 3). d. [ poits] Cosider the taget lies to the parametric curve at the t-values t =,, 3, 4, 5. Are ay of these lies perpedicular to each other? If so, list ay two t-values for which the taget lies are perpedicular. If ot, write NO. The taget lies correspodig to t = ad t = 4 are perpedicular. e. [ poits] As t rages from to 5, the correspodig part of the parametric curve itersects the lie y = x exactly oce. Which iterval cotais the t-value for which the curve itersects the lie y = x? Circle your aswer. You do ot eed to show ay work. (, ) (, 3) (3, 4) (4, 5)
9 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 9 9. [ poits] Read the followig parts carefully, ad circle the appropriate aswer(s). Some parts may have more tha oe correct aswer. a. [3 poits] Circle the value(s) of x for which the followig idetity holds: = x 3 + x6! + x9 3! + x + 4! 3 l() 3 l(3) (l()) 3 e 3 l( 3 3) e 3 3 b. [3 poits] Raymod Gree s pet aacoda Sheela grew 5 m i legth over the past moth. The veteriaria says that each moth, the icrease i Sheela s legth will be 40% of the icrease the moth before. How much loger (i meters) will Sheela be oe year from ow? Circle all that apply. k=0 5(0.4) k ( (0.4) ) 0.4 k= 5(0.4) k 5( (0.4) ) 0.4 ( (0.4) 3 ) 0.4 ( ) + α c. [3 poits] Let α > 0 be a costat. What is the value of the coverget series ()!? cos(α) cos( α) cos(α) α cos( α) cos( α) cos(α) α d. [3 poits] Which of the followig series coverge absolutely? Circle all that apply. si 99 () = ( ) + l() 8 + ( ) 0 9 = = ( ) + (l()).0 ( ) =
10 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 0 0. [4 poits] The coical tak pictured below has a base of radius 4 meters ad a height of 5 meters. It is filled to the top with a toxic liquid, PGM, which has a costat desity of 000kg/m 3. The gravitatioal costat is g = 9.8 m/s. 5 m 4 m a. [4 poits] Write a expressio for the volume of a circular slice of thickess h, a distace h meters from the base. The volume is π(4 4 5 h) h m 3. b. [3 poits] Dr. Durat is tryig to take over Shamcorp by makig a device usig PGM. Dr. Durat must pump all the PGM to the top of the coe to complete his device. Write a expressio ivolvig itegrals which gives the work Dr. Durat does pumpig all of the liquid to the top of the cotaier. The work doe is 9.8π 5 0 ( h ) (5 h) dh kj. c. [3 poits] Write a expressio ivolvig itegrals for A, the average radius of a circular slice of the coe for 0 h 5. We have A = (4 45 ) h dh m. d. [4 poits] Raymod Gree is also buildig a PGM device, but he has a cylidrical cotaier, orieted so that its circular base is o the groud, filled to the top with PGM. The cotaier has a height of 5 meters ad has radius A (the average radius of Dr. Durat s cotaier). Whoever does the least amout of work pumpig the PGM to the top of their cotaier will rule Shamcorp. Will it be Dr. Durat or Raymod Gree? Give a brief justificatio for your aswer. The volume of Durat s cotaier is greater tha that of Gree s cotaier; moreover, the ceter of mass of Durat s cotaier is closer to the groud tha that of Gree s cotaier. This meas that Durat will do more work, so Gree will rule ShamCorp.
11 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page. [0 poits] Perhaps sesig that the ed is ear, Steph is preparig her evetual legacy. While experimetig with a accelerat called Equiate, Steph foud that she could alter its heat of combustio by agig the Equiate i barrels. Let H(t) be the heat of combustio of Equiate, measured i hudreds of millios of Joules per kilogram, after the Equiate has bee aged for t years. A graph of the derivative H (t) is below; ote that H (t) is liear for < t < 3 ad 4 < t < 5. Let R > 0 be the area of the regio betwee the t-axis ad the graph of H (t) for 0 t. Let P > 0 be the correspodig area for t 4. H (t) R P t a. [3 poits] The heat of combustio of Equiate, after agig for 5 years, is 00 millio J/kg. What is the heat of combustio of Equiate that has ot bee aged at all? Your aswer may iclude R ad P. The aswer is R + P hudred millio J/kg. b. [3 poits] Steph is storig four barrels of Equiate i the ShamCorp basemet. Barrel A has ot bee aged; barrel B has bee aged for years; barrel C has bee aged for 4 years; ad barrel D has bee aged for 5 years. I the spaces provided, list the barrels A, B, C, ad D i icreasig order of heat of combustio. C < A < D < B c. [4 poits] At some time betwee 4 ad 5 years of agig, the heat of combustio of Equiate is the same as if it had ot bee aged at all. After how may years of agig does this occur? Your aswer may iclude R ad P. This occurs after 4 + P R years of agig.
12 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page. [9 poits] Three itervals are give below. I the space ext to each iterval, write the letter(s) correspodig to each power series (A)-(I) (below) whose iterval of covergece is exactly that iterval. There may be more tha oe aswer for each iterval. If there are itervals below for which oe of the power series (A)-(I) coverge o that iterval, write NONE i the space ext to the iterval. You do ot eed to show your work. a. [3 poits] (, ) : B b. [3 poits] (0, 0] : C c. [3 poits] [0, ) : NONE (A) x 4+! (B) ( ) (x) 4 (C) ( ) (x 5) 5 (D) (x 5) (E) x! (F) x (G) ( x) (H) (x 5) 5 (I) x
13 Math 6 / Fial (December 9, 06) DO NOT WRITE YOUR NAME ON THIS PAGE page 3 Kow Taylor series (all aroud x = 0): si(x) = ( ) x + ( + )! = x x3 3! + + ( ) x + + for all values of x ( + )! ( ) x cos(x) = ()! = x! + + ( ) x + for all values of x ()! e x = x! = + x + x! + + x + for all values of x! ( ) + x l( + x) = = x x + x3 3 + ( )+ x + for < x ( + x) p = + px + p(p )! x + p(p )(p ) 3! x 3 + for < x < x = x = + x + x + x x + for < x <