MARK BOX problem poits 0 0 0 2-3 60=2x5 4 0 5 0 % 00 HAND IN PART NAME: Solutios PIN: 7 INSTRUCTIONS This exam comes i two parts. () HAND IN PART. Had i oly this part. (2) STATEMENT OF MULTIPLE CHOICE PROBLEMS. Do ot had i this part. You ca take this part home to lear from ad to check your aswers oce the solutios are posted. O Problem 0, fill i the blaks. As you kow, if you do ot make at least half of the poits o Problem 0, the your score for the etire exam will be whatever you made o Problem 0. For TrueFalse/MultipleChoice problems 3, circle your aswer(s) o the provided chart. No eed to show work. The statemet of multiple choice problems will ot be collected. For problems > 3, to receive credit you MUST: () work i a logical fashio, show all your work, idicate your reasoig; o credit will be give for a aswer that just appears; such explaatios help with partial credit (2) if a lie/box is provided, the: show you work BELOW the lie/box put your aswer o/i the lie/box (3) if o such lie/box is provided, the box your aswer. The mark box above idicates the problems alog with their poits. Check that your copy of the exam has all of the problems. Upo request, you will be give as much (blak) scratch paper as you eed. Durig the exam, the use of uauthorized materials is prohibited. Uauthorized materials iclude: books, electroic devices, ay device with which you ca coect to the iteret, ad persoal otes. Uauthorized materials (icludig cell phoes) must be i a secured (e.g. zipped up, sapped closed) bag placed completely uder your desk or, if you did ot brig such a bag, give to Prof. Girardi to hold for you durig the exam (ad they will be retured whe you leave the exam). This meas o electroic devices (such as cell phoes) allowed i your pockets. At a studet s request, I will project my watch upo the projector scree. Durig this exam, do ot leave your seat uless you have permissio. If you have a questio, raise your had. Whe you fiish: tur your exam over, put your pecil dow ad raise your had. This exam covers (from Calculus by Thomas, 3 th ed., ET): 8.7 8.8, 0. 0.7. Hoor Code Statemet I uderstad that it is the resposibility of every member of the Carolia commuity to uphold ad maitai the Uiversity of South Carolia s Hoor Code. As a Caroliia, I certify that I have either give or received uauthorized aid o this exam. I uderstad that if it is determied that I used ay uauthorized assistace or otherwise violated the Uiversity s Hoor Code the I will receive a failig grade for this course ad be referred to the academic Dea ad the Office of Academic Itegrity for additioal discipliary actios. Furthermore, I have ot oly read but will also follow the istructios o the exam. Sigature : Prof. Girardi Page of Math 42
0. Fill-i-the boxes. All series are uderstood to be =, uless otherwise idicated. 0.. Geometric Series. Fill i the boxes with the proper rage of r R. The series r coverges if ad oly if r satisfies r <. 0.2. p-series. Fill i the boxes with the proper rage of p R. The series p coverges if ad oly if p >. 0.3. State the Direct Compariso Test for a positive-termed series a. 0 a If c whe 7 ad c coverges, the a coverges. (oly a c is also ok b/c give a 0) If 0 d a (eed 0 d part here) whe 7 ad d diverges, the a diverges. Hit: sig the sog to yourself. 0.4. State the Limit Compariso Test for a positive-termed series a. Let b > 0 ad L = lim a b. If 0 < L <, the [ b coverges a coverges ] If L = 0, the [ b coverges = a coverges ]. If L =, the [ b diverges = a diverges ]. Goal: cleverly pick positive b s so that you kow what b does (coverges or diverges) ad the sequece { a b } coverges. 0.5. Helpful Ituitio Fill i the 3 boxes usig: e x, l x, x q. Use each oce, ad oly oce. Cosider a positive power q > 0. There is (some big umber) N q > 0 so that if x N q the l x x q e x.. Circle T if the statemet is TRUE. Circle F if the statemet if FALSE. To be more specific: circle T if the statemet is always true ad circle F if the statemet is NOT always true. Scorig: 2 pts for correct aswer, 0 pts for a icorrect aswer, pt for a blak aswer (idicated by a circled B). T F B If lim a 0, the a diverges. T F B If lim a = 0, the a coverges. T F B If a 0 for all N, the a is either absolutely coverget or diverget. T F B If a coverges, the a coverges. T F B If (a + b ) coverges, the a coverges ad b coverge. Prof. Girardi Page 2 of
TABLE FOR YOUR ANSWERS TO MULTIPLE CHOICE PROBLEMS Idicate (by circlig) directly i the table below your solutio to each problem. You may choice up to 2 aswers for each problem. The scorig is as follows. For a problem with precisely oe aswer marked ad the aswer is correct, 5 poits. For a problem with precisely two aswers marked, oe of which is correct, 3 poits. For a problem with othig marked (i.e., left blak) poit. All other cases, 0 poits. Fill i the umber of solutios circled colum. (Worth a total of poit of extra credit.) Your Solutios Do Not Write Below problem 2 2a 2b 2c 2d 2e 3 3a 3b 3c 3d 3e 4 4a 4b 4c 4d 4e 5 5a 5b 5c 5d 5e 6 6a 6b 6c 6d 6e 7 7a 7b 7c 7d 7e 8 8a 8b 8c 8d 8e 9 9a 9b 9c 9d 9e 0 0a 0b 0c 0d 0e a b c d e 2 2a 2b 2c 2d 2e 3 3a 3b 3c 3d 3e umber of solutios circled 2 B x Extra Credit: Prof. Girardi Page 3 of
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2sol. Fall 206 Exam 2 x= x= dx = π. From our textbook, page 506, Example 2. +x 2 3sol. Prof. Girardi Page 7 of
4sol. From our textbook, page 52. 5sol. 6sol. Prof. Girardi Page 8 of
7sol. 8sol. (+2)(+7) a lim b big ()() =. So let b = ad a = = lim ( + 2)( + 7) = lim ( ) (+2)(+7). The 2 = lim ( + 2)( + 7) 2 ( + 2)( + 7) = = a Sice 0 < lim b <, by the LCT, b ad a do the same thig ad we kow that b is the harmoic series so b is diverges. So a diverges. 9sol. Now let u =. Sice 0 u 0, by the AST, ( ) u coverges. (+2)(+7) Now look at the choices. Prof. Girardi Page 9 of
0sol. sol. Let a = ( + ) ( + 2) ( + 3) ( + 4) ( + 5) For sufficietly big, a = ( + ) ( + 2) ( + 3) ( + 4) ( + 5) whe is big () () () () () = = 5/2. 3/2 So we let b = ( ) 3/2 ad compute a b = = 3/2 ( + ) ( + 2) ( + 3) ( + 4) ( + 5) [ 5 ( + ) ( + 2) ( + 3) ( + 4) ( + 5) [ () () () () () ] /2 =. Sice 0 < lim a b ] /2 = [ = 5/2 [ ( + ) ( + 2) ( + 3) ( + 4) ( + 5) ] /2 ( + ) ( + 2) ( + 3) ( + 4) ( + 5) <, the LCT says the a ad b do the same thig. Sice b is a p-series with p = 3 2 >, the b coverges. So the a coverges. ] /2 Prof. Girardi Page 0 of
2sol. Basically, apply the itegral test. Note that the computatio of dx = l l u + C, x l x 3sol. as see by usig a u-du subsitutio with u = l x. Let s see what happes if we try the ratio test. Let a = l. The lim By L Hopital s Rule, Sice lim a + a a + a = lim ( + ) l ( + ) lim l l ( + ) = lim + =, the Ratio Test is icoclusive. l = lim = lim + + =. l l ( + ). Prof. Girardi Page of