Name: Sectio: Recitatio Istructor: READ THE FOLLOWING INSTRUCTIONS. Do ot ope your exam util told to do so. No calculators, cell phoes or ay other electroic devices ca be used o this exam. Clear your desk of everythig excepts pes, pecils ad erasers. If you eed scratch paper, use the back of the previous page. Without fully opeig the exam, check that you have pages 1 through 8. Fill i your ame, etc. o this first page. Show all your work. Write your aswers clearly! Iclude eough steps for the grader to be able to follow your work. Do t skip limits or equal sigs, etc. Iclude words to clarify your reasoig. Do first all of the problems you kow how to do immediately. Do ot sped too much time o ay particular problem. Retur to difficult problems later. If you have ay questios please raise your had ad a proctor will come to you. There is o talkig allowed durig the exam. You will be give exactly 90 miutes for this exam. I have read ad uderstad the above istructios:. SIGNATURE Page 1 of 8
Multiple Choice. Circle the best aswer. No work eeded. No partial credit available. 1. (5 poits) Which of the followig itegrals gives the legth of the graph y = si( x) betwee x = a ad x = b, where 0 < a < b? A. B. C. D. b a b a b a b a 1 + 1 4x cos2 ( x) dx x + cos 2 ( x) dx 1 + cos 2 ( x) dx si 2 ( x) + 1 4x cos2 ( x) dx. 2. (5 poits) Which of the followig series coverge to 2? (I) 2 + 3 (II) 8 ( 3) (III) 1 2 A. I oly B. II oly C. III oly D. I ad III oly E. II ad III oly 3. (5 poits) For what values of x does the series 1 + 2 x + 3 x + 4 x +... + x coverge as? A. No values of x B. x < 1 C. x 1 D. x > 1 E. All values of x Page 2 of 8
4. For each of the followig series tell whether the idicated test determies whether the series coverges or diverges. Be careful as o partial credit is give. ( ) 1 (a) (5 poits) si A. The th term test cocludes that the series coverges. B. The th term test cocludes that the series diverges. C. The th term test hypotheses are ot met by this series, so it caot be applied. D. The th term test hypotheses are met by this series however the test is icoclusive. (b) (5 poits)!e A. The ratio test cocludes that the series coverges. B. The ratio test cocludes that the series diverges. C. The ratio test hypotheses are ot met by this series, so it caot be applied. D. The ratio test hypotheses are met by this series however the test is icoclusive. (c) (5 poits) =2 + 2 2 A. The limit compariso test with b = 1 2 cocludes that the series coverges. B. The limit compariso test with b = 1 cocludes that the series diverges. C. The limit compariso test with b = 2 cocludes that the series coverges. D. The limit compariso test hypotheses are ot met by this series, so it caot be applied. Fill i the Blaks. No work eeded. No partial credit available. 5. (10 poits) Determie whether the followig sequeces coverge or diverge. If a sequece coverges, fid the limit. If the sequece diverges write DNE. l( + 1) (a) lim = 0 Apply L Hospitals (b) lim 2 + 1 1 3 = DNE Page 3 of 8
Stadard Respose Questios. Show all work to receive credit. Please BOX your fial aswer. 6. (18 poits) Use series to evaluate the limit lim e x (1 + x) x 2 (Note: Other methods such as L Hospital s Rule will ear at most 2 poits.) Solutio: e x (1 + x) lim x 2 =2 x 2 =2 [ =3 x! (1 + x) x! x 2! x 2 x 2! + 1 2! ] = 1 2 7. (14 poits) Fid the sum of the series 3 2 +1 7 Solutio: 3 2 +1 7 = 3 7 2+1 7 ( ) ( 1 2 = 3 2 7 7 ( ) 1 = 3 2 1 1/7 ( ) ( ) 7 7 = 3 2 6 5 ) ( 1 1 2/7 = 7 2 14 5 ) Page 4 of 8
8. (20 poits) Determie whether the followig series coverge or diverge. You must justify your aswer with work ad explicitly state which test(s) you use! l( 2 ) (a) Solutio: Cosider the itegral test which is applicable sice l(2 ) is positive ad decreasig: [ l(x 2 ) 1 ( dx l(x 2 ) ) N [ 2] 1 ( l(n 2 ) ) ] 2 [0] 1 x N 4 N 1 4 which teds to ifiity as N goes to ifiity. Therefore the series diverges by the itegral test. (b) 1 ( + 1) Solutio: Cosider the limit compariso test with sequece 1/ (which are both positive). We see that lim 1/ ( + 1) 1/( ( + 1)) + 1 + 1 = 1 So therefore 1 ( + 1) diverges sice 1/ diverges. Page 5 of 8
9. (14 poits) Fid the Maclauri series for f(x) = x cos x Solutio: The Maclauri series for cos(x) is f(x) = x cos x is ( 1) x 2. So therefore the Maclauri series for (2)! ( 1) ( x) 2 f(x) = x (2)! ( 1) (x) = x (2)! = ( 1) x +1 (2)! or equivaletly ( 1) 1 (2 2)! x 10. (14 poits) Fid all values of x for which the series (Hit: Make sure to test ed poits of the iterval.) ( + 1) 2 (x 2) coverges. Solutio: Cosider the ratio test: lim a +1/a ( + 2) 2 (x 2) +1 3 ( + 1) 3 ( + 1) 2 (x 2) ( + 2) 2 ( + 1) 3 (x 2) 2 ( + 1) 3 = x 2 < 1 which is true o the iterval (1, 3). Now test edpoits at x = 1 ( + 1) 2 (x 2) = 3 3 ( + 1) 2 ( 1) 3 Which is coverget by the alteratig series test (sice at x = 3 ( + 1)2 3 0 as ). ( + 1) 2 (x 2) = 3 ( + 1) 2 Which diverges by the limit compariso test to 1/ ad the p-series test. So therefore the series coverges o [1, 3). 3 Page 6 of 8
11. (14 poits) Fid the radius of covergece of the power series 2 5 x Solutio: Cosider the ratio test: lim a +1/a x +1 ( + 1) 2 5 5 +1 x 2 x ( + 1)2 5 2 = x < 1 5 Which is equivalet to x < 5 givig us the solutio R = 5. 12. (16 poits) Fid the Taylor polyomial of degree 3 geerated by f(x) = e x2 cetered at a = 2. Solutio: Calculate f(x) = e x2 = f(2) = e 4 f (x) = 2xe x2 = f (2) = 4e 4 f (x) = 2e x2 + 4x 2 e x2 = f (2) = 2e 4 + 16e 4 = 18e 4 f (x) = 4xe x2 + 8xe x2 + 8x 3 e x2 = f (2) = 8e 4 + 16e 4 + 64e 4 = 88e 4 Now we put them together to get: T 3 (x) = e 4 4 (x 2) 4 (x 2)2 4 (x 2)3 + 4e + 18e + 88e 1! 2! 3! = e 4 + 4e 4 (x 2) + 9e 4 (x 2) 2 + 44 3 e4 (x 2) 3 or = e 4 (1 + 4(x 2) + 9(x 2) 2 + 44 3 (x 2)3 ) Page 7 of 8
Cogratulatios you are ow doe with the exam! Go back ad check your solutios for accuracy ad clarity. Make sure your fial aswers are BOXED. Whe you are completely happy with your work please brig your exam to the frot to be haded i. Please have your MSU studet ID ready so that is ca be checked. DO NOT WRITE BELOW THIS LINE. Page Poits Score 2 15 3 25 4 32 5 20 6 28 7 30 Total: 150 Page 8 of 8