Name: Sectio: Recitatio Istructor: INSTRUCTIONS Fill i your ame, etc. o this first page. Without fully opeig the exam, check that you have pages through. Show all your work o the stadard respose questios. 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. You will be give exactly 90 miutes for this exam. Remove ad utilize the formula sheet provided to you at the ed of this exam. ACADEMIC HONESTY Do ot ope the exam booklet util you are istructed to do so. Do ot seek or obtai ay kid of help from ayoe to aswer questios o this exam. If you have questios, cosult oly the proctors). Books, otes, calculators, phoes, or ay other electroic devices are ot allowed o the exam. Studets should store them i their backpacks. No scratch paper is permitted. If you eed more room use the back of a page. Ayoe who violates these istructios will have committed a act of academic dishoesty. Pealties for academic dishoesty ca be very severe. All cases of academic dishoesty will be reported immediately to the Dea of Udergraduate Studies ad added to the studet s academic record. I have read ad uderstad the above istructios ad statemets regardig academic hoesty:. SIGNATURE Page of
Stadard Respose Questios. Show all work to receive credit. Please BOX your fial aswer.. 8 poits) Fid the legth L of the curve give by y 6 x + 4) 3/ for x 3. y 6 x + 4) 3/ 3 Solutio: We have: dy dx x + 4 x). So usig the arc legth formula we get 4 L 3 3 3 + ) x 4 + 4 x) dx + 4 x x + 4)dx + 4 x4 + x dx 3 ) x + dx 3 ) x + dx 3 ) x + dx [ ] 3 6 x3 + x [ ] 6 33 + 3 [ ] ) + 6 6 3 Page of
. 8 poits) For each sequece, fid the limit ad the state whether the sequece coverges or diverges. a) a 3 + 4 Solutio: Therefore the sequece diverges l 7 ) b) b ) 4 si Solutio: So we may apply L Hopital s rule. l lim si Therefore the sequece coverges. lim a lim 3 + 4 lim b lim 7 4 lim 3 + 4 lim + 4 3 ) l ) lim lim ) 4 7 ) 0 4 0 si 7/ 7 ) ) 4 4 cos 4 7 ) cos 7 4 0) cos 0) 7 4 7 ) 4 Page 3 of
3. 8 poits) Determie if the followig series coverge or diverge. You must show all of your work, show your reasos for decidig to use a certai test, ad support your coclusios. 8 a) 8 Solutio: Use the ratio test: Sice all terms are positive for, we do t eed absolute value a + + )8 8 a 8 + 8 + )8 8 8 ) 8 + 8 + ) 8 8 b) a + So L lim a 8 + 3 5 + 5 < so the series coverges. Solutio: Cosider the limit compariso test: All terms are positive for, so we may apply the test. Let a + 3 5 + 5 ad b 5/ / a + 3 lim lim b 5 + 5 / 5/ + 3 3/ lim /5/ 5 + 5 / 5 lim + 3 + 5 5 + 0 + 0 Therefore, by the limit compariso test, either both a ad b coverge or they both diverge. But b is a p-series with p which diverges. Therefore the series diverges. Page 4 of
4. 8 poits) Cosider the fuctio fx) x 9 + x. a) Fid the power series represetatio of f i sigma otatio. Solutio: x x 9 + x 9 + x 9 a r So fx) is the sum of a geometric series with a x ad r x 9 9 Therefore, fx) ar x ) ) ) x + 0 0 9 9 0 ) x + 9 b) Fid the radius of the covergece to your aswer i part a). Solutio: A geometric series coverges whe r <, so r x 9 < Therefore, the radius of covergece is 3 x < 9 x < 3 Page 5 of
5. 8 poits) Fid the 4th degree Taylor polyomial of fx) six) cetered at a π 4 Solutio: π ) fx) si x f 4 π ) f x) cos x f 4 π ) f x) si x f 4 π ) f x) cos x f 4 π ) f x) si x f 4 So the Taylor polyomial is: + x π ) x π 4 4! ) x π ) 3 4 + 3! x π 4 4! ) 4 Page 6 of
Multiple Choice. Circle the best aswer. No work eeded. No partial credit available. No credit will be give for choices ot clearly marked. ) 6. 7 poits) Which statemet is true about the series cos A. The series coverges by the th term test. B. The series diverges by the th term test. 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. E. Noe of the above are true. 7. 7 poits) Which statemet is true about the series si + A. The itegral test cocludes that the series coverges. B. The itegral test cocludes that the series diverges. C. The itegral test hypotheses are ot met by this series, so it caot be applied. D. The itegral test hypotheses are met by this series however the test is icoclusive. E. Noe of the above are true. 8. 7 poits) What does the followig series coverge to? 5 0 3 + 0 9 40 7 A. 3 B. 4 C. 0 D. 65 7 E. Page 7 of
9. 7 poits) Fid the radius of covergece of A. /4 B. /3 C. 3 D. 4 E. + k0 3) k x k k + 0. 7 poits) Determie whether the followig series are absolutely coverget, coditioally coverget, or diverget: ) cos + ad ) ) A. ) is absolutely coverget; ) is diverget. B. ) is coditioally coverget; ) is diverget. C. ) is absolutely coverget; ) is coditioally coverget. D. ) is diverget; ) is coditioally coverget. E. ) ad ) are coditioally coverget.. 7 poits) Which statemet is true about the series A. The series coverges by the itegral test, but the ratio test is icoclusive. B. The series diverges by the itegral test, but the ratio test is icoclusive. l C. Both the itegral test ad the ratio test show the series coverges. D. Both the itegral test ad the ratio test show the series diverges. E. Noe of the above are true. Page 8 of
. 7 poits) Fid the sum of the series A. 7 5 B. 4 5 C. 5 D. 9 5 E. It diverges 4 5 3 ) 3. 7 poits) Which statemet is true cocerig the followig series: ) + + + 3 ad ) 3 A. Both coverge. B. ) coverges, ) diverges. C. ) diverges, ) coverges. D. Both diverge. E. Caot be determied. 4. 7 poits) The first 4 ozero terms of the Taylor series at x 0 for fx) 4x cos x 3 + x 3 A. fx) + 4x 4x3! B. fx) 4x 4x3! + 4x5 4! C. fx) 3 + 4x x3! D. fx) 3 + 4x + 4x5 4! E. fx) 3 + 4x 3 4x5 4! + 4x5 4! 4x7 6! + 4x5 4! 4x7 6! + 4x7 6! Page 9 of
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 8 3 8 4 8 5 8 6 8 7 8 9 Total: 53 No more tha 50 poits may be eared o the exam. Page 0 of
Itegrals FORMULA SHEET PAGE Derivatives Volume: Suppose Ax) is the cross-sectioal area of the solid S perpedicular to the x-axis, the the volume of S is give by d d sih x) cosh x dx Iverse Trigoometric Fuctios: cosh x) sih x dx V b a Ax) dx d dx si x) x d dx csc x) x x Work: Suppose fx) is a force fuctio. The work i movig a object from a to b is give by: W b dx l x + C x ta x dx l sec x + C a fx) dx sec x dx l sec x + ta x + C a x dx ax l a + C for a Itegratio by Parts: u dv uv Arc Legth Formula: v du d dx cos x) x d dx ta x) d dx sec x) d + x dx cot x) x x + x If f is a oe-to-oe differetiable fuctio with iverse fuctio f ad f f a)) 0, the the iverse fuctio is differetiable at a ad f ) a) f f a)) Hyperbolic ad Trig Idetities Hyperbolic Fuctios sihx) ex e x coshx) ex + e x cschx) sih x sechx) cosh x L b a + [f x)] dx tahx) sih x cosh x cosh x sih x si x cos x) cos x + cos x) six) si x cos x cothx) cosh x sih x si A cos B [sia B) + sia + B)] si A si B [cosa B) cosa + B)] cos A cos B [cosa B) + cosa + B)] Page of
Series th term test for divergece: If lim a does ot exist or if lim a 0, the the series a is diverget. The p-series: ad diverget if p. Geometric: If r < the FORMULA SHEET PAGE is coverget if p > p ar 0 a r The Itegral Test: Suppose f is a cotiuous, positive, decreasig fuctio o [, ) ad let a f). The i) If ii) If the the fx) dx is coverget, a is coverget. fx) dx is diverget, a is diverget. The Compariso Test: Suppose that a ad b are series with positive terms. i) If b is coverget ad a b for all, the a is also coverget. ii) If b is diverget ad a b for all, the a is also diverget. The Limit Compariso Test: Suppose that a ad b are series with positive terms. If a lim c b where c is a fiite umber ad c > 0, the either both series coverge or both diverge. Alteratig Series Test: If the alteratig series ) b satisfies i) 0 < b + b ii) lim b 0 for all the the series is coverget. The Ratio Test i) If lim a + a L <, the the series a is absolutely coverget. ii) If lim a + a L > or lim a + a, the the series a is diverget. iii) If lim a + a, the Ratio Test is icoclusive. f ) 0) Maclauri Series: fx)! 0 x Taylor s Iequality If f +) x) M for x a d, the the remaider R x) of the Taylor series satisfies the iequality R x) M + )! x a + for x a d Some Power Series e x x! 0 si x cos x ) x+ + )! 0 0 l + x) ) x )! ) x R R R R x x R 0 Page of