MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. (a) Fid the mass of the plate. Iclude uits. Mass = Desity Area = ρ Fid the ceter of mass of the plate. Iclude uits. 0 x dx = 2 3 ρ gm x = ρ 0 x x dx 2 3 ρ = 3 5 cm, y = ρ 0 2 ( x) 2 dx 2 3 ρ = 3 8 cm 2. (Exercise 7 from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Suppose that 2J of work is eeded to stretch a sprig from its atural legth of 30 cm to a legth of 42 cm. (a) How much work is eeded to stretch the sprig from 35 cm to 40 cm? How far beyod its atural legth will a force of 30 N keep the sprig stretched? 3. (Exercise from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Show how to approximate the required work by a Riema sum. The express the work as a itegral ad evaluate it: a heavy rope, 50 ft log, weighs 0.5 lb/ft ad hags over the edge of a buildig 20 ft high. (a) How much work is doe i pullig the rope to the top of the buildig? How much work is doe i pullig half the rope to the top of the buildig? 4. (Exercise 5 from Sectio 6.6 i Stewart s Calculus Cocepts ad Cotexts) Show how to approximate the required work by a Riema sum. The express the work as a itegral ad evaluate it: a leaky 0-kg bucket is lifted from the groud to a height of 2m at a costat speed with a rope that weighs 0.8 kg/m. Iitially the bucket cotais 36 kg of water, but the water leads at a costat rate ad fiishes draiig just as the bucket reaches the 2-m level. How much work is doe? 5. Complete Exercise 9 from Sectio 6.6 o pg. 473 i Stewart s Calculus Cocepts ad Cotexts Fid the aswers to exercises 5-8 (ad all odd-umbered exercises i the textbook) i the back of the textbook. 6. Fid the limit of all the sequeces i the sequece activity: http://math.colorado.edu/math2300/projects/sequecespractice.pdf
7. Does {a }, where a =, coverge? If so, what does it coverge to? I wo t be fooled - a is a sequece, ot a series. I am just beig asked if the sequece has a limit. Yes, it coverges to 0. lim = 0. If I had bee asked about the covergece of the series, the that aswer would have bee diverget, sice it is the harmoic series. 8. Decide whether each of the followig sequeces coverges. If a series coverges, what does it coverge to? If ot, why ot? (a) The sequece whose -th term is a =. Coverges to. (The part goes to zero.) The sequece whose -th term is b = +. Coverges to 0: ( ) + + + = ( + ) = + = + + + + + +, ad the deomiator of this expressio grows as gets large, so it approaches zero. (c) The sequece whose -th term is c = cos(π). Diverges. The sequece is (,,,,... ) which oscillates. (d) The sequece {d }, where d = 2 ad d = 2d for >. Diverges. The sequece is (2, 4, 8, 6,... ), a geometric sequece with r >, so the terms go to ifiity. 9. Fid the sum of the series. For what values of the variable does the series coverge to this sum? (a) + x 2 + x2 4 + x3 8 + 2 (a), for x < 2 2 x y y 2 + y 3 y 4 + y, for y < y + (c) 4 + z + z2 3 + z3 9 + (c) z 2, for z < 3 z 3 0. For each of the followig series, determie whether or ot they coverge. If they coverge, determie what they coverge to. ( ) 2 (a) 5 This is a geometric series with first term a = 5 ad ratio r = 2 3 3. r <, so by the geometric series test the series coverges to =2 a r = 5 2 3 = 5. 3 4+ This is a geometric series. To get the first term I substitute = 2 to get a = 5 4 3 4 3 /5 2 = 3 64 25 = 4800. The ratio is r = 4 5. r < so this series coverges by the geometric series test to = 24000. a r = 4800 4 5
(c) (d) =3 7( π) 2 e 3+ = =3 7( π) 2 π e e 3 = 7(( π) 2 ) π e (e 3 ) =3 =3 7(π 2 ) π e (e 3. Writte i this form it ) is clear that this is a geometric series. Substitute = 3 to fid that the first term is a = 7π5. e 7 r = π2 <, so by the Geometric Series Test, this series coverges to a e 3 r = 7π5 e 7 π2 e 3 8 4(.07) + This is a fiite geometric series with the first term a = 4(.07) 3 ad r =.07. The =2 sum is (e) first term first uadded term r = 4(.07)3 4(.07) 0.07. 2 + 3 + 2. Usig partial fractios decompositio, the series equals ( + ). + 2 Calculatig partial sums for this telescopig series: s = 2 3, s 2 = 2 3 + 3 4 = 2 4. More geerally s = 2 +2. Takig limits, ( 2 + 3 + 2 = lim 2 ) = + 2 2 (See separate file for solutios to the ext three problems). For each of the followig series, determie if it coverges absolutely, coverges coditioally, or diverges. Completely justify your aswers, icludig all details. (a) (c) (d) (e) (f) (g) (h) (i) (j) =2 =2 =2 ( ) + 5 (l ) 2 (l ) 2 2 2 ( 3)! ( 4 2 ( 3) + + ) 2 ( ) 3 + 2 + 3 5 2 7 + 3 arcta si 2
(k) (l) ( ) +! 2. Cosider the series ( )! (a) Cofirm usig the Alterate Series Test that the series coverges. How may terms must be added to estimate the sum to withi.000? (c) Estimate the sum to withi.000. 3. How may terms of ( ) should be added to estimate the sum to withi.0? No calculators. 4. Check whether the followig series coverge or diverge. I each case, give the aswer for covergece, ad ame the test you would use. If you use a compariso test, ame the series b you would compare to. (a) (c) (d) (e) (f) (g) (h) (i) ( 2 + ) diverges use itegral or limit compariso test, comparig to ( 2 + ) 2 coverges term-size compariso test with 3 ( + ) diverges divergece test (th term test) (that is, the the term does ot go to zero) 4 + 5 2 diverges limit compariso test or th term test ( ) si 2 (hit: cosider 2 coverges ratio test! (2)! diverges ratio test ( + 3)!! ( + 2)! coverges simplify to compariso test.! coverges ratio test ) coverges limit compariso test 2 (+)(+2) ad the use term-size compariso test or limit 5. Cosider the series l. Are the followig statemets true or false? Fully justify your aswer.
(a) The series coverges by limit compariso with the series The series coverges by the ratio test. False (c) The series coverges by the itegral test. False 6. Cosider the series aswer.. False ( ) l. Are the followig statemets true or false? Fully justify your (a) The series coverges by limit compariso with the series The series coverges by the ratio test.false (c) The series coverges by the itegral test. False (d) The series coverges by the alteratig series test.true (e) The series coverges absolutely. False.False 7. Suppose the series a is absolutely coverget. Are the followig true or false? Explai. (a) a is coverget.true, because if a series coverges absolutely, it must coverge The sequece a is coverget. lim = 0. (c) ( ) a is coverget.true, i fact it coverges absolutely. true, because a coverges, so by the Divergece Test, (d) The sequece a coverges to. False, the sequece coverges to 0 by the Divergece Test. (e) a is coditioally coverget. False (f) a coverges. True, this ca be show usig the term-size compariso test. 8. Does the followig series coverge or diverge? 3 2 You must justify your aswer to receive credit. Yes, by the ratio test, because, if a deotes the th term of this series, the the limit of a + /a as is zero (which is less tha ). 9. A ball is dropped from a height of 0 feet ad bouces. Assume that there is o air resistace. Each bouce is 3 4 of the height of the bouce before. () Fid a expressio for the height to which the ball rises after it hits the floor for the th time. H() = 0( 3 4 ) (2) Fid a expressio for the total vertical distace the ball has traveled whe it hits the floor for the th time. D() = 0 + (2 0 (3/4)) (3/4) (3/4) (3) Usig without proof the fact that a ball dropped from a height of h feet reaches the groud i h/4 secods: Will the ball bouce forever? If ot, how log it will take for the ball to come to rest?!
The ball will ot bouce forever. The total time it bouces is give by with a = (/2) 0(3/4) ad r = 3/4. ( 0/4) + a/( r)
Wat more practice? Here s some more! 20. I theory, drugs that decay expoetially always leave a residue i the body. However i practice, oce the drug has bee i the body for 5 half-lives, it is regarded as beig elimiated. If a patiet takes a tablet of the same drug every 5 half-lives forever, what is the upper limit to the amout of drug that ca be i the body? Let P represet the percetage of the drug i the body after the th tablet. The P = (00 percet) P 2 = (.5.5.5.5.5) + =.0325 P 3 =.0325(.0325) + =.03223 So, P = (.0325).0325. As, P.0323, so this is the maximum amout of the drug i the body. 2. Let {f } be the sequece defied recursively by f = 5 ad f = f + 2 + 4. (a) Check that the sequece g whose -th term is g = 2 +3+ satisfies this recurrece relatio, ad that g = 5. (This tells us g = f for all.) We check: ad plaily g = 5 g + 2 + 4 = (( ) 2 + 5( ) ) + 2 + 4 = 2 + 5 = g Use the result of part (a) to fid f 20 quickly. f 20 = 20 2 + 5 20 = 499 22. Fid the values of a for which the series coverges/diverges: (a) (c) (d) (e) ( 2a ( a 2 ( ) 2 a (l a) ) ) (l ) a
(f) (g) (h) (i) ( + a ) ( + a) l a a l (a) a > /2 a 0 (c) a > (d) e < a < e (e) a > (f) diverges for all a (g) 2 < a < 0 (h) 0 < a < e (i) 0 < a < e 23. Determie if these sequeces coverge absolutely, coverge coditioally or diverge. (a) cos 2 coverges absolutely, take the absolute value ad compare to usig the term-size 2 compariso test ( ) 2 coverges coditioally. Take the absolute value, the use limit compariso, + comparig to to show it does ot coverge absolutely. Use alteratig series test to show origial series coverges (ad thus coverges coditioally) For each of the followig statemets, determie if it is true Always, Sometimes or Never. 24. If a sequece a coverges, the the sequece ( ) a also coverges. Sometimes. Try a = ad a = 0. 25. If a sequece ( ) a coverges to 0, the the sequece a also coverges to 0. Always 26. The average value of a fuctio is egative. Sometimes. Try f(x) = o [0, ], ad f(x) = o [0, ]. 27. The geometric series 5r coverges to 5 r Sometimes - true if r <, false if r.
c 5c 28. The geometric series coverges to Always, by the geometric series test. 5 4 29. If a coverges, the 30. If a series 3. If the series a coverges. Sometimes. Try a = ( ) / 2, ad a = ( ) /. a is coverget, the lim a = 0 Always, by the divergece test. ( ) a is coditioally coverget, the the sequece a coverges. Always, The series coverges ad so by the divergece test, {( ) a } must coverge to 0, ad so {a } also coverges to 0. 32. If the series ( ) a is coverget, the the series ( ) / ad ( ) / 2. a is coverget. Sometimes. Try 33. If lim a + a =, the a coverges coditioally. Sometimes. Try a = ( ) / (cod. cov.) ad a = ( ) / 2 (abs. cov.) ad a = ( ) (div.) 34. If the series a diverges, the the series a diverges. Sometimes. Try ( ) / ad /