Calculus 2 Test File Fall 2013

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Bennett Lambert
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1 Calculus Test File Fall 013 Test #1 1.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1..) Cosider the followig solid. Triagle ABC is perpedicular to rectagle BDEC. Poit A is over the midpoit of edge BC. The base, BDEC, is a square with side legths 4 feet. The height of the solid is 5 feet. Use itegratio ad cross-sectios to fid the volume. For problems #3-5, cosider the solid created whe the regio bouded y = 8- ad y = 7 ad = is revolved aroud the -ais. 3.) Set up but do ot evaluate the itegral(s) ecessary for fidig the volume usig cylidrical shells. 4.) Set up but do ot evaluate the itegral(s) ecessary for fidig the volume usig disks or washers. 5.) Evaluate the itegral(s) from either 4.) or 5.) to fid the volume (NOT BOTH). Do ot use your calculator. 6.) Cosider the regio bouded by y = 8- ad y = 7. Suppose the regio is revolved aroud the lie y = 10. a.) Draw the regio icludig a represetative rectagle formed by your b.) partitio. Set up AND USE YOUR CALCULATOR TO EVALUATE a itegral for fidig the volume of the resultig solid. Be sure to write dow the itegral you used. 7.) A tak has the shape of a right circular cylider (with vertical ais). The height of the tak is 4 meters ad the diameter is 3 meters. If the tak is half full of water, how much work is doe i emptyig the tak through a pipe that pumps the water to a poit 1 meter above the top of the tak? The desity of water is 1000 kg per cubic meter. Oce you set up the itegral, you may use your calculator to evaluate it. 8.) The rate at which people eter a auditorium for a rock cocert is modeled by the fuctio R give by R(t ) = 1380t 675t 3 for 0 t hours; R(t ) is measured i people per hour. No oe is i the auditorium at time t = 0, whe the doors ope. The doors close ad the cocert begis at time t =. How may people are i the auditorium whe the cocert begis? Test # 1.) ( 3 + 1) 1 d.) d (use partial fractio decompositio) 9

2 3.) e d 4.) 5.) e l d 6.) 1 si3 cos d ) sec ta d 8.) d ) d ) Set up BUT DO NOT EVALUATE the partial fractio decompositio for: 11.) For this problem, begi with dp/dt = kp, where P is populatio ad t is time. I 1880, Moutai Home, Arkasas had a populatio of 137. By 1890 it had doubled. Assumig the populatio cotiued to grow epoetially at the same rate, what would the populatio be i 011? (Note: the actual 000 populatio was 11,01.) Test #3 Use the itegral formulas attached to this test to work these three problems. Write the umber of ay formula you use. d ).) si d 3.) csc d 4 9 Evaluate the followig itegrals. Do NOT use the tables of itegrals or your calculator. Show all work. d d 4.) 5.) ) Use Simpso's Rule AND NO CALCULATOR to approimate the followig itegral. Use = 4. Give a upper boud for the error. (you may use your calculator to tur the error boud ito a decimal) 7.) Use the Trapezoidal Rule to approimate the followig itegral. Use = 0. You may use your calculator if you like. You might wat to write dow the calculator code you use. 8.) Solve the followig differetial equatio. dy, y() d y 9.) O the aes below, there are 11 marked poits [ote that the origi is marked]. Recogizig that all coordiates are iteded to be itegers, draw the lie

3 segmets for the part of the directio field for each marked poit for the differetial equatio below. y' = - y 10.) Solve the followig differetial equatio. Test #4 dy d y, y(1) = 1 I #1-, determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. Be sure to show all ecessary work. (-1 ) + si 1.).) e ) Let a 1 =, a = 3 ad a = (a )/a -, >. a.) Fid a. b.) Fid a 3. c.) Fid a 4. I problems #4-5, determie covergece or divergece for the give series. Show your work. Give clear reasos for your coclusio. If the series coverges, fid the sum. 4.) 1 ta 5.) k 1 1 k 1 3 k 6.) If possible, use the itegral test to determie covergece or divergece for the followig series. 1. l For #7-8, for the give series, determie divergece, coditioal covergece or absolute covergece. Be sure to show work justifyig your aswer. 7.) ( 1) cos(1/ ) 8.) ) Usig a geometric series, fid the fractio that produces the decimal ) Fid a geeral formula that fits the followig sequece.

4 7, 16, 7, 40, 55, ) A ball is dropped from a height of 8 feet. It bouces up, reaches a peak, ad falls back to the groud where it bouces agai. Each time it bouces it reaches a peak 3/4 of its previous peak. Fid the total distace traveled by the ball. Test #5 1.) Determie covergece or divergece for the followig series. Be sure to make your reasos clear. 1 1 cos.) Determie covergece or divergece of the give series. 1 ( 1) l ) Determie covergece (absolute or coditioal) or divergece for the followig series. Be sure to make your reasos clear.! 1 4.) Determie covergece (coditioal or absolute) or divergece for the followig series. Be sure to make your reasos clear. 1 1 l 1 5.) Fid the third degree Taylor Polyomial for f() = ta at c = 0. 6.) Determie covergece (absolute or coditioal) or divergece for the give series. Be sure to make your reasos clear. k 13 k 1 1 k1! 4 7.) For the followig power series, fid the iterval of covergece. Do ot worry about the edpoits. 1! 3 8.) Determie covergece (absolute or coditioal) or divergece for the give series. Show your work. Give clear reasos for your coclusio. 1 1 ( 1) For #9 ad 10, cosider the followig series ) Use the limit compariso test to determie covergece or divergece for this series. 10.) Use the basic compariso test to determie covergece or divergece for this series. k

5 Test #6 1.) Use a familiar power series to fid a power series for the followig fuctio. The fid the iterval of covergece. Iclude edpoits. 1 f ( ) (1 ).) Use the first 5 terms of the Maclauri series for e to approimate e ) Use a kow power series to fid a power series for f() = ta ) Fid the Taylor Series for f() = e -1 aroud c = 1. [Note: this is ot the usual power series for the epoetial fuctio.] 5.) Use the Maclauri series for si to evaluate the followig limit. Do NOT use L'Hopital. 3 3 si lim ) For the followig power series, fid the iterval of covergece. Do ot forget to check the edpoits ) For the followig power series, fid the iterval of covergece. Do ot worry about the edpoits. 1!3 8.) Fid the first four terms of the Biomial Series for f() =. Use it to approimate. Test #7 1.) Graph the parametric equatios. = cos(3t) y = cos(t/) si(t).) Set up the itegral AND USE YOUR CALCULATOR TO EVALUATE THE INTEGRAL to fid the area iside the limaço r = 3 + cos (see graph below)

6 3.) Cosider the followig set of parametric equatios. = cos t y = si t a.) Covert the set of equatios to a ice rectagular form (i.e., o trig fuctios). b.) Graph the resultig equatio from part a.). c.) Make ay ecessary chages to your graph from b.) to make it a graph of the parametric equatios. d.) Are the graphs from b.) ad c.) the same? If ot, eplai why ot. 4.) Graph the followig coic ad fill i the blaks. For ay blak that is ot applicable to the particular coic, write "DNA." 1 y Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s) 5.) Graph the followig coic. Fill i all blaks. If a blak does ot apply, write "DNA." y + 3 = 0 Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s) 6.) Graph the followig rotated coic. Fill i all the blaks. If a blak does ot apply, write "DNA." 3 y 1 Give each of the followig i the rotated coordiate system ad the usual coordiate system. Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, - itercept(s), y-itercept(s) 7.) Idetify the kid of curve for each of the followig equatios. a.) + y + C + Dy + E = 0 b.) + 5y + C + Dy + E = 0 c.) 5-4y + C + Dy + E = 0 d.) y - + Dy + E = 0 e.) C + y + E = 0 f.) - - 5y + C + Dy + E = 0 g.) A r h.) si 1 A r si

7 i.) A r j.) + 4y + y + D + Ey + F = 0 si 1 k.) + 4y - y + D + Ey + F = 0 l.) + 4y + 6y + D + Ey + F = 0 8.) Graph the followig coic ad fid all of the followig (for each coordiate system) that apply. If oe does ot apply, put "DNA." Give all poits i both polar ad rectagular form. 4 r 1 si Give each of the followig i the polar coordiate system ad the usual coordiate system. Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, - itercept(s), y-itercept(s) Fial Eam 1.) Cosider the regio bouded by y = 8- ad y = 7. Suppose the regio is revolved aroud the lie y = 10. a.) Draw the regio icludig a represetative rectagle formed by your partitio. b.) Set up AND USE YOUR CALCULATOR TO EVALUATE a itegral for fidig the volume of the resultig solid. Be sure to write dow the itegral you used. Evaluate the followig itegrals..) e cos d 1 d 4.) 3.) 5.) d 6.) l d 9 7.) Without usig your calculator, fid the eact area betwee the curves f() = 4 - ad g() = si(), -1 < < 1. 8.) Suppose the populatio of a give tow follows a model i which the rate of chage of the populatio is always proportioal to the preset populatio. Suppose the populatio i 1910 was 50 ad the populatio i 1918 was 485. What is the populatio i 011? Be sure to show ALL work. 9.) Solve the followig iitial value problem. y " = si - +, y ' (0) = 4, y (0) = ) A tak has the shape of a right circular cylider (with vertical ais). The height of the tak is 8 meters ad the diameter is 6 meters. If the tak is full of water, how much work is doe i emptyig the tak through a pipe that pumps the water to a poit 6 meter above the top of the tak? The desity of water is 1000 kg per cubic meter. Oce you set up the itegral, you may use your calculator to evaluate it. 11.) Graph the followig. r = si( cos()) 1.) Write out the first four o-zero terms of the Taylor series for f() = cos cetered aroud a = /.

8 13.) Determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. (-1 ) + si 1 14.) Determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. e 1 15.) Determie covergece or divergece for the give series. Show your work. Give clear reasos for your coclusio ) Determie covergece or divergece for the give series. Show your work. Give clear reasos for your coclusio ) Determie covergece (absolute or coditioal) or divergece for the give series. Show your work. Give clear reasos for your coclusio. 1 ( 1) ) For the followig power series, fid the iterval of covergece. Do ot worry about the edpoits. 1 ta!3 19.) The followig set of parametric equatios describes the motio of a particle i the -y plae. Chage them to rectagular form ad graph the resultig equatio. Label the positio of the particle at t = 0, /4 ad /. = si t y = cos t, 0 t 0.) Graph the followig coic ad fid all of the followig that apply. If oe does ot apply, put "DNA." 1 y Type of coic, Focus, Focus, Edpoit Mior Ais, Edpoit Mior Ais, Verte, Verte, Asymptote, Asymptote, Ceter, Directri, -itercept(s), y-itercept(s)

Calculus 2 Test File Spring Test #1

Calculus 2 Test File Spring Test #1 Calculus Test File Sprig 009 Test #.) Without usig your calculator, fid the eact area betwee the curves f() = - ad g() = +..) Without usig your calculator, fid the eact area betwee the curves f() = ad