Calculus 2 Test File Spring Test #1
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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 g() = -, for - < <. 3.) Cosider the followig solid. Triagle ABC is perpedicular to rectagle BDEC. Poit A is over the midpoit of edge BC. Edges BD ad CE are feet log. Edges BC ad DE are 4 feet log. The height of the solid is 5 feet. Use itegratio ad cross-sectios to fid the volume. For problems #4-6, cosider the solid created whe the regio bouded y =, y = 3 ad = is revolved aroud the -ais. 4.) Set up but do ot evaluate the itegral(s) ecessary for fidig the volume usig cylidrical shells. 5.) Set up but do ot evaluate the itegral(s) ecessary for fidig the volume usig diss or washers. 6.) Evaluate the itegral(s) from either 4.) or 5.) to fid the volume. Do ot use your calculator. For #7-8, cosider the regio bouded by y = 4 si -, y = 4, = / ad = 3/ (see the figure)
2 7.) Suppose the regio is revolved aroud the lie y = 6. 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. 8.) Suppose the regio revolved aroud the lie =. 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. 9.) A ta has the shape of a right circular cylider. The height of the ta is feet ad the diameter is 0 feet. If the ta is half full of water, how much wor is doe i emptyig the ta through a pipe that pumps the water to a poit 3 feet above the top of the ta? The desity of water is 6.4 pouds per cubic foot. Oce you set up the itegral, you may use your calculator to evaluate it. Test #.) Fid the average value of the fuctio f() = si o the iterval [0, ]. For #-0, do the give itegral. You may ot use your calculator for ay part of these problems. All ecessary wor must be show..) 4 si ) sec ta 4.) si cos 4 5.) e l 6.) e cos( ) 7.) e 4 8.) ( 3 + ) 9.) 0.) Test #3.) 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)
3 5.) Use the Trapezoidal Rule to approimate the followig itegral. Use = 0. Give a upper boud for the error. You may use your calculator if you lie. You might wat to write dow the calculator code you use. si Do the followig itegrals. You may ot use your calculator for ay part of these problems. All ecessary wor must be show. 3.) ( 3 + ) 4.) 5.) ) 6.) (use partial fractio decompositio) 9 8.) 4 9.) Set up, BUT DO NOT SOLVE, the partial fractio decompositio for ( + ) (3-4 ) (+) Use the itegral formulas i your boo to wor these three problems. Write the umber of ay formula you use. l 0.).) l Test #4.) Cosider the fuctio f() = m defied o the iterval [0, ]. a.) Fid the umber, m, such that that f defies a probability desity fuctio (pdf) o the iterval. b.) Fid the mea,..) Solve the followig differetial equatio.
4 dy, y() y 3.) Cosider the regio bouded by y = - ad the -ais. Write the itegrals ecessary for fidig the -coordiate of the ceter of mass of the regio if the desity is give by () =. Use your calculator to evaluate the itegrals ad give the -coordiate of the ceter of mass. 4.) A ta cotais 000L of pure water. Brie that cotais 0.05 g of salt per liter of water eters the ta at a rate of 5 L/mi. Brie that cotais 0.04 g of salt per liter of water eters the ta at a rate of 0 L/mi. The solutio is ept thoroughly mied ad drais from the ta at a rate of 5 L/mi. How much salt is i the ta after 0 miutes? 5.) For this problem, begi with dp/dt = P, where P is populatio ad t is time. I 880, Moutai Home, Arasas had a populatio of 37. By 890 it had essetially doubled. Assumig the populatio cotiued to grow epoetially at the same rate, what would the populatio be i 007? (Note: the actual 000 populatio was,0.) 6.) Fid the orthogoal trajectories for the family of curves give by y = 3. For #7 ad 8, cosider the regio bouded by f() = si, = 0, =. 7.) Cosider the top edge of the regio. (the part formed by the sie curve) Set up the itegral eeded for fidig the legth of that edge. USE YOUR CALCULATOR TO EVALUATE IT. 8.) Cosider the top edge of the regio. (the part formed by the sie curve) Set up the itegral eeded for fidig the area of the surface formed by revolvig that curve aroud the lie y = 3. USE YOUR CALCULATOR TO EVALUATE IT. Evaluate the followig itegrals. 9.) 0.) Test #5.) Show that y = - - is a solutio of the differetial equatio y' + y =..) O the aes below, there are mared poits [ote that the origi is mared]. Recogizig that all coordiates are iteded to be itegers, draw the lie segmets for the part of the directio field for each mared poit for the differetial equatio below. y' = - y
5 3.) Write the itegral (ad use your calculator to evaluate it) for the arc legth of the cardioid r = + cos. 4.) Write the solutio of the iitial-value problem (showig all the wor, icludig the solutio of the differetial equatio) dp dt = 0.08 P 000 ad use it to fid the populatio at t = 40., P 0 = 00 5.) Fid the solutio of the iitial-value problem y' + y =, > 0, y() =. 6.) Fid the area iside r = 3 si ad outside r = + si. 7.) Cosider the set of parametric equatios below. = (t si t ) y = ( cos t ) a.) Draw the graph. b.) Fid a equatio for the taget lie whe t = /3. 8.) Cosider r = si cos. a.) Covert the equatio to rectagular coordiates. b.) Graph the equatio. 9.) Cosider the followig set of parametric equatios. = - si t y = cos 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.) Mae ay ecessary chages to your graph from b.) to mae it a graph of the parametric equatios. d.) Are the graphs from b.) ad c.) the same? If ot, eplai why ot.
6 Test #6.) Graph the followig coic. Fill i all blas. If a bla does ot apply, write "DNA." y Type of coic Focus Focus Edpoit Mior Ais Edpoit Mior Ais Verte Verte Asymptote Asymptote Ceter Directri -itercept(s) y-itercept(s).) Graph the followig rotated coic. Fill i all the blas. If a bla does ot apply, write "DNA." SHOW ALL WORK! Type of coic 3 y Rotated Coordiate System Regular Coordiate System Focus Focus Verte Verte Ceter Asymptote Asymptote 3.) Graph the followig coic. Fill i all blas. If a bla does ot apply, write "DNA." Give everythig i both polar ad rectagular coordiates. r si Type of coic polar rectagular Focus Focus Edpoit Mior Ais Edpoit Mior Ais Verte Verte Asymptote Asymptote Ceter Directri -itercept(s) y-itercept(s) 4.) Idetify the id of curve for each of the followig equatios.
7 a.) + y + C + Dy + E = 0 b.) + 5y + C + Dy + E = 0 c.) 5-4y + C + Dy + E = 0 d.) y - + y + E = 0 e.) C + y + E = 0 f.) - - 5y + C + Dy + E = 0 g.) h.) i.) j.) A r si r A A r si A r si 5.) Suppose a = ad a = (4 + a - )/5. Fid a, a 3, ad a 4. 6.) Determie covergece or divergece for the give sequece. If the sequece coverges, give its limit. Be sure to show all ecessary wor. (- ) + si a.) b.) e a where a = c.) = L. 7.) Determie covergece or divergece for the series. If it coverges, fid its sum. 8.) Determie covergece or divergece for the series. If it coverges, fid its sum.
8 9 0 9.) Determie covergece or divergece for the followig series. Be sure to mae your reasos clear. Test #7 cos I #-4, determie covergece or divergece for the followig series. Be sure to mae your reasos clear..) l 4.) 4.) 3.) ! 4 5.) Cosider the followig series. If possible, use the itegral test to determie covergece or divergece for the followig series. l I #6-7, determie covergece or divergece for the give series. If it coverges, determie if the covergece is absolute or coditioal. Be sure to mae your reasos clear. 6.)! 7.) /3 8.) For the followig power series, use the root test to fid the iterval of covergece. Do ot forget to chec the edpoits ) For the followig power series, fid the radius of covergece. 35 ( )!3 0.) Use the first 5 terms of the power series for e to approimate e -. The, fid a upper boud for the error of this estimate.
9 Fial Eam.) Without your calculator, use cylidrical shells to fid the volume of the regio bouded by f() = 5 - ad the -ais whe it is revolved aroud the y-ais. I # - 7, evaluate the give itegral without usig your calculator. 3.) 3 3.) e cos() 4.) si ) si 6.) 7.) ) Determie covergece or divergece for the give sequece. If it coverges, give its limit. Be sure to show all ecessary wor. (- ) + si I problems #9-0, determie covergece or divergece for the give series. Show your wor. Give clear reasos for your coclusio. If the series coverges, fid the sum. 9.) ta 0.) 3.) Determie covergece (absolute or coditioal) or divergece for the series ( ) l..) Determie covergece (absolute or coditioal) or divergece for the series ( ) ) Determie covergece or divergece for the series 4 3 3! ) Solve the followig differetial equatio. dy y, y() = 5.) A coe of height, h, ad base radius, r, ca be created by rotatig a lie aroud the -ais. a.) Set up the itegral ecessary for fidig the lateral surface area of a coe of height, h, ad base radius, r. (That meas the surface area of the coe, ecludig the bottom circle. See the shaded portio of the picture) 3
10 b.) Usig your calculator, fid the value of the itegral from part a.) if r = 9 iches ad h = foot. 6.) Determie whether or ot y = satisfies the differetial equatio give by y' + y = ) For the followig power series, fid the RADIUS of covergece.!3 8.) Use a ow power series ad itegratio to fid a power series for ta -. 9.) Graph r = si. 0.) Graph the followig coic. Fill i all blas. If a bla does ot apply, write "DNA." y Type of coic Ceter Focus Focus Edpoit Mior Ais Edpoit Mior Ais Verte Verte
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