MATH 129 FINAL EXAM REVIEW PACKET (Spring 2014)

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1 MATH 9 FINAL EXAM REVIEW PACKET (Sprig 4) The followig questios ca be used as a review for Math 9. These questios are ot actual samples of questios that will appear o the fial eam, but the will provide additioal practice for the material that will be covered o the fial eam. Whe solvig these problems keep the followig i mid: Full credit for correct aswers will ol be awarded if all work is show. Eact simplified values must be give uless a approimatio is required. Credit will ot be give for a approimatio whe a eact value ca be foud b techiques covered i the course.. Suppose the rate at which people get a particular disease (measured i people per moth) ca π be modeled b rt ( ) = π si t +. Fid the total umber of people who will get the disease durig the first three moths ( t ).. If f ( w) dw = 7, fid the value of f (5 ) d.. Evaluate t a) dt b) t + + A dz c) z e d d) arcta d + 4. Evaluate l ( z + ) a) dz b) arcsi( ) d z c) g ( ) d where g is twice differetiable with g () = 6, g () = 5, ad g () =. 5. Evaluate (You will receive a cop of the itegratio table durig the fial eam.) a) cos (θ + )dθ b) 4t 9 dt c) d d) si(4 α) dα cos (4 α) cos(4 α) 6. Evaluate + 5 5z 8 d a) d b) dz + c) 6z + z 4 (5 ) d) t dt + t

2 7. Let f be a differetiable fuctio with the followig values: Evaluate the itegrals. ( ) a) f ( ) e f d e f (l ) b) d e f( ) 5 7 f ( ) The velocit v of the flow of blood at a distace r from the cetral ais of a arter with radius R is proportioal to the differece betwee the square of the radius of the arter ad the square of the distace from the cetral ais. Fid a equatio for v usig k as the proportioalit costat. Fid the average rate of flow of blood. Recall that the average value of a fuctio over [a, b] is b give b ( ) b a f d. a 9. I the stud of probabilit, a quatit called the epected value of X is defied as 7 e E( X ) = f ( ) d. Fid EX ( ) if f( ) = 7. <. Fid a approimatio of t e dt usig the midpoit rule with =. (Show our work).. Below is the graph of a fuctio = f( ). Assume the fuctio is icreasig for < <, cocave up for < <, ad cocave dow for < <. Which of the followig are true for a umber of subdivisios? Select all that appl. a) b) c) Left( ) < f ( ) d < Right( ) d) Mid( ) < f ( ) d < Trap( ) 5 Right( ) < f ( ) d < Left( ) e) Trap( ) < f ( ) d < Mid( ) 5 Trap( ) < f ( ) d < Mid( ) f) Mid( ) < f ( ) d < Trap( ) 5 5 b. Approimatios usig Left(), Right(), Trap(), ad Mid() were made for f ( ) d. a If f ( ) ad f ( ) are positive o [ ab,, ] match the results to the rules , 7.67, 6.867, 5.667

3 . Determie if the improper itegral coverges or diverges. Show our work/ reasoig. If the itegral coverges, evaluate the itegral. a) + 4 d b) d e c) d e π si d) d e) π 6 cos ( ) ( ) d du f) 5 u 6 4. Accordig to a book of mathematical tables, substitutio to fid m e s m d. Assume s >. t e dt = π. Use this formula ad 5. Suppose f is cotiuous for all real umbers ad that f ( ) d coverges. Determie which of the followig coverge. Eplai or show our work clearl. Assume a >. a) a f ( ) d b) f ( a ) d c) ( a + f ( ) ) d d) f ( a + ) d 6. Determie if the improper itegral coverges or diverges. Justif our aswer. a) dθ b) θ + + si d ( + ) c) (+ si ) d d) + e 5 + d 7. If the fuctio f( ) satisfies f( ) > for < < 4, but 4 lim b ( ) b 4 iequalities would impl that g( ) d also diverges? (select all that appl). a) g ( ) < f( ) for < < 4 c) g ( ) > f( ) for < < 4 b) g ( ) < f( ) for < < 4 d) g ( ) < f( ) for < < f d = +, which 8. Use the cocept of slicig ad the variable show to aswer the followig about the solid. a) Write a formula for the volume of the slice. b) Write a Riema sum that approimates the volume of the solid. c) Write a itegral for the volume of the solid.

4 9. Cosider the regio bouded b = + 6, =, ad the -ais. Sketch ad shade i this regio. Set up the itegral(s) eeded to fid the area if we use the followig: a) slices that are perpedicular to the -ais. b) slices that are perpedicular to the -ais.. Cosider the regio bouded b = 5e, = 5, ad =. Fid the volume of the solid obtaied b rotatig the regio aroud the followig: a) the -ais b) the lie = 5. Cosider the regio bouded b =, = 8, ad =. Fid the volume of the solid obtaied b rotatig the regio aroud the followig: a) the -ais b) the lie =. Cosider the shaded regio bouded b = f( ) ad = g ( ) as show. Set up the itegral eeded to fid the volume of the solid obtaied b rotatig the regio aroud the -ais.. Cosider the regio bouded b the first arch of = si ad the -ais. Fid the volume of the solid whose base is this regio ad whose cross-sectios perpedicular to the -ais are the followig: a) squares. b) semi-circles 4. The circumferece of a tree at differet heights above the groud is give i the table below. Assumig all of the horizotal cross-sectios are circular, estimate the volume of the tree. Height (iches) 4 5 Circumferece (iches) Set up, but do ot evaluate the itegrals eeded to fid the volumes of the solids. a) The solid obtaied b rotatig the regio bouded b = ad = aroud the -ais. b) The solid obtaied b rotatig the regio uder f( ) = + for aroud the -ais. 6. A metallic rod 5 cm i legth is made from a miture of several materials so that its desit chages alog its legth. Suppose the desit of the rod at a poit cm from oe ed is give b δ ( ) = +.5cosh grams per cm of legth. Fid the total mass of the rod.

5 7. Suppose a cit is roughl circular with a radius of 8 miles ad the desit of people ca be modeled b some fuctio δ ( ) i people per square mile. Set up a itegral to fid the total populatio if is the distace i miles from a) the ceter of the cit b) Mai Street 8. A clidrical form is filled with slow-curig cocrete to form a colum. The radius of the form is feet ad the height is 5 feet. While the cocrete hardes, gravit causes the desit to var so that the desit at the bottom is 9 pouds per cubic foot ad the desit at the top is 5 pouds per cubic foot. Assume that the desit varies liearl from top to bottom. Fid the total weight (i pouds) of the cocrete colum. 9. a) Fid a formula for the geeral term of the sequece 4 6 8,,,,, b) Determie if the sequeces coverge or diverge. If the sequece coverges, fid its limit. + 5( + )! a = b = 5 ( )!. Fid the followig sums: k 4 b) a) k = k = k 4. A mg dose of a particular medicie is give ever 4 hours. Suppose 5% of the dose remais i the bod at the ed of 4 hours. Let P represet the amout of medicie that is i the th bod right before the dose is take. Let Q represet the amout of medicie that is i the bod right after the th dose is take. Epress P ad Q i closed-form.. Use the itegral test to determie if the series coverges or diverges. (You will ot alwas be told which series test to use durig the fial eam.) + a) b) = (l ) = + +. Use the ratio test to determie if the series coverges or diverges. + e (!) a) b) ( ) ( )! = =

6 6 4. What does the ratio test tell us about the covergece or divergece of the series? 5 = + 5. Determie if k ( ) is absolutel coverget, coditioall coverget, or diverget. (l ) k = 5 k k 6. Which test(s) could be used to prove that the series a) The p series test. b) The itegral test. c) The ratio test. d) The compariso test, usig the series. = = coverges? (select all that appl) 7. Idicate whether the followig statemets are True or False. a) If a b ad a coverges, the b coverges. b) If a b ad a diverges, the b diverges. c) If a coverges, the a coverges. d) If a coverges, the lim a =. e) If lim a =, the a coverges. 8. Determie the radius of covergece ad the iterval of covergece (ou do ot eed to ivestigate covergece at the edpoits): (+ )( + 4) a) b) ! 4! 6! 8! = 4 ( )!( )!( ) 4!( ) c) Suppose that C ( ) coverges whe = 4 ad diverges whe = 6. Which of the = followig are True, False, or impossible to determie? a) The power series diverges whe =. b) The power series coverges whe =. c) The power series diverges whe = Fid the Talor polomial of degree two about Use our polomial to fid a approimatio for f (). a = for the fuctio f( ) ( 7) = +.

7 4. Suppose P( ) = c + c+ c is the secod degree Talor polomial for a fuctio f( ) where f( ) is alwas icreasig ad cocave dow. Determie the sigs of c, c, ad c. k+ k! k 4. Cosider the fuctio give b f( ) = ( ) ( ). k = ( k)! a) Fid f (). b) Fid f (). c) Fid f (). d) Fid the Talor series for f( ) about =. Iclude a epressio for the geeral term of the series. 4. Fid the eact value of f ( ) d if k 4 = + = f( ) ( k ) 4 5 k = 44. Write out the first four ozero terms of the Talor series for cos( θ ) about π θ =. 45. B recogizig each series as a Talor series evaluated at a particular value of, fid the followig sums, if possible. (You will be give a short table of Talor series for well-kow fuctios durig the fial.) k k+ k ( ) (.5) π a) + + b)! 5! 7! c) + k = k + k = e 46. Use the Talor series for f( ) = si ear = to fid the value of si cotiuous fuctio g ( ) =. = g () () for the 47. Fid the Talor series about for the followig fuctios (iclude the geeral term): a) f( ) = l(+ ) b) f( ) = e a 48. Epad about i terms of the variable r ( a+ r) a ver small whe compared to a. where a is a positive costat ad r is

8 49. Idicate whether the followig statemets are True or False. a) If f( ) ad ghave ( ) the same Talor polomial of degree two ear =, the f( ) = g ( ). f () g () b) The Talor series for f( gabout ) ( ) = is f() g() + f () g () + +.! c) The Talor series for f coverges everwhere f is defied. 5. Write out the first four ozero terms of the Talor series about = for ( ) = ta ( ) f t dt 5. Idetif the equilibrium solutio(s) for ustable. Justif our aswer. dq QQ Q dt = ( )( + 4) ad classif each as stable or 5. Match the followig differetial equatios with oe of its solutios. d a) 4 d = i) = + d 5 b) = ii) = e + e d + d c) 6 d = + iii) = 5 d d) l d = + iv) = e + kt 54. Fid the values of A ad k so that () t = Ae is a solutio to through the poit (, e). d d 4 + = ad passes dt dt 55. Solve the differetial equatios subject to the iitial coditios: d a) 4 d =, () = b) d cos ( θ ) dθ =, ( π ) = d c) 4 t( ) dt =, () =

9 56. A particular drug is kow to leave a patiet s sstem at a rate directl proportioal to the amout of the drug i the bloodstream. Previousl, a phsicia admiistered 9 mg of the drug ad estimated that 5 mg remaied i the patiet s bloodstream 7 hours later. a) Write a differetial equatio for the amout of drug i the patiet s bloodstream at time t. b) Solve the differetial equatio i part a). c) Fid the approimate time whe the amout of drug i the patiet s bloodstream was. mg. 57. Match the differetial equatio with the slope field (assume a is a positive costat): d a) a d = + b) d a d = c) d ( 4)( a) d = d d) ( 4)( a) d = e) d a d = f) d a d = + (i) (ii) (iii) (iv) (v) (vi) 58. Dead leaves accumulate o the floor of a forest at a cotiuous rate of 4 grams per square cetimeter per ear. At the same time, these leaves decompose cotiuousl at the rate of 6% per ear. a) Write a differetial equatio for the quatit of leaves (i grams per square cetimeter) at time t. Solve this differetial equatio. b) Fid the equilibrium solutio ad give a practical iterpretatio. Is the solutio stable?

10 d 59. The differetial equatio f(, ) d = has slope field at the right: Match the related differetial equatio with its slope field below: d a) f(, ) d = b) d = d f (, ) d c) ( f(, )) d = d) d f (, d = ) (i) (ii) (iii) (iv) 6. A room with a souther eposure heats up durig the morig. The temperature of the room icreases liearl so that it rises Ffor ever 5 miutes. Earl i the morig, a cup of coffee with a temperature of 8 F is placed i the room whe the room temperature is 6 F. Newto s Law of Coolig states that the rate of chage i the temperature of the coffee should be proportioal to the differece i temperature betwee the coffee ad the room. a) Write a formula for the temperature of the room t miutes after the coffee is placed there. b) Write a iitial value problem for the temperature of the coffee as a fuctio of time. 6. The area that a bacteria colo occupies is kow to grow at a rate that is proportioal to the square root of the area. Assume the proportioalit costat is k =.6 Write a differetial equatio that represets this relatioship. Solve the differetial equatio. 6. Two models, based o how iformatio is spread through a populatio, are give below. Assume the populatio is of a costat size M. a) If the iformatio is spread b mass media (TV, radio, ewspapers), the rate at which iformatio is spread is believed to be proportioal to the umber of people ot havig the iformatio at that time. Write a differetial equatio for the umber of people havig the iformatio b time t. Sketch a solutio assumig that o oe (ecept the mass media) has the iformatio iitiall.

11 b) If the iformatio is spread b word of mouth, the rate of spread of iformatio is believed to be proportioal to the product of the umber of people who kow ad the umber who do't. Write a differetial equatio for the umber of people havig the iformatio b time t. Sketch the solutio for the cases i which i) o oe kows iitiall, ii) 5% of the populatio kows iitiall, ad iii) 75% of the populatio kows iitiall. 6. A tak of water has the shape of a right circular coe show below. I each case, set up the itegral for the amout of work eeded to pump the water out of the tak uder the give coditios. Your itegral must correspod to the variable idicated i the picture. (The desit of water is 6.4 pouds per cubic foot.) a) The tak is full, the tak will be emptied, ad the water is pumped to a poit at the top of the tak. b) The tak is full, the tak will be emptied, ad the water is pumped to a poit feet above the top of the tak. c) The tak is full, the water will be pumped util the level of the water i the tak drops to 5 feet, ad the water is pumped to a poit at the top of the tak. d) The iitial water level of the tak is feet, the tak will be emptied, ad the water is pumped to a poit feet above the top of the tak. 64. Workers o a platform 45 feet above the groud will lift a block of cocrete weighig 5 pouds from the groud to the platform. The block is attached to a chai that weighs pouds per foot. Fid the amout of work required. 65. A flag i the shape of a right triagle is hug over the side of a buildig as show. It has a total mass of 8 kg ad uiform desit. Set up the itegral eeded to fid the work doe i rollig up the flag to the top of the buildig. Use g = 9.8 m/s. 66. A dam is feet log ad 5 feet high. If the water is 4 feet deep, fid the force of the water o the dam.

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