MATH 9 FINAL EXAM REVIEW PACKET (Revised Sprig 8) 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 exam, but they will provide additioal practice for the material that will be covered o the fial exam. Whe solvig these problems eep the followig i mid: Full credit for correct aswers will oly be awarded if all wor is show. Exact values must be give uless a approximatio is required. Credit will ot be give for a approximatio whe a exact value ca be foud by techiques covered i the course. The aswers, alog with commets, are posted as a separate file o http://math.arizoa.edu/~calc.. Suppose the rate at which people get a particular disease (measured i people per moth) ca π be modeled by rt () = π si t +. Fid the total umber of people who will get the disease durig the first three moths ( t ).. If f( u) du = 7, fid the value of f (5 xdx ).. Evaluate t dt t + 4. Use the method of itegratio by parts: a) Evaluate b) Evaluate ( z + ) l z dz arcsi( x x ) dx. c) Let g be twice differetiable with g () = 6, g () = 5, ad g () =. Fid x g ( x) dx. 5. Evaluate the followig itegrals (you ca use the table of itegrals): a) cos (θ + )dθ b) 4t 9 dt c) y + 8y+ 5 6. Use the method of partial fractios to evaluate y + 5y. y + y
dx 7. Use the method of trigoometric substitutio to evaluate. (5 x ) 8. The velocity v of the flow of blood at a distace r from the cetral axis of a artery with radius R is proportioal to the differece betwee the square of the radius of the artery ad the square of the distace from the cetral axis. Fid a equatio for v usig as the proportioality costat. Fid the average rate of flow of blood. Recall that the average value of a fuctio over [a, b] is b give by ( ) b a f xdx. a 9. I the stu of probability, a quatity called the expected value of X is defied as x 7 e x EX ( ) = xf( xdx ). Fid E( X ) if f( x) = 7. x < t. a) Fid a approximatio of e dt usig the midpoit rule with =. (Show your wor). b) Does the midpoit rule give a overestimate or uderestimate for the itegral i part a?. Determie if the improper itegral coverges or diverges. Show your wor/ reasoig. If the itegral coverges, evaluate the itegral. x a) x + 4 dx b) dx e π si x c) dx d) x x dx π 6 e cos x ( ). Accordig to a boo of mathematical tables, substitutio to fid x m e s m d x. t e dt = π. Use this formula ad. Suppose f is cotiuous for all real umbers ad that f ( xdx ) coverges. Determie which of the followig coverge. Explai or show your wor clearly. Assume a >. a) a f( x) dx b) ( ) f ( ax ) dx c) a+ f( x) dx d) f ( a+ x) dx 4. Use a appropriate compariso to determie if the improper itegral coverges or diverges. If the itegral coverges, fid a upper boud for the itegral (see p56). dθ + si x (+ si xx ) a) b) dx c) dx θ + ( x + ) x +
5. Use the cocept of slicig ad the variable show at the right to set up the defiite itegral eeded to fid the volume of the solid. 6. Cosider the regio bouded by y = x+ 6, y = x, ad the x-axis. Setch 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 x-axis. b) slices that are perpedicular to the y-axis. 7. Cosider the regio bouded by y = 5e x, y = 5, ad x =. Fid the volume of the solid obtaied by rotatig the regio aroud the followig: a) the x-axis b) the lie y = 5 8. Cosider the regio bouded by y = x, y = 8, ad the y-axis. Fid the volume of the solid obtaied by rotatig the regio aroud the y-axis. 9. Cosider the regio bouded by the first arch of y = si x ad the x-axis. Fid the volume of the solid whose base is this regio ad whose cross-sectios perpedicular to the x -axis are the followig: a) squares. b) semi-circles. 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) 6 8 6. The soot produced by a garbage icierator spreads out i a circular patter. The depth, Hr (), i millimeters, of the soot deposited each moth at a distace of r ilometers from the r icierator is give by Hr ( ) =.5e. Write a defiite itegral givig the total volume of soot deposited withi 5 ilometers of the icierator each moth. Evaluate the itegral ad express your fial aswer i cubic meters.. A metallic rod 5 cm i legth is made from a mixture of several materials so that its desity chages alog its legth. Suppose the desity of the rod at a poit x cm from oe ed is give by δ ( x) = +.5coshx grams per cm of legth. Fid the total mass of the rod.. A cylidrical 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, gravity causes the desity to vary so that the desity at the bottom is 9 pouds per cubic foot ad the desity at the top is 5
pouds per cubic foot. Assume that the desity varies liearly from top to bottom. Fid the total weight (i pouds) of the cocrete colum. 4. A ta of water has the shape of a right circular coe show below. I each case, set up the itegral for the amout of wor eeded to pump the water out of the ta uder the give coditios. Your itegral must correspod to the variable idicated i the picture. (The desity of water is 6.4 pouds per cubic foot.) a) The ta is full, the ta will be emptied, ad the water is pumped to a poit at the top of the ta. b) The ta is full, the ta will be emptied, ad the water is pumped to a poit feet above the top of the ta. c) The ta is full, the water will be pumped util the level of the water i the ta drops to 5 feet, ad the water is pumped to a poit at the top of the ta. d) The iitial water level of the ta is feet, the ta will be emptied, ad the water is pumped to a poit feet above the top of the ta. 5. Worers o a platform 45 feet above the groud will lift a bloc of cocrete weighig 5 pouds from the groud to the platform. The bloc is 5 feet tall ad is attached to a 4 foot chai that weighs pouds per foot. Fid the amout of wor required. 6. A flag i the shape of a right triagle is hug over the side of a buildig as show at the right. It has a total weight of 4 pouds ad uiform desity. Fid the wor eeded to lift the flag oto the roof of the buildig. ft Roof 6 ft 4 6 8 7. a) Fid a formula for the geeral term of the sequece,,,,, 9 6 5 6 49 + b) Determie if the sequece a = coverges or diverges. If it coverges, fid its limit. 5 8. Fid the followig sums: a) = 4 b) = 4 9. A mg dose of a particular medicie is give every 4 hours. Suppose 5% of the dose remais i the bo at the ed of 4 hours. Let represet the amout of medicie that is i the th bo right before the dose is tae. Let Q represet the amout of medicie that is i the th bo right after the dose is tae. Express P ad Q i closed-form. P
. Use the itegral test to determie if the series coverges or diverges: + a) b) = (l ) = + +. Use the ratio test to determie if the series coverges or diverges: + e (!) a) b) ( ) ( )! = =. 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.. Determie the radius of covergece ad the iterval of covergece (you do ot eed to ivestigate covergece at the edpoits): 4 6 8 (+ )( x+ 4) x x x x a) b) + + + +! 4! 6! 8! = 4 ( x )!( x )!( x ) 4!( x ) c) + + + + + 4 8 6 4. Suppose that C ( ) x coverges whe x = 4 ad diverges whe x = 6. Which of the = followig are True, False, or impossible to determie? a) The power series diverges whe x =. b) The power series coverges whe x =. c) The power series diverges whe x = 5. 5. Fid the Taylor polyomial of degree two that approximates the fuctio ear x =. f x = + ( ) x 6. Suppose P( x) = c + c x+ c x is the secod degree Taylor polyomial for a fuctio f ( x ) where f ( x ) is always icreasig ad cocave dow. Determie the sigs of c, c, ad c.
+! 7. Cosider the fuctio give by f( x) = ( ) ( x ). = ( )! a) Fid f (). b) Fid f (). c) Fid f (). d) Fid the Taylor series for f ( x ) about x =. Iclude a expressio for the geeral term of the series. 8. Fid the exact value of f ( xdx ) if 4 = + = + + + + + f( x) ( ) x x x 4x 5x = 9. Match the fuctio with its Taylor series ear x =. a) f( x) = x i) x ( ) = ( )! x b) f ( x) = e ii) c) f ( x ) = cosx iii) d) f ( x ) = arcta x iv) = = x + x ( ) + x =! 4. By recogizig each series as a Taylor series evaluated at a particular value of x, fid the followig sums, if possible. a) + + b)! 5! 7! + ( ) (.5) c) + = = π + e 4. a) State the Taylor series for f ( x) = sixear x =. b) Use the series i part a) to fid the value of g () () for the cotiuous fuctio si x x gx ( ) = x. x = 4. Fid the Taylor series about for the followig fuctios (iclude the geeral term): a) f ( x) = xl(+ x) b) f ( x) = e x a 4. Expad about i terms of the variable r ( a+ r) a very large whe compared to a. where a is a positive costat ad r is
44. Idicate whether the followig statemets are True or False. a) If f ( x) ad gx ( ) have the same Taylor polyomial of degree two ear x =, the f ( x) = g( x). f () g () b) The Taylor series for f ( xgxabout ) ( ) x = is f() g() + f () g () x+ x +.! c) The Taylor series for f coverges everywhere f is defied. 45. a) Write Euler s Formula for b) Express the complex umber i e θ. 4 c) Express the complex umber e i i the form i Re θ. π i i the form a+ bi. ( 4 ) d) Express the complex umber e + it i the form at () + bti (). 46. Match the followig differetial equatios with oe of its solutios. a) y 4 dx = i) y = x + x dx y 5 x y + b) = ii) x x y = e + e c) y x 6x dx = + iii) y = 5 d) y l y x dx = + iv) x y = e + x 47. Cosider the iitial value problem approximate x y dx = +, () y(.4). Complete the followig table of values. y =. Use Euler s method to x y Δx Δy.....4
dx = +, y () =. 48. Solve the differetial equatio subject to the iitial coditio: x( y 4) 49. A particular drug is ow to leave a patiet s system at a rate directly proportioal to the amout of the drug i the bloodstream. Previously, a physicia 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 approximate time whe the amout of drug i the patiet s bloodstream was. mg. 5. Match the differetial equatio with the slope field (assume a is a positive costat): a) d) ax y dx = + b) x a dx = c) ( y 4)( y dx = a ) ( x 4)( y a) dx = e) a dx = f) y a dx = + (i) (ii) (iii) (iv) (v) (vi)
5. The differetial equatio f ( xy, ) dx = has slope field: Match the related differetial equatio with its slope field: a) f ( xy, ) dx = b) dx = f ( x, y) c) ( f ( xy, )) dx = d) f ( x, y dx = ) (i) (ii) (iii) (iv) 5. Dead leaves accumulate o the floor of a forest at a cotiuous rate of 4 grams per square cetimeter per year. At the same time, these leaves decompose cotiuously at the rate of 6% per year. a) Write a differetial equatio for the quatity 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? 5. A room with a souther exposure heats up durig the morig. The temperature of the room icreases liearly so that it rises F for every 5 miutes. Early 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.
54. The area that a bacteria coloy occupies is ow to grow at a rate that is proportioal to the square root of the area. Assume the proportioality costat is =.6 Write a differetial equatio that represets this relatioship. Solve the differetial equatio. 55. It is of cosiderable iterest to policy maers to model the spread of iformatio through a populatio. Two models, based o how the iformatio is spread, are give below. Assume the populatio is of a costat size M. a) If the iformatio is spread by 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 by time t. Setch a solutio assumig that o oe (except the mass media) has the iformatio iitially. b) If the iformatio is spread by word of mouth, the rate of spread of iformatio is believed to be proportioal to the product of the umber of people who ow ad the umber who do't. Write a differetial equatio for the umber of people havig the iformatio by time t. Setch the solutio for the cases i which i) o oe ows iitially, ii) 5% of the populatio ows iitially, ad iii) 75% of the populatio ows iitially. 56. A populatio is modeled by a fuctio Pt ( ) that satisfies the logistic differetial equatio dp P P = dt 6 5. a) Cosider the solutio with the coditio P () = 8. Is the solutio icreasig or decreasig? Fid l im Pt ( ). t b) Cosider the solutio with the coditio () 7 P =. For what value of P is the populatio growig the fastest?