MATH CALCULUS II Objectives and Notes for Test 4
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1 MATH 44 - CALCULUS II Objectives ad Notes for Test 4 To do well o this test, ou should be able to work the followig tpes of problems. Fid a power series represetatio for a fuctio ad determie the radius ad iterval of covergece. Use differetiatio or itegratio to fid a power series represetatio of a fuctio. Fid a Maclauri or Talor series for a fuctio usig the defiitio. (The defiitio will NOT be provided.) Fid a Maclauri or Talor series for a fuctio usig a kow series of a similar fuctio. (Several importat series from Table o page 6 will be provided.) Use a series to evaluate a limit. Fid the sum of a series b matchig the series to its fuctio ad evaluatig. Fid a th degree Talor polomial, use it to approimate a fuctio, ad estimate the magitude of the error. Fid the slope of the taget lie to a polar curve at a poit specified b a give agle. Fid the area of the regio eclosed b a polar curve. Fid the area of a regio that lies betwee two polar curves. Fid the legth of a polar curve. Verif a fuctio is a probabilit desit fuctio. Fid a probabilit desit fuctio that models a epoetiall distributed radom variable or a ormall distributed radom variable (ou do ot eed to memorize the ormal probabilit desit fuctio, just be able to use it). Give a probabilit desit fuctio, fid the probabilit of a specified evet. (Kow how to use our calculator to evaluate a itegral that ou caot evaluate b had! Come b the learig lab if ou eed calculator help. I will ot aswer calculator questios durig the test.) Verif that a fuctio is a solutio to a differetial equatio ad/or fid costat values so that a fuctio satisfies a differetial equatio. Give a directio (or slope) field, graph a particular solutio curve to a differetial equatio with a specified iitial coditio. Use the graph to estimate the value of the solutio at a give poit. Be able to idetif equilibrium solutios from a slope field. Fid geeral ad particular solutios to first-order separable differetial equatios. Use Euler s method. Solve miig (tak) problems with differetial equatios.
2 MATH 44, Review for Test 4 Page of 7 Some sample problems are provided below for our review. Please ote this is ot a comprehesive list of ever tpe of problem ou ma fid o the eam. I would recommed that ou also review homework problems ad eamples worked i class. Feel free to me if ou fid a tpos or errors. I will revise ad repost as ecessar. * beside item umber meas the item has bee corrected or revised. CHAPTER 8:. Fid a power series represetatio ad its iterval of covergece for the fuctio f ( ) = + 5 For -, evaluate the idefiite itegral as a power series.. d +. e d (Hit: Write out the first several terms for e, divide b, ad itegrate before writig as a summatio.) si 4. Use series to evaluate the limit: lim 0 For 5-7, fid the Maclauri series ad its radius of covergece. You ma use the defiitio or use a kow series ( e, si, geometric, ) to obtai a ew series. 5. f ( ) = e 6. f ( ) = l ( ) 7. f ( ) = arcta ( ) 8. a) Write the 4 th degree Talor polomial T4 ( ) for ( ) si b) Estimate the accurac of the approimatio f ( ) T ( ) SECTIONS H. ad H. f = cetered at 0. for < <. π *9. Fid the slope of the taget lie to the polar curve r = siθ whe θ = 0. Fid the area iside the polar curve r = + cosθ. Fid the area of the regio that lies iside both curves. r = + cosθ, r = + siθ. 4
3 MATH 44, Review for Test 4 Page of 7 SECTION 6.8. Suppose that f ( ) is the probabilit desit fuctio for the weight of a female college studet, where is measured i pouds. 40 a) What is the meaig of f ( ) 0 d b) Write a epressio for the mea of this desit fuctio. c) How do we fid the media of this desit fuctio? = for 0 ad f ( ) = 0for all other values of. 9 a) Verif that f is a probabilit desit fuctio.. Let f ( ) ( 4 ). b) Fid P( X ) c) Fid the mea. 4. I Austi, the legth of time spet waitig for a cit bus is modeled b a epoetial desit fuctio with a mea of 7 miutes. a) Write the probabilit desit fuctio that models this situatio. b) What is the probabilit that a perso waits less tha 5 miutes for the bus? c) Fid the media wait time. 5. The time t i miutes that it takes a studet to work a certai math problem is modeled b the probabilit desit fuctio ( ) f t t te for t 0 = 4. Fid the probabilit that it takes 0 for t <0 a radoml chose studet more tha oe-half miute to work the problem. 6. The height of America me betwee the ages of 8 ad 4 is a ormall distributed radom variable. I 99, the mea height was 70 iches ad the stadard deviatio was iches. a) If a ormall distributed radom variable is modeled b the probabilit desit fuctio f ( ) = e, write the fuctio that models this problem. σ π ( µ ) ( σ ) b) If a 8- to 4- ear old ma was chose at radom, what was the probabilit that he was 75 iches or taller? Set up a itegral that will solve the problem, the use our calculator to evaluate the itegral.
4 MATH 44, Review for Test 4 Page 4 of 7 CHAPTER 7 7. For what values of k does the fuctio = e k satisf the differetial equatio + 5 = 0? = is show. 8. A directio field for the differetial equatio ( )( 4) a) Sketch the graphs of the solutios that satisf the iitial coditios: i) ( 0) = ii) ( 0) =.5 iii) ( ) 0 = 4. b) i) For the iitial coditio ( 0) =, what is lim ( ) ii) For the iitial coditio ( 0) =.5, what is lim ( ) iii) For the iitial coditio ( 0) = 4., what is lim ( ) c) What are the equilibrium solutios? 9. Fid the geeral solutio of the differetial equatio. If a iitial coditio is give, fid the particular solutio that satisfies the give iitial coditio. d, 0 5 e + cos = + e si, 0 = 0 a) ( ) d + = = b) ( ) ( ) ( ) d + = d) d c) ( )??? = ( ) d π π ta = +, 0 < <, = d e) ( ) d f) = d e
5 MATH 44, Review for Test 4 Page 5 of 7 0) A slope field for the differetial equatio = is show. a) Sketch a graph of the solutio that satisfies the iitial coditio ( ) sketch to estimate ( 0.6). b) Use Euler s Method with step size h = 0. to estimate ( 0.6). c) Fid the eact solutio to the differetial equatio ( 0) =. The use our equatio to fid ( 0.6) estimates ou foud i parts a) ad b). 0 = ad use our = with iitial coditio ad compare that value with the ) A tak cotais 000 liters of pure water. Brie that cotais 0.05 kg/l eters the tak at a rate of 8 L/mi. Brie that cotais 0.04 kg/l eters the tak at a rate of 5 L/mi. The solutio drais from the tak at a rate of L/mi. a) Fid the amout of salt i the tak at time t. b) How much salt is i the tak after ½ hour? c) How much salt is i the tak after a ver log time?
6 MATH 44, Review for Test 4 Page 6 of 7 ANSWERS: + 5, ( 5,5) 5 = 0 = *) ( ) OR ( ) + I = ) ( ) = 0 + C + ) l + + C 4) -/6 5)! = +, R =! = 0 6), = +, + R = 7) ( ) 4 = T4 = 6 R = 8a) ( ) 4 8b) Alteratig series so ( ) ( ) 6 6 = = Error < ! 5! 9) 0) 9 π ) π a) The probabilit that a radoml chose female studet weighs betwee 0 pouds ad 40 pods. b) f ( ) d c) solve the equatio f ( ) a) Prove ( 4 ) = b) P ( X ) 4a) f ( ) 0 9 d 7 e for 0 = 7 0 for < 0 5) P( X 0.5) a) ( ) f = e 6b) π ( 70 ) ( ) d = for m. m c) µ =.75 4b) c) 7 l 4.85miutes ( ) ( ) e π d ) k =, 8b) i) ii) iii) 8c) = 0, =, = 4 9a) ( ) = 5e 9b) cos = l + cos 9c) = Ae arcta 9d) = l + C 9e) 4 = + si 9f) = l + C
7 MATH 44, Review for Test 4 Page 7 of 7 0a) from graph: about.5 0b) from Euler s method:.666 = 0.6 =.565 Estimate from graph ot too far off. Euler s method gives a uderestimate (this is because the graph is cocave up ad the taget lies sit below the curve. If the graph were cocave dow, Euler s method would give a overestimate.) Note that we would get a better estimate with Euler s method if we decreased the step size. 0c) ( ) d 600 ( 000 ) t = 0.6 = e dt 000 a) ( t) b) ( ) ( e 00 ) 0 =.97 kg c) 600 ( 000 ) 600 lim t e kg t =
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