Math 132 Spring 2003 Exam 2 Solutions No calculators with a CAS are allowed Be sure your calculator is set for radians not degrees if you do any calculus computations with trig functions Part I Multiple Choice 5 points problem blacken your answers on the answer card 1 Find the value of B B B A F B C D G H 2 I 2e E diverges to J diverges to Let B B B so B B Then B B B B G G Then B B B lim lim B B B lim B l 2 For a certain radioactive isotope the amount remaining after 3 hours is 78 of the original amount What is its half life Round your answer to 2 decimal places A 6 21 hrs B 8 37 hrs C 8 85 hrs D 9 12 hrs E 9 24 hrs F 9 56 hrs G 9 75 hrs H 9 93 hrs I 10 12 hrs J 10 35 hrs 5 Let C amount of isotope present at time Then C 5C and C C Since 0 78C C 5 we get that 0 78 5 so ln 5 If 2 half life we have C C 52 Then ln ln 52 so 2 ln ln hrs ln 5 3 Suppose we perform the partial fraction decomposition B B B B B E B F B G B What is the value of F A F E F I F B B B B B C F G F B F F F J F E B F B G B D F H F E B B FB B GB B B B B Equating numerators gives B B E B B FB B GB B Setting B gives F so F 4 During Lab 2 LoggerPro software collected the following data for time in sec and velocity in m sec Use Simpson s Rule to estimate the change in position displacement of the moving person for Round your answer to 2 decimal places A m E 0 13 m I m B m F m J 22 m C m G m D m H m The displacement is given by The interval is divided into 4 subintervals with 0 1 Simpson s approximation is W m 5 Find the value of B B A 17 08 F B if the integral converges B 19 03 G C H D I The integral is improper type II because If we let B B B so Then B B B lim B B B B B B B E diverges to J diverges to has a vertical asymptote at B lnl l G ln l B l G B lim ln l B l l lim ln ln e 6 Suppose a radioactive isotope is emitting radiation at a rate of millirems min beginning at time What is the total amount of radiation it will emit over all time that is if the process continues forever A millirems E millirems I 40 millirems B millirems C millirems F millirems G millirems J an infinite amount Total amount emitted lim lim l lim D millirems H 10 millirems millirems 7 You use Simpson s rule with 8 to estimate the value of B B Using the error control formula we get ERROR l B B W l Find the best possible answer with this information round your answer to 6 decimal places A 0 000013 B 0 000042 C 0 000067 D 0 000093 E 0 000131 F 0 000147 G 0 000158 H 0 000169 I 0 000176 J 0 000187 l B B W l Q where Q is chosen so that l0 B l Q on We have 0 B B 0 w B B 0 B B B Since and decreasing on 0 B has its max value at the point B so 0 B This is the best choice of Q we can make Therefore l B B W l B is positive 8 According to one simple physiological model an adult male athlete needs 20 calories per day per pound of body weight to maintain his weight If he consumes more or fewer calories than those required to maintain his weight his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight If an athlete weighs 160 lbs and consumes 3000 calories day which differential equation describes the athlete s weight at time In each equation k is a constant A 5 3000 B 5 C 29 5 D 5 150 8 E 5 3000 F 5 G 5 3000 H J I 5 160 5 3000 160 5 number of calories consumed number needed to maintain current weight 5 9 Find the area in the first quadrant bounded by the curves C ln B C B and B 2 A 4 ln B ln F ln G ln C 1 H 2 ln D ln I 2 ln E J ln The figure shows the positions of the curves and the region shaded The area of the region is E B ln B B Using integration by parts we get ln B B ln B B B ln B B B B B ln B B G Then B ln B B B B B ln B B l B B ln B l ln ln 10 Find W the Simpson approximation with 4 subdivisions for B B B A F B G C H Method I longer We have B D I E J so W Method II shorter Simpson s Rule approximates the integrand with pieces of parabolas Since the integrand B B is itself a parabola Simpson s approximation W8 with any choice of even 8 will actually give the exact value of the integral W B B B B B l 11 Suppose the velocity m sec of a point moving along a line satisfies 1 What is the position of the particle when A m F 27 m B m G 22 m C m H 17 m D m I 12 m E 35 m J 8 m Separating variables in the d e and integrating both sides we get ln l l ln l l H ln l l H so that l l H l l Then H l l Gl l where G H Since we have G that is G Therefore l l so m so 12 The temperature X of a body C at time min is changing at a rate of X X The slope field direction field for the differential equation is shown below some solution curves were added here on the solutions sheet Over the time interval which of the following are true you may need to use either the slope field or the differential equation itself to answer i If the temperature is 3 when then the body is cooler when ii No matter what the temperature is when the body is cooler after time iii The temperature of the body at time depends only on not …
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