tapia jat4858 HW03 clark 52990 This print out should have 18 questions Multiple choice questions may continue on the next column or page find all choices before answering 001 10 0 points If the points 1 3 0 7 4 1 2 3 2 5 2 2 lie on the graph of a continuous function y f x use the trapezoidal rule and all these points to estimate the integral Z 2 I f x dx 0 15 correct 2 3 I 8 4 I 29 4 5 I 31 4 0 1 I 26 3 2 I 25 correct 3 3 I 49 6 4 I 8 5 I 17 2 I 25 3 keywords Simpson s Rule integral graph Explanation The trapezoidal rule estimates the integral I as h 1 3 f 0 2f 2f 1 2f f 2 2 2 2 1 With h and the given values of f there2 fore the area is estimated by 15 I 2 002 lie on the graph of a continuous function y f x use Simpson s Rule and all these points to estimate the integral Z 2 I f x dx Explanation Simpson s Rule estimates the integral I as 1 3 h f 0 4f 2f 1 4f f 2 3 2 2 1 With h and the given values of f there2 fore the area is estimated by 1 I 7 2 I 1 10 0 points If the points 1 3 0 1 5 1 6 3 2 5 2 2 003 10 0 points Below is the graph of a function f 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 6 4 2 3 2 1 1 2 3 tapia jat4858 HW03 clark 52990 Estimate the integral I Z 2 keywords midpoint rule subinterval integral graph 3 f x dx 004 3 using the Midpoint Rule with six equal subintervals 1 I 9 10 0 points A farmer wishes to estimate the area of a river front pasture He knows some calculus so he takes measurements every 100 feet and draws the following graph 2 I 7 3 I 6 4 I 8 correct 40 60 50 80 60 5 I 5 Explanation Since 3 3 is subdivided into six equal subintervals each of these will have length 1 and the six corresponding rectangles are shown as the gray shaded areas in 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 What value does he obtain if he uses the trapezoidal rule and all the data to estimate the area 1 Area 24050 sq feet 2 Area 24000 sq feet correct 3 2 1 6 3 Area 24150 sq feet 4 4 Area 24100 sq feet 2 5 Area 24200 sq feet 1 2 3 Explanation By the trapezoidal rule Area 1 100 y0 2y1 2y2 2y3 y4 2 50 40 120 100 160 60 Consequently The heights of the rectangles are midpoint sample values of f that can be read off from the graph Thus with midpoints I 4 2 3 1 3 5 8 Area 24000 sq feet keywords word problem estimate area trapezoidal rule 005 10 0 points tapia jat4858 HW03 clark 52990 A radar gun was used to record the speed of a runner during the first 5 seconds of a race as shown in the table t secs 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 3 Reading off the values of v t from the table we thus see that I 43 75 meters vel m sec 0 4 7 3 8 9 7 10 1 10 6 10 7 10 8 10 88 10 99 006 10 0 points Which of the following integrals are improper I1 Z 1 I2 Use Simpson s Rule and all the given data to estimate the distance the runner covered during those 5 seconds I3 2 x 2 dx x 1 0 Z 1 dx 1 x2 1 1 dx x 1 I2 only 2 dist 43 77 meters 2 I1 and I2 only 3 dist 43 81 meters 3 none of them 4 dist 43 73 meters 4 I1 and I3 only 5 dist 43 79 meters Explanation The distance covered during those 5 seconds is given by the integral I Z 0 1 dist 43 75 meters correct Z 100 5 all of them 6 I1 only 5 7 I3 only v t dt 0 where v t is the velocity of the runner at time t Simpson s Rule estimates this integral as I 1n 3 2v 1 4v 2 2 7 5 2v 3 4v 2v 2 4v 2 2 o 9 v 5 2v 4 4v 2 6 v 0 4v 1 8 I2 and I3 only correct Explanation An integral I Z b f x dx a is improper when one or more of the following conditions are satisfied tapia jat4858 HW03 clark 52990 i the interval of integration is infinite i e when a or b or both ii f has a vertical asymptote at one or more of x a x b or x c for some a c b Consequently Z 100 exists To evaluate It set u x 4 Then du dx while 1 dx 1 x2 1 is not improper x 0 u 4 x t u t 4 In this case Z 2 x 2 dx x 1 0 is improper Z 1 0 1 dx x It t 4 Z so that lim It lim t 007 10 0 points Determine if the improper integral Z 6 dx I x 4 2 0 is convergent and if it is find its value 1 I is not convergent 2 I 2 5 3 I 2 9 4 I 4 7 4 3 correct 6 I 2 Explanation The integral is improper because the interval of integration is infinite To test for convergence we thus have to determine if Z t 6 dx lim It It 2 t 0 x 4 h 6 it 4 6 du u2 u 4 4 is improper 5 I 4 t 3 3 6 2 t 4 2 Consequently the integral is convergent and I 3 2 008 10 0 points Determine if the improper integral I Z 2xe x 3 dx 2 converges and if it does compute its value 2 1 I 30e 3 2 2 I 6e 3 3 I does not converge 2 4 I 6e 3 2 5 I 30e 3 correct Explanation The integral Z 2 2xe x 3 dx tapia jat4858 HW03 clark 52990 is improper because of the infinite range of integration To overcome this we restrict to a finite interval of integration and consider the limit Z t lim It It 2xe x 3 dx t 2 To evaluate It we integrate by parts Then Z t it h x 3 6 e x 3 dx It 6xe 2 2 h 6 x 3 e x 3 it 2 2 30e 3 6 t 3 e …
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