6.055J/2.038J (Spring 2010)Homework 6Submit your answers and explanations online by 10pm on Wednesday, 14 Apr 2010.Open universe: Collaboration, notes, and other sources of information are encouraged. However, avoidlooking up answers to the problem, or to subproblems, until you solve the problem or have tried hard. Thispolicy helps you learn the most from the problems.Homework is graded with a light touch: P (made a decent effort), D (made an indecent effort), or F (did notmake an effort).Problem 1 Guessing an integral using easy casesUse easy cases to choose the correct value of the integralZ∞−∞e−ax2dx. (1)√πa√π/aProblem 2 Differential-equation solutionWhich sketch shows a solution of the differential equationdydt= Ay(M − y),where A and M are positive constants?ytDCBA00Curve ACurve BCurve CCurve DHomework 6 / 6.055J/2.038J: Art of approximation in science and engineering (Spring 2010) 2Problem 3 FogFog is a low-lying cloud, perhaps 1 km tall and made up of tiny water droplets (radius r ∼ 10 µm).By estimating the terminal speed of fog droplets, estimate the time that the cloud takes to settle tothe ground.10±s or 10. . .sTo include in the explanation box: What is the everyday consequence of this settling time?Problem 4 Hyperbolic-function sketchWhich graph is ln cosh x (where cosh x ≡ (ex+ e−x)/2)?yx0 303CDBACurve ACurve BCurve CCurve DProblem 5 Guessing an integralChoose the correct value of the integralZ∞−∞1a2+ x2dx, (2)where a is a positive constant.πaπ/a√πa√π/aHomework 6 / 6.055J/2.038J: Art of approximation in science and engineering (Spring 2010) 3Problem 6 DebuggingUse special (i.e. easy) cases of n to decide which of these two C functions correctly computes thesum of the first n odd numbers:Program A:int sum_of_odds (int n) {int i, total = 0;for (i=1; i<=2*n+1; i+=2)total += i;return total;}Program B:int sum_of_odds (int n) {int i, total = 0;for (i=1; i<=2*n-1; i+=2)total += i;return total;}Problem 7 Damped, driven springA damped, driven spring–mass system (e.g., in 18.03, 2.003, 2.004, 6.003, and maybe also 8.01) isdescribed by the differential equationmd2xdt2+ bdxdt+ kx = F0eiωt, (3)where m is the mass of the object, b is the damping constant, k is the spring constant, x is thedisplacement of the mass, ω is the (angular) frequency of the driving force, and F0is the amplitudeof the driving force. The solution has the formx = x0eiωt, (4)where x0is the (possibly complex) amplitude.Which graph, on log–log axes, correctly shows the transfer function F0/x0? Don’t solve the differentialequation – use an approximation method to guess the answer!Curve AωCurve BωCurve
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