MAT126, Paper Homework \"Nab\" 1. After being kidnapped on a trip to Canada, little Jimmy was able to see the speedometer of the kidnapper's car, and carefully noted down the speeds (in km/hr) every tenth of an hour (six minutes). Jimmy was able to send these speeds to his friend Juan via a message tied to the leg of a pigeon. Juan wants to figure out how far away Jimmy is from where they nabbed him. 0 30 time t speed v(t) .1 .2 .3 .4 .5 .6 .7 60 81 93 94 80 70 40 .8 .9 1 52 20 0 Using the midpoint rule, estimate the total distance (in km) that Jimmy was taken, that is, estimate R1 the integral 0 v(t) dt. (Use only the data provided- don't average to guess at times not listed.) km km 00 Using the assumption 10 hr 2 v (t) 10 hr 2 , determine the maximum error in your estimate. Z 2. Approximate 1 ln(x) cos(x) dx using Simpson's rule with n = 4 intervals (that is, use 5 data points), 0 and give an estimate on the error of your approximation. You can get around the problem of ln(0) cos(0) not being defined as follows. First integrate by parts, and then it is reasonable to take sin(0)/0 = 1 since limx0 sin(x) = 1. x Similarly, you can use L'Hospital's rule to determine that limx0+ ln(x) sin(x) = 0. So that you don't kill yourself taking derivatives, I'll tell you that the 4th derivative of sin(x)/x is ((x4 12x2 + 24) sin(x) + (4x3 24) cos(x))/x5 , which is between 1/5 and 1/5 for any value of x. MAT126, Paper Homework \"Bldg\" 1. Find the area of the bounded region lying between the curves y = x2 and y = 4x x2 . 2. An architect has designed a building with a circular base in such a way that the height of the roof is the same at each point with the same x-coordinate. That is, cross-sections with a fixed x-coordinate are squares. If the diameter of the base is 100 feet, what is the volume of the building