A car traveling along a straightnroad is clocked at a number of points. The data

Question

A car traveling along a straightnroad is clocked at a number of points. The data from the observations are given: Time (sec): 30 33 35 38 43 Distance (feet) 60 285 443 683 1053 Speed (ft/sec): 75 77 80 74 72 A. Use the Hermite polynomial to predicitnthe position of the car and its speed when t=40s. Use the derivative of the Hermite polynomial to determine whether the car exeeds a 55mi/h speedmlimit on the road. If so , what is the first time the vicar exceeds this speed? What is the predicted maximum speed for the car? (use at least 3 data points) B. Use Lagrange interpolstion to oredicitnthe position of the car and its speed when t=40s (ignore speed in your input). Use the derivative the Lagrange polynomial to determine the differences in the predicted and the actual speed, and to determine whether the can ever exceeds 55mi/h limitmon the road. If so, what is the time the car exceeds this speed? What is the predicted maximum speed for the car? (use at least 3 data points) I can use Lagrange and Hermite to predict the postion, however I can't figure out how to the last part of A and B using the derivative to find speed or to find the predicted max. A is similar to question question 10 from section 3.4 in the textbook

Explanation / Answer

The Hermite polynomial generated by the data in the table is p(x) = 75x + 0.222222x2(x ? 3) ? 0.0311111x2(x ? 3)2 ? 0.00644444x2(x ? 3)2(x ? 5)+ 0.00226389x2(x ? 3)2(x ? 5)2 ? 0.000913194x2(x ? 3)2(x ? 5)2(x ? 8)+ 0.000130527x2(x ? 3)2(x ? 5)2(x ? 8)2? 0.0000202236x2(x ? 3)2(x ? 5)2(x ? 8)2(x ? 13). (a) From the Hermite polynomial above, the predicted value of the position of the car is p(10) = 742.502839 ? 743 feet,and the value of the speed of the car is p (10) = 48.381736 ? 48 feet/sec. From the table the approximation to the position seems reasonable, but the approximation to the speed is highly suspect. (b) Using the derivative of the Hermite interpolating polynomial, we found the first time the car exceeds 55 mph = 80.6 ft/sec by finding the smallest value of t for which p (t) ? 80.6. This value of t ist0 ? 5.648802541922,for which p (t0) ? 80.666666666666. (c) The maximum speed was estimated to bep (12.371870) = 119.417338 ft/sec ? 81.2084 mph.The algorithm from the text to generate the Hermite polynomial and its derivative is implemented inC and is given below

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