6 |
Accumulating Change: Limits of Sums and the Definite Integral |
Project 6.1: Acceleration, Velocity, and Distance |
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Setting According to tests conducted by Road and Track, a 1993 Toyota Supra Turbo accelerates from 0 to 30 mph in 2.2 seconds and travels 1.4 miles (1320 feet) in 13.5 seconds, reaching a speed of 107 mph. Road and Track reported the data given in Table 6.53. TABLE 6.53
Tasks 1. Convert the speed data to feet per second, and find a quadratic model for velocity (in feet per second) as a function of time (in seconds). Discuss how close your model comes to predicting the 107 mph reached after 13.5 seconds. 2. Add the data point for 13.5 seconds, and find a quadratic model for velocity v(t). 3. Use four rectangles and your model from Task 2 to estimate the distance traveled during acceleration from rest to a speed of 50 mph and the distance traveled during acceleration from a speed of 50 mph to a speed of 100 mph. Repeat the estimate using twice as many rectangles. 4. Use nine rectangles to approximate the distance traveled during the first 13.5 seconds. How close is your estimate to the reported value? 5. Find the distances traveled during
Compare these answers to your estimates in Tasks 3 and 4. Explain how estimating with areas of rectangles is related to calculating the definite integral.
Prepare a written report of your work. Include scatter plots, models, graphs, and discussions of each of the above tasks. |
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Project 6.2: Estimating Growth |
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Setting Table 6.54 lists the rate of growth of a typical male from birth to 18 years. TABLE 6.54
Tasks
Prepare a report that presents your findings in Tasks 1 through 6. Explain the different methods that you used, and discuss why these methods should all give similar results. Attach your mathematical work as an appendix to your report. |
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