6

  Accumulating Change: Limits of Sums and the Definite Integral


Project 6.1: Acceleration, Velocity, and Distance

 

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

Time

(seconds)

Speed reached from rest

(mph)

0

0

2.2

30

2.9

40

4.0

50

5.0

60

6.5

70

8.0

80

9.0

90

11.8

100

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

a. acceleration from rest to a speed of 50 mph

b. acceleration from a speed of 50 mph to a speed of 100 mph

c. the first 13.5 seconds of acceleration

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.


Reporting

Prepare a written report of your work. Include scatter plots, models, graphs, and discussions of each of the above tasks.

 

Project 6.2: Estimating Growth

 

Setting

Table 6.54 lists the rate of growth of a typical male from birth to 18 years.

TABLE 6.54

Rate of growth

Age (years)

(centimeters per year)

0

18.0

1

16.0

2

12.0

3

6.5

4

6.0

5

6.25

7

6.0

8

5.75

9

5.25

10

5.0

11

4.5

12

5.0

13

6.75

14

9.0

15

7.0

16

2.0

17

0.75

18

0.50

26. Based on data collected in the Berkeley Growth Study.

Tasks

  1. Use the data and right rectangles to approximate the height of a typical 18-year-old male.
  2. Sketch a smooth, continuous curve over a scatter plot of the data. Remember that such a curve should exhibit only the curvature implied by the data. Find a piecewise model for the data. Use no more than three pieces.
  3. Use your piecewise model and limits of sums to approximate the height of a typical 18-year-old male. Convert centimeters to feet and inches, and compare your answer to the estimate you obtained using right rectangles. Which is likely to be the more accurate approximation? Why?
  4. Use your piecewise model and what you know about definite integrals to find the height of a typical 18-year-old male in feet and inches. Compare your answer with the approximation you obtained in Task 3.
  5. Randomly choose ten 18-year-old male students, and determine their heights. (Include your data&emdash;names are not necessary, only the heights.) Discuss your selection process and why you feel that it is random. Find the average height of the 18-year-old males in your sample. Compare this average height with your answer to Task 4. Discuss your results.
  6. Refer to your sketch of the rate-of-growth graph in Task 2, and draw a possible graph of the height of a typical 18-year-old male from birth to age 18.


Reporting

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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