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Precalculus Project # 3

(Due: )

Goals:

  1. Students will understand the concepts of linear, quadratic, exponential and trigonometric regressions.
  1. Students will understand the representation of real data as a function.
  1. Students will understand how to derive the above functions manually from the real data.
  1. Students will understand how to use spreadsheet to organize and graph real data.
  1. Students will understand how to use spreadsheet to graph and compute all the above regressions.
  1. Students use real data and mathematical functions, derived from step 3, to make real- life predictions.
  1. Students will understand how to express mathematical concepts in writing through publishing a written report documenting their work.

Problem statement:

In this project, you will find and use models relating to the carbon dioxide level of earth’s atmosphere.

Since 1958, the Mauna Loa Climate Observatory in Hawaii has been collecting data on the carbon dioxide level of earth’s atmosphere. The table shows the average readings for January of each year from 1959 through 1994. The readings measure the carbon dioxide concentration in parts per million.

1959 / 1960 / 1961 / 1962 / 1963 / 1964 / 1965 / 1966 / 1967 / 1968 / 1969 / 1970
315.4 / 316.3 / 316.7 / 317.8 / 318.6 / 319.4 / 319.3 / 320.5 / 322.2 / 322.4 / 323.8 / 324.9
1971 / 1972 / 1973 / 1974 / 1975 / 1976 / 1977 / 1978 / 1979 / 1980 / 1981 / 1982
326.0 / 326.6 / 328.4 / 329.2 / 330.2 / 331.6 / 332.8 / 334.8 / 336.1 / 337.8 / 339.1 / 340.6
1983 / 1984 / 1985 / 1986 / 1987 / 1988 / 1989 / 1990 / 1991 / 1992 / 1993 / 1994
341.2 / 343.5 / 344.8 / 346.1 / 347.8 / 350.3 / 352.6 / 353.5 / 354.6 / 355.9 / 356.6 / 358.3

PART I

a)Manually graph the 36 data points. Let x represents the number of years and the y represents the carbon dioxide in parts per million.

b)Find manually the linear, quadratic and exponential functions F(x) (from the scatter plot).

c)Use the TI to perform linear, quadratic and exponential regressions.

d)Select a year from the data points to calculate the percentage errors from both the manual calculations and the TI regressions. Record the correlation coefficient R from all the regressions and the percentage errors from the manual calculations.

e)Compare the results of questions b through d. Make your conjecture about the best fit based on the error analysis and percentage errors.

f)Use Microsoft word and Excel to produce hard copy of your project.

g)Make predictions of how many parts per million will the carbon dioxide be in years 2000, 2010 and 2020?

Additional problem statement:

The data in the table represents the carbon dioxide levels for January of each year. Throughout each year, the level oscillates as follows:

1)In April the average reading was about 2.5 parts per million higher than the average reading given by the models in parts (b & c) above.

2)In July, the average reading was the same as the average reading given by the models in parts (b & c) above.

3)In October, the average reading was about 2.5 parts per million lower than the average reading given by the model in parts (b & c) above.

PART II

a)Use a sine function to rewrite the best fit models found in parts (b & c) above so that the models incorporate the described oscillations.

b)Manually graph the revised models.

c)Use a graphing calculator to sketch the graph of the revised model.

d)Make careful sketches of the models in (b & c) for 1 year.

e)Are the revised models you found periodic? Explain your reasoning.

f)Compare the results of questions c through e. Make your conjecture about the best fit based on the error analysis and percentage errors.

g)Use the best revised model to predict the level of carbon dioxide in earth’s atmosphere in the years 2000, 2010 and 2020.

PART III

Use at least one source to research the following question:

a)What physical factors on the earth would contribute to the oscillation in the carbon dioxide level during the year?

General guidelines

1)Finally, your project must have at least cover page, introduction, manual calculations, spreadsheet calculations (the same as the TI) and conclusion.

2)Be sure to include your data and results for each item in parts I, II & III.

Good luck!