Chapter 4

For the data sets in Problems \(1-4\), construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 2 & 8 & 24 & 56 & 110 & 192 & 308 & 464 \end{array} $$

For the data sets in Problems \(1-4\), construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 23 & 48 & 73 & 98 & 123 & 148 & 173 & 198 \end{array} $$

For Problems 2 and 3, find the natural cubic splines that pass through the given data points. Use the splines to answer the requirements. \begin{tabular}{c|cccccccccc} \(x\) & \(3.0\) & \(3.1\) & \(3.2\) & \(3.3\) & \(3.4\) & \(3.5\) & \(3.6\) & \(3.7\) & \(3.8\) & \(3.9\) \\ \hline\(y\) & \(20.08\) & \(22.20\) & \(24.53\) & \(27.12\) & \(29.96\) & \(33.11\) & \(36.60\) & \(40.45\) & \(44.70\) & \(49.40\) \end{tabular} a. Estimate the derivative evaluated at \(x=3.45\). Compare your estimate with the derivative of \(e^{x}\) evaluated at \(x=3.45\). b. Estimate the area under the curve from \(3.3\) to 3.6. Compare with $$ \int_{3.3}^{3.6} e^{x} d x $$

Find the natural cubic splines that pass through the given data points. Use the splines to answer the requirements. \begin{tabular}{l|ccccccccc} \(x\) & \(3.0\) & \(3.1\) & \(3.2\) & \(3.3\) & \(3.4\) & \(3.5\) & \(3.6\) & \(3.7\) & \(3.8\) & \(3.9\) \\ \hline\(y\) & \(20.08\) & \(22.20\) & \(24.53\) & \(27.12\) & \(29.96\) & \(33.11\) & \(36.60\) & \(40.45\) & \(44.70\) & \(49.40\) \end{tabular} a. Estimate the derivative evaluated at \(x=3.45 .\) Compare your estimate with the derivative of \(e^{x}\) evaluated at \(x=3.45\) b. Estimate the area under the curve from \(3.3\) to \(3.6 .\) Compare with $$ \int_{3.3}^{3.6} e^{x} d x $$

For the data sets in Problems \(1-4\), construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 7 & 15 & 33 & 61 & 99 & 147 & 205 & 273 \end{array} $$

Find the natural cubic splines that pass through the given data points. Use the splines to answer the requirements. $$ \begin{array}{l|lllllll} x & 0 & \pi / 6 & \pi / 3 & \pi / 2 & 2 \pi / 3 & 5 \pi / 6 & \pi \\ \hline y & 0.00 & 0.50 & 0.87 & 1.00 & 0.87 & 0.50 & 0.00 \end{array} $$

For the data sets in Problems \(1-4\), construct a divided difference table. What conclusions can you make about the data? Would you use a low-order polynomial as an empirical model? If so, what order? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 1 & 4.5 & 20 & 90 & 403 & 1808 & 8103 & 36,316 \end{array} $$

In the following data, \(X\) represents the diameter of a ponderosa pine measured at breast height, and \(Y\) is a measure of volume-number of board feet divided by \(10 .\) Make a scatterplot of the data. Discuss the appropriateness of using a 13 th-degree polynomial that passes through the data points as an empirical model. If you have a computer available, fit a polynomial to the data and graph the results. \begin{tabular}{c|cccccccccccccc} \(X\) & 17 & 19 & 20 & 22 & 23 & 25 & 31 & 32 & 33 & 36 & 37 & 38 & 39 & 41 \\ \hline\(Y\) & 19 & 25 & 32 & 51 & 57 & 71 & 141 & 123 & 187 & 192 & 205 & 252 & 248 & 294 \end{tabular}

The Cost of a Postage Stamp - Consider the following data. Use the procedures in this chapter to capture the trend of the data if one exists. Would you eliminate any data points? Why? Would you be willing to use your model to predict the price of a postage stamp on January \(1,2010 ?\) What do the various models you construct predict about the price on January \(1,2010 ?\) When will the price reach \$1? You might enjoy reading the article on which this problem is based: Donald \(\mathrm{R}\). Byrkit and Robert E. Lee, "The Cost of a Postage Stamp, or Up, Up, and Away," Mathematics and Computer Education 17, no. 3 (Summer 1983): \(184-190\). \begin{tabular}{lcl} \hline \multicolumn{1}{c} { Date } & \multicolumn{1}{c} { First-class stamp } \\\ \hline \(1885-1917\) & \(\$ 0.02\) \\ \(1917-1919\) & \(0.03\) & (Wartime increase) \\ 1919 & \(0.02\) & (Restored by Congress) \\ July 6, 1932 & \(0.03\) & \\ August 1, 1958 & \(0.04\) & \\ January 7, 1963 & \(0.05\) & \\ January 7,1968 & \(0.06\) & \\ May 16, 1971 & \(0.08\) & \\ March 2, 1974 & \(0.10\) & \\ December 31, 1975 & \(0.13\) & (Temporary) \\ July 18, 1976 & \(0.13\) & \\ May 15, 1978 & \(0.15\) & \\ March 22, 1981 & \(0.18\) & \\ November 1, 1981 & \(0.20\) & \\ February 17, 1985 & \(0.22\) & \\ April 3, 1988 & \(0.25\) & \\ February 3, 1991 & \(0.29\) & \\ January 1, 1995 & \(0.32\) & \\ January 10, 1999 & \(0.33\) & \\ January 7, 2001 & \(0.34\) & \\ June 30, 2002 & \(0.37\) & \\ January 8, 2006 & \(0.39\) & \\ May 14, 2007 & \(0.41\) & \\ May 12, 2008 & \(0.42\) & \\ May 11, 2009 & \(0.44\) & \\ January 22, 2012 & \(0.45\) & \\ \hline \end{tabular}

Construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers? Construct a divided difference table. Is smoothing with a low-order polynomial appropriate? If so, choose an appropriate polynomial and fit using the least-squares criterion of best fit. Analyze the goodness of fit by examining appropriate indicators and graphing the model, the data points, and the deviations. In the following data, \(X\) is the Fahrenheit temperature and \(Y\) is the number of times a cricket chirps in \(1 \mathrm{~min}\) (see Problem 3, Section \(4.2\) ). \begin{tabular}{l|cccccccccc} \(X\) & 46 & 49 & 51 & 52 & 54 & 56 & 57 & 58 & 59 & 60 \\ \hline\(Y\) & 40 & 50 & 55 & 63 & 72 & 70 & 77 & 73 & 90 & 93 \\ \(X\) & 61 & 62 & 63 & 64 & 66 & 67 & 68 & 71 & 72 \\ \hline\(Y\) & 96 & 88 & 99 & 110 & 113 & 120 & 127 & 137 & 132 \end{tabular}

Construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers? Construct a divided difference table. Is smoothing with a low-order polynomial appropriate? If so, choose an appropriate polynomial and fit using the least-squares criterion of best fit. Analyze the goodness of fit by examining appropriate indicators and graphing the model, the data points, and the deviations. In the following data, \(X\) represents the diameter of a ponderosa pine measured at breast height, and \(Y\) is a measure of volume-number of board feet divided by 10 (see Problem 4, Section 4.2). \begin{tabular}{l|llllllllllllll} \(X\) & 17 & 19 & 20 & 22 & 23 & 25 & 31 & 32 & 33 & 36 & 37 & 38 & 39 & 41 \\ \hline\(Y\) & 19 & 25 & 32 & 51 & 57 & 71 & 141 & 123 & 187 & 192 & 205 & 252 & 248 & 294 \\ \hline \end{tabular}

Construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers? Construct a divided difference table. Is smoothing with a low-order polynomial appropriate? If so, choose an appropriate polynomial and fit using the least-squares criterion of best fit. Analyze the goodness of fit by examining appropriate indicators and graphing the model, the data points, and the deviations. The following data represent the population of the United States from 1790 to 2000 . $$ \begin{array}{lc} \hline \text { Year } & \text { Observed population } \\ \hline 1790 & 3,929,000 \\ 1800 & 5,308,000 \\ 1810 & 7,240,000 \\ 1820 & 9,638,000 \\ 1830 & 12,866,000 \\ 1840 & 17,069,000 \\ 1850 & 23,192,000 \\ 1860 & 31,443,000 \\ 1870 & 38,558,000 \\ 1880 & 50,156,000 \\ 1890 & 62,948,000 \\ 1900 & 75,995,000 \\ 1910 & 91,972,000 \\ 1920 & 105,711,000 \\ 1930 & 122,755,000 \\ 1940 & 131,669,000 \\ 1950 & 150,697,000 \\ 1960 & 179,323,000 \\ 1970 & 203,212,000 \\ 1980 & 226,505,000 \\ 1990 & 248,709,873 \\ 2000 & 281,416,000 \\ \hline \end{array} $$

The following data measure two characteristics of a ponderosa pine. The variable \(X\) is the diameter of the tree, in inches, measured at breast height; \(Y\) is a measure of volume-the number of board feet divided by \(10 .\) Fit a model to the data. Then express \(Y\) in terms of \(X\)

The following data represent the length and weight of a set of fish (bass). Model weight as a function of the length of the fish. \begin{tabular}{l|llllll} Length (in.) & \(12.5\) & \(12.625\) & \(14.125\) & \(14.5\) & \(17.25\) & \(17.75\) \\ \hline Weight (oz) & 17 & \(16.5\) & 23 & \(26.5\) & 41 & 49 \end{tabular}

Construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers? Construct a divided difference table. Is smoothing with a low-order polynomial appropriate? If so, choose an appropriate polynomial and fit using the least-squares criterion of best fit. Analyze the goodness of fit by examining appropriate indicators and graphing the model, the data points, and the deviations. The following data represent the length of a bass fish and its weight. \begin{tabular}{l|llllll} Length (in.) & \(12.5\) & \(12.625\) & \(14.125\) & \(14.5\) & \(17.25\) & \(17.75\) \\ \hline Weight (oz) & 17 & \(16.5\) & 23 & \(26.5\) & 41 & 49 \end{tabular}

The following data give the population of the United States from 1800 to \(2000 .\) Model the population (in thousands) as a function of the year. How well does your model fit? Is a one-term model appropriate for these data? Why? \begin{tabular}{l|cccccc} Year & 1800 & 1820 & 1840 & 1860 & 1880 & 1900 & 1920 \\ \hline Population (thousands) & 5308 & 9638 & 17,069 & 31,443 & 50,156 & 75,995 & 105,711 \\ Year & 1940 & 1960 & 1980 & 1990 & 2000 & \\ \hline Population (thousands) & 131,669 & 179,323 & 226,505 & 248,710 & 281,416 & \end{tabular}

Construct a scatterplot of the given data. Is there a trend in the data? Are any of the data points outliers? Construct a divided difference table. Is smoothing with a low-order polynomial appropriate? If so, choose an appropriate polynomial and fit using the least-squares criterion of best fit. Analyze the goodness of fit by examining appropriate indicators and graphing the model, the data points, and the deviations. The following data represent the weight-lifting results from the 1976 Olympics. $$ \begin{array}{lc|ccc} \hline \multicolumn{2}{c|}{\text { Bodyweight class (lb) }} & \multicolumn{3}{|c}{\text { Total winning lifts (lb) }} \\ \hline \multicolumn{2}{c|}{\text { Max. weight }} & \text { Snatch } & \text { Jerk } & \text { Total weight } \\ \hline \text { Flyweight } & 114.5 & 231.5 & 303.1 & 534.6 \\ \text { Bantamweight } & 123.5 & 259.0 & 319.7 & 578.7 \\ \text { Featherweight } & 132.5 & 275.6 & 352.7 & 628.3 \\ \text { Lightweight } & 149.0 & 297.6 & 380.3 & 677.9 \\ \text { Middleweight } & 165.5 & 319.7 & 418.9 & 738.5 \\ \text { Light-heavyweight } & 182.0 & 358.3 & 446.4 & 804.7 \\ \text { Middle-heavyweight } & 198.5 & 374.8 & 468.5 & 843.3 \\ \text { Heavyweight } & 242.5 & 385.8 & 496.0 & 881.8 \\ \hline \end{array} $$