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Chapter 4: Problem 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? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 1 & 4.5 & 20 & 90 & 403 & 1808 & 8103 & 36,316 \end{array} $$

Short Answer

Expert verified
Use a 6th-degree polynomial as an empirical model, based on constant 6th divided differences.

Step by step solution

01

Set up the divided difference table

Start by arranging the given data points in the first two columns. The first column consists of the \(x\) values \([0, 1, 2, 3, 4, 5, 6, 7]\), and the second column consists of the corresponding \(y\) values \([1, 4.5, 20, 90, 403, 1808, 8103, 36316]\).
02

Compute first divided differences

First divided differences are calculated using the formula: \(f[x_i, x_{i+1}] = \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i}\). Fill the third column with these calculations. For example, \(f[x_0, x_1] = \frac{4.5 - 1}{1-0} = 3.5\). Repeat for each interval.
03

Compute second divided differences

The second divided differences use the formula: \(f[x_i, x_{i+1}, x_{i+2}] = \frac{f[x_{i+1}, x_{i+2}] - f[x_i, x_{i+1}]}{x_{i+2} - x_i}\). Fill in the calculations: for instance, \(f[x_0, x_1, x_2] = \frac{15.5 - 3.5}{2-0} = 6\). Continue this process for each available set.
04

Continue with higher order differences

Proceed by calculating third, fourth, and higher divided differences using the same approach until you are unable to calculate more. Use the general formula \(f[x_i, ..., x_{i+n}] = \frac{f[x_{i+1}, ..., x_{i+n}] - f[x_i, ..., x_{i+n-1}]}{x_{i+n} - x_i}\).
05

Analyze table for patterns

Examine the divided difference table for a constant difference at one of the levels. The degree of the polynomial that fits the data well is determined by the constant last non-zero difference in the differences table. Look for the first row where the difference becomes constant.
06

Draw conclusions on polynomial order

If you observe a level in the divided difference table where differences become constant, the order of that difference indicates the degree of polynomial suitable as a model. Analyze if it aligns with real or near-constant differences. Suggest a polynomial degree based on the smallest such level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Degree
The concept of polynomial degree is a critical aspect when working with divided difference tables in mathematical data analysis. Essentially, the degree of a polynomial tells us the highest power of the variable in the polynomial equation.

When constructing a divided difference table to analyze data, one of the objectives is to determine the appropriate polynomial degree that best fits the data points. As you compute successive divided differences, the aim is to reach a row where the differences become constant.

  • If this happens at the first level, a linear (degree 1) polynomial is suitable.
  • If at the second level, a quadratic (degree 2) polynomial would be ideal.
  • If further levels are needed, you may need a cubic (degree 3) or higher degree polynomial.
Identifying the correct polynomial degree helps ensure the empirical model accurately represents the trend in the data without unnecessary complexity.
Data Analysis
Data analysis using divided difference tables is a powerful technique to uncover patterns and trends within a set of data points. It involves observing how data values change incrementally as you move through the table columns.

Setting up the divided difference table involves listing the data points and applying specific formulas to calculate successive differences. Each level of divided differences provides deeper insight into the data's behavior.

  • First-order differences can show linear trends.
  • Second-order differences may reveal quadratic patterns.
  • Higher-order differences can expose more complex relationships.
By analyzing these differences, you can spot at which level differences stabilize, thus guiding you to the appropriate polynomial order that best models the data.
Empirical Model
An empirical model is a mathematical representation based on observed data, rather than derived theoretically. In data analysis, using a polynomial as an empirical model involves choosing the polynomial degree based on observed trends.

After constructing your divided difference table, observe where the differences become constant. This indicates the degree of the polynomial that can effectively capture the data's relationship.

  • Empirical models are chosen for their simplicity and the accuracy of fit they offer.
  • Lower-order polynomials are preferred as they tend to be simpler and less overfitted to specific data points.
  • The right order balances accuracy with simplicity, ensuring the model generalizes well to future data.
Thus, understanding both the data patterns through a divided difference table and selecting the appropriate polynomial degree are crucial steps in building a reliable empirical model to make predictions or understand data behavior.

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Most popular questions from this chapter

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 \(X\)

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. $$ \begin{array}{lc|ccc} \hline \multicolumn{2}{c|}{\text { Bodyweight class (lb) }} & \multicolumn{3}{|c}{\text { Total winning lifts (lb) }} \\ \hline \multicolumn{2}{c|}{\text { Max. weight }} & \text { Snatch } & \text { Jerk } & \text { Total weight } \\ \hline \text { Flyweight } & 114.5 & 231.5 & 303.1 & 534.6 \\ \text { Bantamweight } & 123.5 & 259.0 & 319.7 & 578.7 \\ \text { Featherweight } & 132.5 & 275.6 & 352.7 & 628.3 \\ \text { Lightweight } & 149.0 & 297.6 & 380.3 & 677.9 \\ \text { Middleweight } & 165.5 & 319.7 & 418.9 & 738.5 \\ \text { Light-heavyweight } & 182.0 & 358.3 & 446.4 & 804.7 \\ \text { Middle-heavyweight } & 198.5 & 374.8 & 468.5 & 843.3 \\ \text { Heavyweight } & 242.5 & 385.8 & 496.0 & 881.8 \\ \hline \end{array} $$

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? $$ \begin{array}{l|llllllll} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 23 & 48 & 73 & 98 & 123 & 148 & 173 & 198 \end{array} $$

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}
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