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Chapter 4: Problem 2

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

Short Answer

Expert verified
The data fits a first-degree polynomial (linear model), use a linear model.

Step by step solution

01

Understand Divided Difference Table

A divided difference table helps us determine the polynomial that fits a given set of data points. The first column contains the x-values, and the second column contains the corresponding y-values. Each subsequent column represents divided differences, calculated incrementally.
02

Set Up Initial Table

Write the x-values and the corresponding y-values: \[\begin{array}{c|c}x & y \\hline0 & 23 \1 & 48 \2 & 73 \3 & 98 \4 & 123 \5 & 148 \6 & 173 \7 & 198 \\end{array}\]
03

Calculate First Divided Differences

Compute the first divided differences by subtracting successive y-values and dividing by the difference in x-values: \[\begin{array}{c|c|c}x & y & \text{First Divided Difference} \\hline0 & 23 & 25 \1 & 48 & 25 \2 & 73 & 25 \3 & 98 & 25 \4 & 123 & 25 \5 & 148 & 25 \6 & 173 & 25 \7 & 198 & \\end{array}\]
04

Check Higher Divided Differences

Since the first divided differences are constant, calculate the second differences to ensure they are zero:\[\begin{array}{c|c|c|c}x & y & \text{First} & \text{Second} \\hline0 & 23 & 25 & 0 \1 & 48 & 25 & 0 \2 & 73 & 25 & 0 \3 & 98 & 25 & 0 \4 & 123 & 25 & 0 \5 & 148 & 25 & 0 \6 & 173 & 25 & 0 \7 & 198 & \\end{array}\]
05

Conclusion on Polynomial Order

Since the first differences are constant, the data fits a first-degree polynomial perfectly, meaning a linear relationship of the form \(y = 25x + 23\) exists between x and y.

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Key Concepts

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

Divided Difference Table
A divided difference table is an organized way to determine a polynomial that can model a given dataset. It helps in understanding how data points relate to each other through increments. The table is systematically constructed:
  • The first column lists the x-values of the dataset.
  • The second column contains the corresponding y-values or outputs.
  • Each additional column shows the divided differences.
When you start calculating these differences, you look at successive y-values, subtract them, and then divide by the differences in the x-values. This process continues with higher-order differences, depending on the degree of the polynomial needed to model the data.
A constant set of differences at any column indicates the degree of the polynomial; this is crucial for polynomial modeling.
Empirical Model
An empirical model is a mathematical expression that represents relationships between variables. It is based on observed data rather than theory or perfect assumptions. In the context of the divided difference table, it can help formulate a polynomial that accurately fits the data provided.When choosing a model, we seek simplicity and precision:
  • A good empirical model can predict outcomes with minimal error.
  • It should be complex enough to capture trends but simple enough to be easily interpreted and computed.
  • Utilizing tools like a divided difference table helps identify the right order for our polynomial model.
In the exercise example, the empirical model is identified as a linear relationship (\(y = 25x + 23\)), fitting perfectly due to constant first divided differences.
Polynomial Degree
The degree of a polynomial is the highest power of the variable x in the polynomial expression. It's crucial for understanding the shape and behavior of the polynomial graph. Degrees and what they represent:
  • A first-degree polynomial (linear) is a straight line with a constant rate of change.
  • A second-degree polynomial (quadratic) presents a parabolic shape, with a variable rate of change.
  • Higher-degree polynomials (cubic, quartic, etc.) involve more level of complexity and curvature.
In data modeling, recognizing the degree helps in concluding the nature of relationships between variables. In our example, the constant first difference showcased that the relationship was linear, confirming a first-degree polynomial as adequate.
Data Analysis
Data analysis involves examining, cleaning, transforming, and modeling data to discover useful information that informs decision-making. Utilizing polynomial modeling through a divided difference table is a form of analyzing numerical data to uncover mathematical relationships. The tasks involved may include:
  • Identifying patterns and trends within data.
  • Determining the polynomial degree to best fit the data, ensuring accurate predictions.
  • Verifying the consistency of differences to confirm the polynomial model applied.
Efficient data analysis leads us to clearer insights, allowing predictions with accuracy. In our context, analyzing constant first differences assured us of a consistent linear model, enhancing the reliability of our conclusions.

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Most popular questions from this chapter

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 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}

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 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 $$ \int_{3.3}^{3.6} e^{x} d x $$

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 \(X\)
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