Chapter Two – Practice Exercises
2a)
While the owner in exercise 1a) was happy with the results of using elimination/substitution, she was curious to see if the results would differ using Newton’s Divided Difference (NDD) interpolation. You have decided to assist her by generating a cubic polynomial using NDD. (Solution given) The data is:
ABC Children's Party Company
| Maximum children attending the party | Cost per Child | Total Cost of Party |
|---|---|---|
| 10 | $37 | $370 |
| 25 | $28 | $700 |
| 50 | $22 | $1100 |
| 100 | $15 | $1500 |
Long Description
2b)
Using the same seven data points from the previous chapter select three data points and plug into the grid below to produce a quadratic solution. Simplify the resulting polynomial and put in standard form. Note solution given for the three bracketed points.
(Solution given)
Seven Data Points
| x | y or f(x) |
|---|---|
| -6.2 | -8 |
| [-3] | [-7] |
| -1.5 | -2.2 |
| [1] | [0.7] |
| 3.5 | 3 |
| 4.25 | 5 |
| [7.9] | [11] |
Long Description
Exercise 2b Answer Grid
| - | x | f(x) | 1st divided difference | 2nd divided difference |
|---|---|---|---|---|
| - | - | [latex]b_0[/latex] | [latex]b_1(x - x_0)[/latex] | [latex]b_2(x - x_0)(x - x_1)[/latex] |
| - | - | - | - | - |
| - | - | - | - | - |
| - | - | - | - | - |
| - | - | - | - | - |
| - | - | - | - | - |
Long Description
2c)
Add an additional data point and develop a 3rd degree (cubic) polynomial. Compare this to the solution from 2a) and decide whether or not it improves the interpolation. Note student answers may vary[latex].[/latex]
The table provides pricing for different ranges of numbers of children attending a party. This increased flexibility the solution brings to this exercise will provide specific pricing by exact number attending, rather than a range.
These sample data points are to be used in the accompanying practice problem. Three of the seven are bracketed as they are used to provide a solution to this exercise.
Students should use the grid to record their solution. It includes columns for the independent variable, x the dependent variable f(x), the first divided difference and the second divided difference.
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