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# MTH312

Record all results to at least 5 decimal place accuracy, with rounding.

EXERCISES

1. The power generated by a windmill varies with the wind speed. In an experiment, the following measurements were obtained:
 Wind speed (kph) 22 35 48 61 74 Electric power (W) 320 490 540 500 480
• Construct the Lagrangian interpolating polynomial of degree two, in ascending powers of x, which passes through the first three data points. Use this polynomial to calculate the power generated at a wind speed of 40 kph.
• Construct the Lagrangian interpolating polynomial of degree three (do not simplify the polynomial) which passes through the first four data points. Use this form to calculate the power generated at a wind speed of 40 kph.these polynomials to calculate the power generated at a wind speed of 40 kph. Comment on the results.
• Construct a divided-difference table for this data.
• Find divided-difference polynomials of degrees two, three and four. Use these polynomials to calculate the power generated at a wind speed of 40 kph.
• Construct a forward-difference table for this data.
• Find forward-difference polynomials of degrees two, three and four.Use these polynomials to calculate the power generated at a wind speed of 40 kph. Comment on the results.
• Plot the original data and the interpolating polynomials on the same axes.
• In a study of radiation-induced polymerization, a source of gamma rays was employed to give measureddoses of radiation. The dosage varied with position in the radiation apparatus and the following data was recorded:
 Position 1 1.5 2 3 3.5 Dosage 2.71 2.98 3.2 3.2 2.98

For some reason the reading at 2.5 cm was not reported, however the value of the radiation at this point is required.

• Find interpolating polynomials of degrees two to four using x0 = 1.
• Use these polynomials to approximate the dosage at x = 2.5 and comment on the results.
1. The following table gives the relative viscosity V of ethanol as a function of the percentage of anhydrous solute weight w:
 w 20 30 40 50 60 70 V(w) 2.138 2.662 2.84 2.807 2.542 2.21
• Find the third degree interpolating polynomial, P3(w), based on the nodes 20, 40, 50, 70.
• Use the MATLAB m-file polyfit to verify your result in (a).
• Plot P3(w) and the original data on the same axes.
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