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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

1.0

1.5

2.0

3.0

3.5

Dosage

2.71

2.98

3.20

3.20

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

2.840

2.807

2.542

2.210

  • 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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