MTH312
Record all results to at least 5 decimal place accuracy, with rounding.
EXERCISES
- 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.
- 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.
