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
*x*_{0 }= 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,*P*_{3}(*w*), based on the nodes 20, 40, 50, 70. - Use the MATLAB m-file
**polyfit**to verify your result in (a). - Plot
*P*_{3}(*w*) and the original data on the same axes.

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