The purpose of this lab is to acquaint you with the application of local extreme values as it applies to the method of least-squares.

cp /math/calclab/MA1024/Least_squares_start_B08.mws My_Documents

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Least_squares_start_B08.mws`, and go through it
carefully. Then you can start working on the exercises.

A crucial first step in solving such problems is being able to find and classify local extreme values of a function. What we mean by a function having a local extreme value at a point is that for values of near , for a local maximum and for a local minimum.

In single-variable calculus, we found that we could locate candidates for local extreme values by finding points where the first derivative vanishes. For functions of two dimensions, the condition is that both first order partial derivatives must vanish at a local extreme value candidate point. Such a point is called a stationary point. It is also one of the three types of points called critical points. Note carefully that the condition does not say that a point where the partial derivatives vanish must be a local extreme point. Rather, it says that stationary points are candidates for local extrema. Just as was the case for functions of a single variable, there can be stationary points that are not extrema. For example, the saddle surface has a stationary point at the origin, but it is not a local extremum.

Finding and classifying the local extreme values of a function
requires several steps. First, the partial derivatives must
be computed. Then the stationary points must be solved for, which is not
always a simple task.

Next, one must check for the presence of

If
and
then is a local minimum.

If and then is a local maximum.

If then is a saddle point.

If then no conclusion can be made.

If and then is a local maximum.

If then is a saddle point.

If then no conclusion can be made.

Notice that the function to be minimized is a function of

Another example is the exponential function which again after taking the natural log of both sides results in a linear relationship between and as follows:

X | Y1 | Y2 | Y3 |

1 | 11.3 | 8.04 | 1.81 |

2 | 6.94 | 19.5 | 1.38 |

3 | 5.70 | 17.4 | 1.49 |

4 | 12.1 | 20.7 | 1.34 |

5 | 6.95 | 30.5 | 0.802 |

6 | 20.4 | 23.1 | 0.901 |

7 | 14.6 | 76.8 | 0.985 |

8 | 17.9 | 60.6 | 1.46 |

9 | 26 | 81.2 | 0.828 |

10 | 22.8 | 97.6 | 0.844 |

11 | 22.9 | 114 | 0.727 |

12 | 25.3 | 129 | 0.639 |

13 | 31 | 150 | 0.698 |

14 | 33.7 | 148 | 0.531 |

15 | 31.4 | 190 | 0.0482 |

16 | 34.2 | 193 | 0.510 |

17 | 25.5 | 212 | 0.378 |

18 | 38.7 | 225 | 0.471 |

19 | 38 | 247 | 0.292 |

20 | 44.9 | 270 | 0.360 |

- Use the second column as data values for . Plot the data set . Find the least-squares line that best fits the data and then plot the data set along with the least-squares line.
- Use the third column as data values for . Plot the data set . Find a power-law function that best fits the data. Include a plot of the data set the corresponds to along with the least-squares line and also a plot of the original data set along with the power-law function that best fits the data.
- Use the fourth column as data values for . Plot the data set . Find an exponential funtion that best fits the data. Include a plot of the data set the corresponds to along with the least-squares line and also a plot of the original data set along with the exponential function that best fits the data.

2008-11-24