Math 210 Fall 2008  

Homework Set 5, Due Thurs. Oct. 23 in Class
(Late papers OK until 1:00 Friday)

    Least Squares

  1. Use the method of least squares to find a plane of the form
    z = ax + by + c
    that best fits the following five points in three dimensional space with coordinates (x,y,z):
    (0, 0, 1.1),   (1, 1, 2),   (0, 1, -0.1),   (1,0,3),   (0, -1, 2.1).

  2. a). Some experimental data (xi, yi) is believed to fit a curve of the form y = (1+x)/(a + bx2), where the parameters a and b are to be determined from the data. The method of least squares does not apply directly to this since the parameters a and b do not appear linearly. Show how to find an equivalent equation to which the method of least squares does apply.
    b). Repeat part a) for the logistic curve y = L/(1 + e(a - bt)). Here the constant L is assumed to be known and b > 0.. [Since b > 0, then y converges to L as t increases. Thus the value of L can often be estimated simply by eye-balling a plot of the data for large t.]

  3. The comet Tentax, discovered only in 1968, moves within the solar system. The following are observations of its position  (r, w)   in a polar coordinate system with center at the sun:

    r 2.70 2.00 1.61 1.20 1.02
    w 48 67 83 108 126
    (here w is an angle measured in degrees).

    By Kepler's first law the comet should move in a plane orbit whose shape is either an ellipse or hyperbola (this assumes the gravitational influence of the planets is neglected). Thus the polar coordinates (r,w) satisfy

     
<pre>
                                         p
                            r =  -----------------
                                    1 - e cos w</pre>
    where p and e are parameters describing the orbit. Use the data to estimate p and e by the method of least squares. [Hint: Make some (simple) preliminary manipulation so the parameters p and e appear linearly so one can then apply the method of least squares.]

    Plotting Graphs

  4. See the graph on the right. graph
    a). If the horizontal axis is   x   and the vertical axis  y, what is the equation for   y  as a function of   x?
    b). If the horizontal axis is   log x  and the vertical axis  y, what is the equation for  y  as a function of   x? [In most applications, while not essential, it will be wiser to use "natural" logarithms, since that is what most other people use.]
    c). If the horizontal axis is  x  and the vertical axis log y, what is the equation for  y  as a function of   x?
    d). If the horizontal axis is  log x  and the vertical axis log y, what is the equation for  y  as a function of   x?

  5. For each of the seven closest planets, Kepler, using data from Bruno, knew the distance r from the planet to the sun and the time T it takes to orbit the sun (the length in earth days of a year on that planet).

      Mercury Venus Earth Mars Jupiter Saturn Uranus
    r 60110150230 780 1430 2870
    T 9022536569043301075030650
    (here r is in million km)

    Kepler sought a formula relating r and T. It took him a long time; he did not have logarithms. Guided by the idea of using graphs as in the previous problem, you can do this fairly easily.
    Make four experimental graphs of this data (as in the previous problem just above). The goal is to hope one of these four curves looks roughly like a straight line. If it does, then use least squares to find the "best" straight line -- and then the desired formula for the relation between r and T.
    [Since the data is only approximate and since we anticipate a "simple" answer, you may find it appropriate to use your numerical results to lead you to a simpler formula.]