A common problem in experimental work is to obtain a mathematical relationship
y
=
f
x
between two variables x and y by "fitting" a curve to points in the plane that correspond to experimentally determined values of x and y, say
x
1
,
y
1
,
x
2
,
y
2
,
...
,
x
n
,
y
n
The curve
y
=
f
x
is called a mathematical model of the data. The general form of the function f is commonly determined by some underlying physical principle, but sometimes it is just determined by the pattern of the data. We are concerned with fitting a straight line
y
=
m
x
+
b
to data. Usually, the data will not lie on a line (possibly due to experimental error or variations in experimental conditions), so the problem is to find a line that fits the data "best" according to some criterion. One criterion for selecting the line of best fit is to choose m and b to minimize the function
g
m
,
b
=
∑
i
=
1
n
m
x
i
+
b
−
y
i
2
This is called the method of least squares, and the resulting line is called the regression line or the least squares line of best fit. Geometrically,
m
x
i
+
b
−
y
i
is the vertical distance between the data point
x
i
,
y
i
and the line
y
=
m
x
+
b
.
These vertical distances arc called the residuals of the data points, so the effect of minimizing
g
m
,
b
is to minimize the sum of the squares of the residuals. In these exercises, we will derive a formula for the regression line.
Assume that not all the
x
i
'
s
are the same, so that
g
m
,
b
has a unique critical point at the values of m and b obtained in Exercise 49(c). The purpose of this exercise is to show that g has an absolute minimum value at this point.
(a) Find the partial derivatives
g
m
m
m
,
b
,
g
b
b
m
,
b
,
and
g
m
b
m
,
b
,
and then apply the second partials test to show that g has a relative minimum at the critical point obtained in Exercise 49.
(b) Show that the graph of the equation
z
=
g
m
,
b
is a quadric surface.
(c) It can be proved that the graph of
z
=
g
m
,
b
is an elliptic paraboloid. Accepting this to be so, show that this paraboloid opens in the positive z-direction, and explain how this shows that g has an absolute minimum at the critical point obtained in Exercise 49.