Deep Dives

1. Least Squares (#differentiation, #theoreticaltools) Adapted from Stewart's "Cal-

culus"

Suppose that a scientist has reason to believe that two quantities z and y are related

linearly, that is y = mz+b, at least approximately, for some values of m and b.

The scientist performs an experiment and collects data in the form of points (21,31).

(72.32), (n.), and then plots these points. The points don't lie exactly on

a straight line, so the scientist wants to find constants m and b so that the line

y=mx+b "fits" the points as well as possible.

***

Let dy- (mz, + b) be the vertical deviation of the point (z,y) from the line.

The method of least squares determines m and b so as to minimize of the

sum of the squares of these vertical deviations. Show that, according to this method,

the line of best fit is obtained when m and b satisfy the following system of equations:

z₁+ bn =>

mΣ²²+ b[x₁=[zy

inl