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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Suppose that a scientist has reason to believe that two quantities x and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (x1, x2), (x2, y2) ,…, (xn, yn) 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 (see the figure).

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Let di = yi − (mxi, + b) be the vertical deviation of the point (xi, yi) from the line. The method of least squares determines m and b so as to minimize i = 1 n d i 2 , the sum of the squares of these deviations. Show that, according to this method, the line of best fit is obtained when

m i = 1 n x i + b n = i = 1 n y i

and m i = 1 n x i 2 + b i = 1 n x i = i = 1 n x i y i

To determine

To show: The line of best fit obtained only when mi=1nxi+bn=i=1nyi and mi=1nxi2+bi=1nxi=i=1nxiyi .

Explanation

Given:

Let di=yi(mxi+b) be the vertical deviation of the point (xi,yi) from the line.

The method of least squares determines m and b to minimize i=1ndi2 is the sum of the squares of these deviations.

Proof:

Let di=yi(mxi+b) and to minimize the given equation into i=1ndi2 .

Substitute di=yi(mxi+b) in the function i=1ndi2 .

f(m,b)=i=1n[yi(mxi+b)]2

Note that the above function is a variable of m and b.

Take partial derivative of f(m,b) with respect to m and obtain fm .

fm=m[i=1n[yi(mxi+b)]2]=[i=1n2[yi(mxi+b)]](0xi0)=i=1n2xi[yi(mxi+b)]

Thus, fm=i=1n2xi[yi(mxi+b)] (1)

Set the above partial derivative to zero.

From equation (1),

2i=1nxi[yi(mxi+b)]=0i=1n(xiyimxi2bxi)=0i=1nxiyii=1nmxi2i=1nbxi=0i=1nxiyi=i=1nmxi2+i=1nbxi

Simplify further as follows.

i=1nxiyi=mi=1nxi2+i=1nbxii=1nxiyi=mi=1nxi2+bi=1nxi

Therefore, i=1nxiyi=mi=1nxi2+bi=1nxi is proved.

Take partial derivative of f(m,b) with respect to b and obtain fb .

fb=b[i=1n[yi(mxi+b)]2]=[i=1n2[yi(mxi+b)]](001)=i=1n2[yi(mxi+b)]

Thus, fb=i=1n2[yi(mxi+b)] (2)

Set the above partial derivative to zero.

From equation (2),

2i=1n[yi(mxi+b)]=0i=1n(yimxib)=0i=1nyii=1nmxii=1nb=0i=1nyi=i=1nmxi+i=1nb

Simplify further as follows

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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