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In Exercise $32,$ let $(a, b)$ be the critical point. Show that $R(a, b)-R(a+h, b+k)=6(h-k)^{2}+k^{2} / 2,$ thereby proving $R$ is maximized at the critical point. (Why?) Given $n$ data points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right), \ldots,\left(x_{m}, y_{n}\right),$ let $y=a x+b$be the equation of the least square (regression) line. For each $x$ - value, $x_{i}$ there is the observed $y$ -value, $y_{i}$, and the value of $y$ predicted by the regression line. This difference $y_{i}-\left(a x_{i}+b\right)$ is the error at $x=x_{i}$. Determine $a$ and $b$ so as to minimize the sum of the squares of the errors. If we sum all of the squared errors, we have $f(a, b)=\sum_{i=1}^{n}\left(y_{i}-\left(a x_{i}+b\right)\right)^{2}$$f$ is a function of $a$ and $b$. To find the critical points of this function solve the equations$$\begin{aligned}&f_{a}(a, b)=0\\&f_{b}(a, b)=0\end{aligned}$$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

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Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

01:54

Use the method of Lagrange…

01:09

Given $n$ data points $\le…

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