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Show that the second equation in the previous exercise gives $b=\bar{y}-a \bar{x}$ where $\bar{x}=\frac{\sum_{i=1}^{n} x_{i}}{n}$ and $\bar{y}=\frac{\sum_{i=1}^{n} y_{i}}{n},$ and substituting for $b$ in the first equation and solving for $a$ gives$$a=\frac{\sum_{i=1}^{n} x_{i} y_{i}-\frac{\sum_{i=1}^{n} x_{i} \sum_{i=1}^{n} y_{i}}{n}}{\sum_{i=1}^{n} x_{i}^{2}-\frac{\left(\sum_{i=1}^{n} x_{i}\right)^{2}}{n}}$$Note that in Section 1.8 we wrote this expression as $a=\frac{\overline{s_{x y}}-\bar{x} \cdot \bar{y}}{\overline{s_{x}}-\bar{x}^{2}}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 3

Extrema

Partial Derivatives

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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

01:18

(a) Prove by induction on …

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Show that $\sum_{i=1}^{n}\…

05:11

Proof Prove that$\sum_…

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