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Following the previous exercise, show that the constants $a, b$ and $c$ for the quadratic regression curve $y=a x^{2}+b x+c$ may be found by solving the equations$$\begin{aligned}&\sum_{i=1}^{n} y_{i}=n a+b \sum_{i=1}^{n} x_{i}+c \sum_{i=1}^{n} x_{i}^{2}\\&\sum_{i=1}^{n} x_{i} y_{i}=a \sum_{i=1}^{n} x_{i}+b \sum_{i=1}^{n} x_{i}^{2}+c \sum_{i=1}^{n} x_{i}^{3}\\&\sum_{i=1}^{n} x_{i}^{2} y_{i}=a \sum_{i=1}^{n} x_{i}^{2}+b \sum_{i=1}^{n} x_{i}^{3}+c \sum_{i=1}^{n} x_{i}^{4}\end{aligned}$$for $a, b$ and $c$ Recall the Extreme Value Theorem for a function of a single variable states that continuous function on a closed interval attain their extrema either at critical points or at their endpoints. This theorem generalizes: for a continuous function $z=f(x, y),$ defined on a closed region, the function will attain its extrema at either critical points (within the region) or at boundary points of the region. The critical points are found by the methods of this section. To find the appropriate boundary point, we may use the equation of the boundary to eliminate one variable, reducing the function on the boundary to one of a single variable, and its extrema on the boundary are now found by the methods of Chapter 1.

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

44:28

A common problem in experi…

51:43

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