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In mathematics, a multivariable function is one that depends on two or more variables. For example, in the problem of fitting a line to data using the least squares method, the objective is to minimize the function of two variables, typically represented as \( f(a, b) \). This function often represents some sort of error that we want to reduce.
When fitting a line to a set of data points, the multivariable function of interest is the sum of squared errors. In this specific problem, we are given data points for precipitation in New York City. The function to minimize becomes \( f(a, b) = \sum_{i=1}^{3} (y_i - (ax_i + b))^2 \), where \( a \) and \( b \) represent the slope and y-intercept of the line, and \( x_i \) and \( y_i \) are the data points for month and precipitation respectively.
Minimizing this function allows us to find the best fitting linear model that represents the relationship between the months and precipitation accurately.

Partial derivatives are an essential concept in calculus for dealing with functions of multiple variables. They measure how a multivariable function changes as one of its variables changes while keeping the other variables constant.
In the context of the least squares method, partial derivatives help in finding the values of \( a \) and \( b \) that minimize the function \( f(a, b) \). We compute the partial derivatives with respect to \( a \) and \( b \):\[\frac{\partial f}{\partial a} = -2 \sum_{i=1}^{3}x_i(y_i - ax_i - b)\]\[\frac{\partial f}{\partial b} = -2 \sum_{i=1}^{3}(y_i - ax_i - b)\]
Setting these partial derivatives equal to zero yields the normal equations, which can be solved to find the optimal values of \( a \) and \( b \). This step is crucial because it transitions from the general form of the problem to solving specifically for the parameters that provide the best fit.

The normal equations are derived from the partial derivatives of the multivariable function \( f(a, b) \). These equations form a linear system that can be solved to find the best-fitting line's parameters, \( a \) and \( b \), in the least squares method. In this exercise, the normal equations obtained were:
\[ \sum x_i^2 a + \sum x_i b = \sum x_i y_i \]
\[ \sum x_i a + 3b = \sum y_i \]
The normal equations are a critical step because they simplify the process of finding the line's parameters. By solving this system of equations, we can derive \( a \) and \( b \) for the line \( y = ax + b \) that minimizes the sum of squared differences between the observed values and the values predicted by the line. This results in a practical application of linear algebra that effectively handles the fitting process.

A linear model is a mathematical model that describes a relationship between two variables using a straight line. This model has the form \( y = ax + b \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( a \) is the slope of the line, and \( b \) is the y-intercept.
In the context of the exercise, the linear model aims to represent how precipitation changed over the months of January, February, and March in New York City in 1855. By using the least squares method, we determine the values of \( a \) and \( b \) that provide the best fit line for the given data.
Once the values for \( a \) and \( b \) are found, the linear model \( y = -4.525x + 9.05 \) describes the trend in precipitation. This model, derived from analyzing historical data, helps in understanding patterns and making predictions, illustrating why linear models are instrumental in both academic studies and practical analyses.