Let f(x, y, z)=43.2-2.3 x+11.3 y-4.5 z . Complete the...

Let $f(x, y, z)=43.2-2.3 x+11.3 y-4.5 z .$ Complete the following: An increase of 1 in the value of $y$ causes the value of $f$ to ______ by ______, whereas increasing the value of $x$ by 1 and ______ the value of $z$ by causes a decrease of $11.3$ in the value of $f$.

Key Concepts

Multivariate Linear Function

A multivariate linear function is an algebraic expression where the output is a linear combination of multiple input variables. Each variable is multiplied by a constant coefficient, and these products are summed along with a constant term. This type of function models situations where the effect of each variable on the outcome is independent and additive.

Coefficient Interpretation

In a multivariate linear function, the coefficient of a variable represents the amount by which the function’s output changes when that variable increases by one unit, assuming all other variables remain constant. This interpretation allows us to understand the influence or sensitivity of each independent variable on the dependent variable.

Partial Derivative

The partial derivative of a multivariate function with respect to one of its variables quantifies the rate of change of the function as that variable changes, while holding the other variables fixed. In the context of linear functions, the partial derivative is constant and equal to the coefficient of the variable, which simplifies analysis of the function.

Constant Rate of Change

A key property of linear functions is that the rate of change remains constant. This means that for any given variable, a uniform increase or decrease in that variable results in a consistent change in the function's output, as determined by the corresponding coefficient. This property is critical in predicting how changes in the input affect the overall function.

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