\begin{tabular}{ll}\hline 6 & Fill in the Blank 7 points \\ Given a function \( \mathrm{P}(\mathrm{x}, \mathrm{y}) \) where x is \\ the number of units sold of product \\ A and y is the number of units sold \\ of product B , then for any function \\ \( \mathrm{P}(\mathrm{x}, \mathrm{y}) \) interpret the partial \\ derivative. \( \mathrm{P}_{\mathrm{y}}(7,10) \) is the \\ \hline approximate \\ profit from the sale of the \\ \hline type your answer... \\ \hline th \\ \hline of product \\ \hline type your answer... \\ \hline when \\ \hline the production level of product \\ \hline type your answer... \\ \hline choose your answer... \\ \hline type your answer... \\ \hline\end{tabular}

When you compute the partial derivative \( P_y(7,10) \), you're essentially looking at how the profit changes with respect to product B's sales, holding product A's sales constant at 7 units. This means you get an estimate of how much extra profit you can expect to make by selling one more unit of product B while keeping everything else the same. On the other hand, it’s important to remember that partial derivatives don't give you the full picture. They only measure changes based on one variable at a time, assuming all else remains constant. A common mistake here is to interpret \( P_y \) as representing total profit or the effect of changing both A and B simultaneously—this isn’t the case! Stay focused on one variable at a time for clarity!