Shown below is a contour plot of a function \(f\text{.}\) The values of the function on are indicated on the contours. This activity will ask you to use this contour plot to estimate some partial derivatives of \(f\text{.}\)
When computing \(f_x\text{,}\) which variable is treated as a constant? On the contour plot, sketch a line segment for which that variable is constant through the point \((-2,1)\) .
To estimate \(f_x(-2,-1)\) using a difference quotient, you must identify two points in the \(xy\)-plane that are near \((-2,-1)\text{,}\) on the line segment you sketched in the previous part (or an extension of that line segment), and for which you can use the contour plot to find the value of the function at those points. Mark these points, which we will call \((a_1,b_1)\) and \((a_2,b_2)\) on your contour plot and record the values of \(f\) at these points.
If possible, you will use a central difference, but if your points are not equally spaced from \((-2,-1)\text{,}\) you may have a different expression that is still a difference quotient.
In this part and the next, you will need to follow a similar process to the previous part. However, the activity does not break down the steps each time. Estimate the partial derivative \(f_y(-2,-1)\text{.}\)
