To approximate \(f_z(3, 5)\) using the contour diagram, follow these steps: ### Step 1: Locate the point \((x, y) = (3, 5)\) - Find \(x = 3\) on the horizontal axis. - Find \(y = 5\) on the vertical axis. - Identify the point where these coordinates intersect on the contour diagram. ### Step 2: Determine the contour level at \((3, 5)\) - Observe the contour lines near \((3, 5)\). - Interpolate between the closest contour lines if necessary. ### Step 3: Approximate the partial derivative \(f_z(3, 5)\) - The partial derivative with respect to \(z\) (assuming \(z = y\)) can be approximated by the slope of the contour lines in the \(z\)-direction at the point. - Use the difference in the contour levels around the point and the vertical distance between contours to estimate the rate of change. --- ### Approximate \(f_z(3, 5)\): From the contour diagram: - The contour levels near \((3, 5)\) are approximately 6 and 8. - The change in the contour level (difference in \(f\)) is roughly \(8 - 6 = 2\). - The vertical distance between these contours is approximately 1 unit (based on the grid). Thus: \[ f_z(3, 5) \approx \frac{\Delta f}{\Delta y} \approx \frac{2}{1} = 2 \] ### Final answer: \[ f_z(3, 5) \approx \boxed{2} \] You can input this approximation into the answer box.