Certainly! Let's go through the step-by-step process to approximate \( f_z(3, 5) \) using the contour diagram. --- ## Step 1: Understand the problem - The goal is to estimate the partial derivative of \( f \) with respect to \( z \) at the point \( (x, y) = (3, 5) \). - The partial derivative \( f_z(3, 5) \) measures how much \( f \) changes as \( z \) varies around the point \( (3, 5) \). **Note:** In the contour diagram, the contours represent levels of \( f \). --- ## Step 2: Identify the relevant contours - Locate the point \( (3, 5) \) on the contour diagram. - Find the contour lines immediately above and below the point along the \( y \)-direction (since \( z \) often relates to \( y \) or the variable of change). - From the diagram: - The point \( (3, 5) \) is roughly at the intersection of the contour lines labeled **6** and **4**. - The contour **6** is slightly above \( y = 5 \). - The contour **4** is slightly below \( y = 5 \). ## Step 3: Determine the change in \( f \) with respect to \( y \) - The change in \( f \) across these contours is approximately: \[ \Delta f \approx 6 - 4 = 2 \] - The change in the \( y \)-coordinate (assuming the grid units are consistent) between these contours is approximately: \[ \Delta y \approx 1 \text{ unit} \] --- ## Step 4: Approximate the partial derivative \( f_y(3, 5) \) - Using the finite difference approximation: \[ f_y(3, 5) \approx \frac{\Delta f}{\Delta y} = \frac{6 - 4}{1} = 2 \] **This gives the rate of change of \( f \) with respect to \( y \) at the point (3, 5).** --- ## Step 5: Relation to \( f_z(3, 5) \) - **Note:** The notation \( f_z \) suggests the derivative with respect to \( z \), but in the context of the diagram, it appears the relevant variable is \( y \). - **Assuming** \( z = y \), then: \[ f_z(3, 5) \approx f_y(3, 5) \approx 2 \] --- ## **Final Answer:** \[ \boxed{ f_z(3, 5) \approx 2 } \] This is the approximate rate of change of the function \( f \) with respect to \( z \) at the point \( (3, 5) \). --- If you have any further questions or need clarification, feel free to ask!