Let's solve the problem step by step: **Step 1: Understanding the Problem** We are given a contour map of a function \( f(x, y) \) and asked to estimate \( f_y(2, 6) \), which is the partial derivative of \( f \) with respect to \( y \) at the point \( (2, 6) \). **Step 2: Locate the Point \((2, 6)\) on the Contour Map** - On the x-axis, go to \( x = 2 \). - On the y-axis, go to \( y = 6 \). - Find the intersection point. **Step 3: Identify the Nearest Contour Values Above and Below \( y = 6 \)** At \( x = 2 \): - Just below \( y = 6 \) is \( y = 5 \), and the contour passing through is approximately the 100-contour. - Just above \( y = 6 \) is \( y = 7 \), and the 150-contour is closer to \( x = 2 \) at that height. **Step 4: Estimate \( f_y(2, 6) \) Using the Definition** Recall that the partial derivative with respect to \( y \) is: \[ f_y(2, 6) \approx \frac{f(2, 7) - f(2, 5)}{7 - 5} \] From the contours: - At \( (2, 5) \), \( f \approx 100 \) - At \( (2, 7) \), \( f \approx 150 \) So, \[ f_y(2, 6) \approx \frac{150 - 100}{2} = \frac{50}{2} = 25 \] **Step 5: Final Answer** \[ \boxed{25} \] --- ### **Explanation:** - We estimated the values of the function from the contour lines at \( x = 2 \) for \( y = 5 \) and \( y = 7 \). - The partial derivative with respect to \( y \) is the rate of change of \( f \) in the \( y \)-direction, which is why we used points above and below \( y = 6 \). **So, the correct estimate is \( f_y(2, 6) \approx 25 \).**