To approximate $f_x(3,5)$ using the contour diagram, we need to estimate the rate of change of $f(x,y)$ in the $x$-direction at the point $(3,5)$.

Steps:

1. Identify the contours around (3,5): We locate the point $(3,5)$ on the diagram and observe the contour lines near it.
2. Estimate $f(x,y)$ values along the $x$-axis near $(3,5)$: We look for the values of $f$ along a horizontal line passing through $(3,5)$ and note the contour values on either side.
3. Calculate the approximate partial derivative $f_x(3,5)$: We use the difference between contour values across a small $x$-interval to approximate $f_x(3,5)$.

From the contour plot:

• Near $(3,5)$, moving horizontally in the $x$-direction, we observe contours with values of approximately 4 (closer to $x = 2$) and 6 (closer to $x = 4$).
• If we approximate $f(x,y)$ changing from 4 to 6 over an $x$-interval of about 2 units, then $f_x(3,5) \approx \frac{6 - 4}{4 - 2} = \frac{2}{2} = 1.$

Answer:

$f_x(3,5) \approx 1.$

Would you like more details, or have any questions?

Related Questions:

1. How would you approximate $f_y(3,5)$ using this contour plot?
2. What does it mean if $f_x$ at a point is zero?
3. How do contour lines help in understanding gradients?
4. How would a steeper gradient appear in a contour plot?
5. What can we say about $f(x,y)$ at points where contour lines are equally spaced?

Tip: In contour plots, closely spaced lines indicate a steep gradient, while widely spaced lines indicate a gradual slope.