https://file.aiquickdraw.com/mathbot/file/1730320994177lt9jwryu.jpg

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?

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### 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.

Approximate f_x(3,5) using the contour diagram of f(x,y) shown below.

Learn how to approximate the partial derivative f_x(3,5) using a contour diagram of f(x,y). This problem involves estimating the rate of change in the x-direction at the point (3,5) by analyzing contour values. Suitable for students in Grades 10-12.

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