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Using the contour plot, and the following function ln(x^2+4y^2) estimate which of the following partial derivatives has the largest value.

a.
fx(1,0)

b.
fy(1,0)

c.
fx(2,0)

d.
fy(2,0)

To solve the problem and estimate which partial derivative has the largest value, let's break it down step by step. We're analyzing the function:

\[
f(x, y) = \ln(x^2 + 4y^2)
\]

and evaluating the partial derivatives at specific points using the contour plot provided. From the contour plot, we can observe how the function changes as we move in different directions.

### Step 1: Interpreting the Contour Plot

- The contour plot shows curves of constant values of \( f(x, y) \). The closer the contours, the steeper the change in the function value, indicating a higher magnitude of the derivative.
- From the contour plot, observe the points \( (1, 0) \) and \( (2, 0) \), and look at how quickly the contours change in the \( x \)- and \( y \)-directions.

### Step 2: Analyze the Function Behavior

For the function \( \ln(x^2 + 4y^2) \):
- At \( (1, 0) \): This corresponds to the point where \( x = 1 \) and \( y = 0 \). Here, we estimate the partial derivatives:
  - \( f_x(1, 0) \) is the rate of change of the function as \( x \) changes while keeping \( y = 0 \).
  - \( f_y(1, 0) \) is the rate of change as \( y \) changes while keeping \( x = 1 \).

- At \( (2, 0) \): Similarly, this is where \( x = 2 \) and \( y = 0 \), and we estimate:
  - \( f_x(2, 0) \) and \( f_y(2, 0) \).

### Step 3: Compare Derivatives from the Contour Plot

- From the contour plot, the spacing of the lines near \( (1, 0) \) shows a steeper change in the \( x \)-direction compared to the \( y \)-direction. This suggests that \( f_x(1, 0) \) will be larger than \( f_y(1, 0) \).
- At \( (2, 0) \), the contours are more widely spaced, indicating that the rate of change is slower. However, \( f_y(2, 0) \) seems smaller due to the contours being more circular and less steep in the \( y \)-direction.

### Step 4: Conclusion

Based on the analysis of the contour plot:
- The partial derivative \( f_x(1, 0) \) has the largest magnitude because the contours change most rapidly in the \( x \)-direction near \( x = 1 \) and \( y = 0 \).

Thus, the correct answer is **(a) \( f_x(1, 0) \)**.

---

Would you like further clarification or details? Here are five related questions to deepen your understanding:

1. How can you compute partial derivatives analytically for the function \( \ln(x^2 + 4y^2) \)?
2. What does the spacing between contour lines indicate about the function's behavior?
3. How would the function behave for points where both \( x \) and \( y \) are nonzero?
4. How do contour plots help in estimating gradients and directional derivatives?
5. Can you find the critical points of the function \( f(x, y) = \ln(x^2 + 4y^2) \) using its partial derivatives?

**Tip**: In a contour plot, the direction in which contours are closer indicates a steeper slope, which can help estimate the direction of the largest derivative!

Using the contour plot, and the following function ln(x^2+4y^2) estimate which of the following partial derivatives has the largest value.

a. fx(1,0)

b. fy(1,0)

c. fx(2,0)

d. fy(2,0)

Learn how to estimate which partial derivative of the function ln(x^2 + 4y^2) has the largest value using contour plots. The problem focuses on the changes in fx and fy at points (1,0) and (2,0) using gradient analysis from the plot. Suitable for calculus students.

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