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Consider the following contour map of a function f(x,y).

Which of the following functions corresponds to the contour map?

a.
ln(4x^2+y^2)

b.
ln(x^2+4y^2)

c.
None of the listed options

d.
e^{4x}^2+y^2

e.
4x^2+y^2

f.
x^2+4y^2

g.
e^{x}^2+4y^2

To determine which of the given functions corresponds to the contour map, we need to analyze the contour levels and their spacing. Contour maps display curves along which the function takes constant values, and the behavior of the contours gives us clues about the underlying function.

### Observations:
1. The contour map consists of concentric circles centered at the origin \((0, 0)\), indicating that the function has radial symmetry.
2. The contour levels increase as we move away from the origin. The contours are labeled with values like 0, 1, 2, and 3.
3. The spacing between the contours suggests that the function increases quadratically in both the \(x\)- and \(y\)-directions, but potentially with different coefficients for \(x\) and \(y\).

### Analysis of the options:
Let's analyze the listed functions:

1. **\( \ln(4x^2 + y^2) \)**:
   - This is a logarithmic function. Since logarithmic functions tend to grow more slowly than quadratics, we would expect very tightly spaced contours near the origin. However, the spacing between the contours in the map seems to indicate a quadratic growth rather than logarithmic growth. This function is unlikely to match the contour map.

2. **\( \ln(x^2 + 4y^2) \)**:
   - Similar to the previous function, this is a logarithmic function, with different coefficients for \(x\) and \(y\). Again, the contour map suggests quadratic rather than logarithmic growth, so this is also unlikely.

3. **\( e^{4x^2 + y^2} \)**:
   - This is an exponential function. Exponential functions grow much more rapidly than quadratics, and the contour map doesn't display the extremely rapid growth typical of exponentials. The spacing between the contour lines looks consistent with quadratic growth, not exponential.

4. **\( 4x^2 + y^2 \)**:
   - This is a quadratic function with different coefficients for \(x\) and \(y\). This would produce ellipses as the contours, which are consistent with what we see in the map if the contour levels reflect the sum of squares of \(x\) and \(y\) with different weights. It’s a strong candidate.

5. **\( x^2 + 4y^2 \)**:
   - This is another quadratic function, but with different scaling for the \(y\)-direction. The contours would again be ellipses, but stretched along the \(y\)-axis rather than the \(x\)-axis. The contour map shows the contours as nearly circular rather than highly elliptical, suggesting that this is less likely than option (e).

6. **\( e^{x^2 + 4y^2} \)**:
   - This is another exponential function, which would lead to rapidly increasing values and very tight spacing between contours. The contour map doesn’t display this characteristic.

### Conclusion:
The function that best fits the contour map is likely **\( 4x^2 + y^2 \)**, which is option **(e)**. The contours in the map resemble ellipses, consistent with the behavior of this quadratic function, where the \(x\)-direction grows faster due to the coefficient 4.

Would you like further details or have any additional questions?

### Related Questions:
1. How does the contour map of a quadratic function differ from that of a logarithmic function?
2. What would the contour map of the function \( x^2 + y^2 \) look like?
3. How do the coefficients in quadratic functions affect the shape of the contour lines?
4. What role does symmetry play in determining the form of contour maps?
5. How would the contour map change if the function was \( 4x^2 + 4y^2 \)?

### Tip:
When analyzing contour maps, always check for symmetry and the spacing of the contours to determine the growth behavior of the underlying function.

Consider the following contour map of a function f(x,y). Which of the following functions corresponds to the contour map?

a. ln(4x^2+y^2)
b. ln(x^2+4y^2)
c. None of the listed options
d. e^{4x^2+y^2}
e. 4x^2+y^2
f. x^2+4y^2
g. e^{x^2+4y^2}

This problem involves identifying the correct function based on a given contour map. The options include logarithmic and quadratic functions, and the solution explains how to differentiate between these using symmetry, contour spacing, and radial growth patterns. Suitable for undergraduate multivariable calculus students.

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