Let's work through the contour maps for the functions $f(x, y)$ in exercises 28 and 30.

Problem 28: $f(x, y) = \ln(x^2 + 4y^2)$

To understand the contour map of this function, we need to set $f(x, y) = c$ where $c$ is a constant, which will define the level curves.

1. Starting with the equation: $\ln(x^2 + 4y^2) = c$ By exponentiating both sides, we get: $x^2 + 4y^2 = e^c$ This represents an ellipse, where the constant $e^c$ determines the size of the ellipse. For different values of $c$, we will get ellipses of different sizes.

   The general form of these ellipses is: $\frac{x^2}{e^c} + \frac{y^2}{e^c/4} = 1$ These are elongated in the $y$-direction since the coefficient of $y^2$ is multiplied by 4. The contour map will show a series of nested ellipses, centered at the origin.

Problem 30: $f(x, y) = y \sec x$

For this function, we set $f(x, y) = c$, where $c$ is a constant:

1. Setting the function equal to $c$, we have: $y \sec x = c$ Solving for $y$, we get: $y = c \cos x$ This equation represents a set of cosine curves for different values of $c$. For each constant $c$, we get a cosine curve along the $x$-axis, scaled vertically by the value of $c$.

   The contour map will therefore consist of a series of cosine curves, where $c$ determines the amplitude of the curve.

Summary of Contour Maps:

• Problem 28: The contours are ellipses centered at the origin, with varying sizes.
• Problem 30: The contours are cosine curves with different amplitudes based on the constant $c$.

Would you like more details or visual explanations for these functions?

Here are 5 questions related to these problems:

1. What is the effect of increasing $c$ on the contour map of $\ln(x^2 + 4y^2)$?
2. How do the ellipses in problem 28 change when $x$ and $y$ values vary?
3. What role does the secant function play in shaping the contour map of $y \sec x$?
4. How would the contour map change if the function in problem 30 was $y \tan x$ instead?
5. What does the cosine behavior imply about the periodicity in the contour map of problem 30?

Tip: For functions involving trigonometric terms like $\sec x$, pay attention to the periodic nature of the curves, as they often introduce repeating structures in contour maps.