The contour map in the figure represents colorcoded seismic amplitudes of a fault horizon and a projected contour map, which is used in earthquake studies. (Source: Adapted from Shipman/Wilson/Todd, An Introduction to Physical Science, Eighth Edition)

Do the level curves correspond to equally spaced amplitudes? Explain.

Show that the mixed partial derivatives f_xyy, f_yxy, and f_yyx are equal. f(x, y, z) = 2z / x+y

Find all values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously. f(x, y) = e^x²+xy+y²

Prove that if f is continuous and f(a, b) < 0, then there exists a δ-neighborhood about (a, b) such that f(x, y) < 0 for every point (x, y) in the neighborhood.

For f(x, y), find all values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously. f(x, y) = 3x³ - 12xy + y³

The acidity of rainwater is measured in units called pH. A pH of 7 is neutral, smaller values are increasingly acidic, and larger values are increasingly alkaline. The map shows curves of equal pH and gives evidence that downwind of heavily industrialized areas, the acidity has been increasing. Using the level curves on the map, determine the direction of the prevailing winds in the northeastern United States.

The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder's equity z (in billions of dollars) for Wal-Mart for the years 1998 through 2003.

A model for these data is

$$z=f(x, y)=0.156 x+0.031 y-1.66$$

Simplify the expression for f(x, 55) and interpret its meaning in the context of the problem.

The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder's equity z (in billions of dollars) for Wal-Mart for the years 1998 through 2003.

A model for these data is

$$z=f(x, y)=0.156 x+0.031 y-1.66$$

Which of the two variables in this model has the greater influence on shareholder's equity?

The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder's equity z (in billions of dollars) for Wal-Mart for the years 1998 through 2003.

A model for these data is

$$z=f(x, y)=0.156 x+0.031 y-1.66$$

Use a graphing utility and the model to approximate z for the given values of x and y.

Show that the mixed partial derivatives f_xyy, f_yxy, and f_yyx are equal. f(x, y, z) = e^-x sin yz

Find all values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously. f(x, y) = 1/x + 1/y + xy

Ideal Gas Law According to the Ideal Gas Law, $P V=k T$, where $P$ is pressure, $V$ is volume, $T$ is temperature (in Kelvins), and $k$ is a constant of proportionality. A tank contains $2600$ cubic inches of nitrogen at a pressure of $20$ pounds per square inch and a temperature of $300 \mathrm{~K}$.
Express $P$ as a function of $V$ and $T$ and describe the level curves.

The table shows the net sales x (in billions of dollars), the total assets y (in billions of dollars), and the shareholder's equity z (in billions of dollars) for Wal-Mart for the years 2002 through 2007. (Source: 2007 Annual Report for Wal-Mart)

A model for these data is

$$z=f(x, y)=0.026 x+0.316 y+5.04.$$

Simplify the expression for f(x, 95) and interpret its meaning in the context of the problem.

Ideal Gas Law According to the Ideal Gas Law, $P V=k T$, where $P$ is pressure, $V$ is volume, $T$ is temperature (in Kelvins), and $k$ is a constant of proportionality. A tank contains $2600$ cubic inches of nitrogen at a pressure of $20$ pounds per square inch and a temperature of $300 \mathrm{~K}$.
Determine $k$.