It is difficult to graph a linear function by hand. One method that works if the $x$, $y$, and $z$-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure $12.68$; this shows the part of the plane in the octant where $x \geq 0, y \geq 0, z \geq 0.$ If the intercepts are not all positive, the same method works if the $x$, $y$, and $z$-axes are drawn from a different perspective. Use this method to graph the linear function.

$$z= 6-2x-3y$$

Describe in words the level surfaces of $f(x, y, z)=$ $\sin (x+y+z)$

Based on the given figure, assuming g is linear, find a formula for g(x, y).

Hiking on a level trail going due east, you decide to leave the trail and climb toward the mountain on your left. The farther you go along the trail before turning off, the gentler the climb. Sketch a possible topographical map showing the elevation contours.

Consider the acceleration due to gravity, $g$, at a distance $h$ from the center of a planet of mass $m$.

(b) If $h$ is held constant, is $g$ an increasing or decreasing function of $m$? Why?

The monthly payment, m dollars, for a 30 -year fixed rate mortgage is a function of the total amount borrowed, P dollars, and the annual interest rate, r %. In other words, $m=f(P, r)$.

Interpret the following statement in the context of monthly payment: $f(300,000,5)=1610.46$.

Show that the level surfaces of g(x, y, z)=ax+by+cz where $c \neq 0$ are parallel planes.

What features of the contour diagram of g(x, y) in the figure suggest that g is linear?

Match the functions (a)-(d) with the shapes of their typical level curves (I)-(IV).

(a) $f(x, y)=\frac{y}{x^2+1}$

(b) $f(x, y)=\frac{1}{x^2+2 y^2}$

(c) $f(x, y)=\frac{x^2+1}{y^2+1}$

(d) $f(x, y)=\frac{x}{x^2+y^2+1}$

I. Circles

II. Parabolas

III. Hyperbolas

IV. Ellipses

It is difficult to graph a linear function by hand. One method that works if the $x$, $y$, and $z$-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure $12.68$; this shows the part of the plane in the octant where $x \geq 0, y \geq 0, z \geq 0.$ If the intercepts are not all positive, the same method works if the $x$, $y$, and $z$-axes are drawn from a different perspective. Use this method to graph the linear function.

$$z= 4+x-2y$$

In this exercise, could the contour diagram represent a linear function?

Consider the acceleration due to gravity, $g$, at a distance $h$ from the center of a planet of mass $m$.

(a) If $m$ is held constant, is $g$ an increasing or decreasing function of $h$ ? Why?

Is the statement in the following exercise true or false? Give reasons for your answer.

If $\lim\limits_{(x, y) \rightarrow(a, b)} f(x, y)$ exists then $f(x, y)$ is continuous at $(a, b)$.

The pressure, P atmospheres, of 10 moles of nitrogen gas in a steel cylinder depends on the temperature of the gas, $T$ Kelvin, and the volume of the cylinder, $V$ liters, so $P=f(T, V)$. Interpret the following statement in terms of pressure: $f(300,5)=49.2$.