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.

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

The gravitational force, $F$ newtons, exerted on an object by the earth depends on the object's mass, $m$ kilograms, and its distance, $r$ meters, from the center of the earth, so $F=f(m, r)$. Interpret the following statement in terms of gravitation: $f(100,7000000) \approx 820$.

In the following problem, for the two given points:

Use the given figure, the contour diagram of f(x, y), to find $\Delta f$, the difference between the values of f from the first to the second point.

$(0,-2)$ and $(0,2)$

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

If $\lim\limits_{(x, y) \rightarrow(0,0)} f(x, y)=3$, then the limit of $f(x, y)$ is 3 as $(x, y)$ approaches $(0,0)$ along the $x$-axis.

Use the catalog on page $702$ to identify the surface.

$$x^2+z^2=1$$

In the following problem, use the contours of the linear function z=f(x, y) in the given figure to create possible contour labels for the linear function z=g(x, y) satisfying the given condition.

The graph of $g$ has the same contour as $f$ for the value $z=40$, and a negative slope in the $x$-direction.

In the following problem, for the two given points:

Find the distance h from the first to the second point.

$(0,-2)$ and $(0,2)$

Construct a function $f(x, y)$ with the given property.

Not continuous at the point $(2,0)$; continuous everywhere else.

Describe the level surface $f(x, y, z)=x^2 / 4+z^2=1$ in words.

Give a formula for a function whose graph is described. Sketch it using a computer or calculator.

A cone of circular cross-section opening downward and with its vertex at the origin.

A cable company charges $100 for a monthly subscription to its services and$5 for each special feature movie that a subscriber chooses to watch.

If $R=f(s, m)$, find $f(1000,5000)$ and interpret it in terms of revenue.

A cable company charges $100 for a monthly subscription to its services and$5 for each special feature movie that a subscriber chooses to watch.

Write a formula for the monthly revenue, $R$ in dollars, earned by the cable company as a function of $s$, the number of monthly subscribers it serves, and $m$, the total number of special feature movies that its subscribers view.

A car rental company charges $\$ 40$ a day and $15$ cents a mile for its cars.

(b) If $C=f(d, m)$, find $f(5,300)$ and interpret it.

represent the surface as the graph of a function, $f(x, y)$, and by level surfaces of the form $g(x, y, z)=c$. (There are many possible answers.)

Upper half of unit sphere centered at the origin.