Match the pairs of functions with the given contour diagrams. Which contours represent f and which represent g? (The x - and y-scales are equal.)

$$f(x, y)=x^2-y^2, g(x, y)=x y$$

A relaxed spring of length 0.15 m stands vertically on the floor; its stiffness is 1000 N/m. You release a block of mass 0.4 kg from rest, with the bottom of the block 0.8 m above the floor and straight above the spring. How long is the spring when the block comes momentarily to rest on the compressed spring?

Match the given contour diagrams with the surfaces. Give reasons for your choice.

Sketch a graph of the surface and briefly describe it in words.

$$2 x+4 y+3 z=12$$

Match the given contour diagrams with the surfaces. Give reasons for your choice.

Match the pairs of functions with the given contour diagrams. Which contours represent f and which represent g? (The x - and y-scales are equal.)

f(x, y) = 2x + 3y, g(x, y) = 2x - 3y

Match the pairs of functions with the contour diagrams (I)-(IV). Which contours represent f and which represent g? (The x - and y-scales are equal.)

f(x, y) = x + y, g(x, y) = x - y

A capacitor consists of two large metal disks of radius 1.1 m placed parallel to each other, a distance of 1.2 mm apart. The capacitor is charged up to have an increasing amount of charge +Q on one disk and -Q on the other. At about what value of Q does a spark appear between the disks?

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

A solid metal ball of radius 1.5 cm bearing a charge of $-17 \mathrm{nC}$ is located near a solid plastic ball of radius 2 cm bearing a uniformly distributed charge of $+7 \mathrm{nC}$ on its outer surface. The distance between the centers of the balls is 9 cm. What is the electric field at the center of the metal ball due only to the charges on the plastic ball?

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced.

$$f(x,y) = -x^{2}-y^{2}+1$$

Find the limit as $(x, y) \rightarrow(0,0)$ of $f(x, y)$. Assume that polynomials, exponentials, logarithmic, and trigonometric functions are continuous.

$$f(x, y)=x^2+y^2$$

Sketch graph of the equation in $3$-space.

$$x=-3$$

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

$$x^2+y^2-z=0$$