Question

match the surfaces $z = f(x,y)$ with the verbal description of the level curves $(f(x,y)=k,k = 1,2,3,ldots)$ by placing the letter of the verbal description to the left of the number of the surface.

1. $z = xy$
2. $z=\frac{1}{x - 1}$
3. $z=x^{2}+y^{2}$
4. $z = 2x^{2}+3y^{2}$
5. $z=sqrt{(x^{2}+y^{2})}$
6. $z = 2x+3y$

a. a collection of unequally spaced concentric circles
b. a collection of unequally spaced parallel lines
c. a collection of hyperbolas
d. a collection of concentric ellipses
e. a collection of equally spaced parallel lines
f. a collection of equally spaced concentric circles

Explanation:

Step1: Find level - curves for $z = xy$

Set $xy=k$, then $y = \frac{k}{x}$, which are hyperbolas. So $z = xy$ matches with C.

Step2: Find level - curves for $z=\frac{1}{x - 1}$

Set $\frac{1}{x - 1}=k$, then $x=1+\frac{1}{k}$ ($k
eq0$), which are a collection of parallel lines. Since $k = 1,2,3,\cdots$, they are unequally spaced. So $z=\frac{1}{x - 1}$ matches with B.

Step3: Find level - curves for $z=x^{2}+y^{2}$

Set $x^{2}+y^{2}=k$, which are concentric circles. The distance between consecutive circles is $\sqrt{k + 1}-\sqrt{k}=\frac{1}{\sqrt{k + 1}+\sqrt{k}}$, so they are unequally spaced. So $z=x^{2}+y^{2}$ matches with A.

Step4: Find level - curves for $z = 2x^{2}+3y^{2}$

Set $2x^{2}+3y^{2}=k$, which are concentric ellipses. So $z = 2x^{2}+3y^{2}$ matches with D.

Step5: Find level - curves for $z=\sqrt{x^{2}+y^{2}}$

Set $\sqrt{x^{2}+y^{2}}=k$, then $x^{2}+y^{2}=k^{2}$. The distance between consecutive circles is $(k + 1)-k = 1$, so they are equally - spaced concentric circles. So $z=\sqrt{x^{2}+y^{2}}$ matches with F.

Step6: Find level - curves for $z = 2x+3y$

Set $2x + 3y=k$, then $y=-\frac{2}{3}x+\frac{k}{3}$, which are parallel lines. The distance between consecutive lines is $\frac{|(k + 1)-k|}{\sqrt{2^{2}+3^{2}}}=\frac{1}{\sqrt{13}}$, so they are equally - spaced. So $z = 2x+3y$ matches with E.

Answer:

1. C
2. B
3. A
4. D
5. F
6. E