Question

describe the behavior, as c varies, of the level curve f(x,y) = c for f(x,y) = x^2 + y^2 + 1

Answer 1

Well, when c = 1, what points do you have?
What about when c < 1? And c > 1?

Answer 2

@JamesJ i'm supposed to do this for c = 0 and 2. do i plug 0 in for x and y, then 2? or 0 for x and 2 for y?

Answer 3

If c = 0, then you have what?
x^2 + y^2 + 1 = 0
i.e.,
x^2 + y^2 = -1

What points (x,y) satisfy that equation?

Answer 4

so c= z?

Answer 5

Set aside z for now. The question is what points (x,y) satisfy the equation f(x,y) = c for given values of c.

Start then with c = 0. What values of (x,y) satisfy the equation f(x,y) = 0?

Answer 6

i'm not quite sure how to get 2 square roots to equal -1...

Answer 7

Indeed!
\[ x^2 \geq 0 \ \ \ \ and \ \ \ \ y^2 \geq 0\] for all x and all y.

Hence are there ANY points (x,y) such that f(x,y) = 0?

Answer 8

....nooo?

Answer 9

No, none. There are no points (x,y) such that f(x,y) = 0.

What about when f(x,y) = 1?