Hello, I have a problem that says 'graph the level curve of f(x,y) = sqrt(4 - x^2 - y^2) at c = 0, 1, and 2.'
I graphed these and checked my work with the answer in the back of the book. For c=0, I had a circle centered at the origin with radius 2, which I thought would be the base of the hemisphere. However, the book indicated there would not be a level curve at c=0 at all. I don't understand why this is.
Any help is appreciated. Thank you!
To understand why the level curve for c=0 does not exist, let's first review the concept of a level curve.
In mathematics, a level curve represents the set of points (x, y) in a function's domain where the function takes on a constant value. In other words, all the points on a level curve have the same function value. Graphing the level curve involves plotting these points on a coordinate plane.
Now let's consider the function f(x, y) = sqrt(4 - x^2 - y^2). To graph the level curve for c=0, we need to find all the points (x, y) where f(x, y) = 0.
Substituting c=0 into the function, we get:
0 = sqrt(4 - x^2 - y^2)
To determine if there are any solutions to this equation, we can square both sides to eliminate the square root:
0^2 = (sqrt(4 - x^2 - y^2))^2
0 = 4 - x^2 - y^2
x^2 + y^2 = 4
The equation x^2 + y^2 = 4 represents a circle centered at the origin with radius 2. So, based on your work, you correctly found a circle for c=0.
However, it seems there might be a misunderstanding in the problem statement or in the book's answer. Given the function f(x, y) = sqrt(4 - x^2 - y^2), there should indeed be a level curve for c=0, represented by the circle you found.
To confirm this, double-check the problem statement and make sure you accurately graphed the level curve. If you are still uncertain, consider seeking clarification from your instructor or referring to additional resources for verification.