Find the domain, the range, and describe the level curves for the function f( x, y) = 1 + e^( -x^2 -y^2)
this is easier to see (for me) as 1 + e^-(x^2+y^2)
clearly, the domain is all reals for both x and y.
since f(x,y) is even, its maximum is at f(0,0) = 2, and e^-(x^2+y^2) goes to 0, so the range is [1,2]
The level curves are circles, since we want f(x,y) = c
e^-(x^2+y^2) = c
e^(x^2+y^2) = c
x^2+y^2 = ln c
for suitably chosen values of c
To find the domain and range of the function f(x, y) = 1 + e^(-x^2 - y^2), we need to consider the restrictions on the variables x and y.
Domain:
The function f(x, y) is defined for all real values of x and y. Therefore, the domain of the function is (-∞, ∞) for both x and y.
Range:
The range of a function represents the set of all possible output values. In this case, since the function involves the exponential term e^(-x^2 - y^2), the smallest possible value for this term is 0 when x and y are both equal to 0. Adding 1 to this gives a minimum value of 1 for f(x, y).
As x and y increase positively or negatively, the exponential term e^(-x^2 - y^2) becomes smaller, approaching zero. Thus, as (x, y) moves away from the origin, f(x, y) approaches a maximum value of 1. Therefore, the range of the function is [1, ∞).
Level Curves:
Level curves represent curves on a graph where the function takes on a constant value. In this case, let's consider a few specific values for f(x, y):
- If f(x, y) = 1, it means that e^(-x^2 - y^2) = 0. This is not possible since the exponential term cannot be zero. Hence, there are no level curves for f(x, y) = 1.
- If f(x, y) > 1, it means that e^(-x^2 - y^2) > 0. The level curves for values greater than 1 form ellipses centered at the origin, with the size of the ellipse decreasing as the value of f(x, y) increases.
- As f(x, y) approaches infinity, the level curves become smaller and closer to the origin.
Overall, the level curves of the function f(x, y) = 1 + e^(-x^2 - y^2) are ellipses centered at the origin that increase in size as you move away from the origin and approach a maximum size as f(x, y) approaches infinity.