Given the function f(x,y)=y/(x^2+y^2), give and identify the level curves for k=1, k=1/2, and k=1/4, and draw a contour map showing these level curves. Any help would be immensely appreciated. Thanks

Just set y/(x^2 + y^2) = k

y = kx^2 + ky^2
x^2 + y^2 - y/k = 0
x^2 + (y - 1/2k)^2 = 1/4k^2

That is a circle with center at (0,1/2k) with diameter 1/k.

To find the level curves of a function, we need to set the function equal to a constant value (k) and solve for y in terms of x. Then, we can plot these level curves on a coordinate plane to create a contour map.

Let's find the level curves for k = 1, k = 1/2, and k = 1/4 one by one.

1. For k = 1:
f(x, y) = 1
y / (x^2 + y^2) = 1

Multiplying both sides by (x^2 + y^2), we get:
y = x^2 + y^2

Rearranging the terms:
y - y^2 = x^2

To simplify, we can substitute z = y - y^2:
z = x^2

Now, we have the equation of a parabola (z = x^2).

2. For k = 1/2:
f(x, y) = 1/2
y / (x^2 + y^2) = 1/2

Multiplying both sides by (x^2 + y^2), we get:
y = (1/2)(x^2 + y^2)

Rearranging the terms:
y - (1/2)y^2 = (1/2)x^2

To simplify, we can substitute z = y - (1/2)y^2:
z = (1/2)x^2

Now, we have the equation of a parabola (z = (1/2)x^2).

3. For k = 1/4:
f(x, y) = 1/4
y / (x^2 + y^2) = 1/4

Multiplying both sides by (x^2 + y^2), we get:
y = (1/4)(x^2 + y^2)

Rearranging the terms:
y - (1/4)y^2 = (1/4)x^2

To simplify, we can substitute z = y - (1/4)y^2:
z = (1/4)x^2

Now, we have the equation of a parabola (z = (1/4)x^2).

To draw the contour map showing these level curves, we need to plot equations z = x^2, z = (1/2)x^2, and z = (1/4)x^2 on a coordinate plane.

The x-axis represents the x-values, and the y-axis represents the z-values. The equations above show how the z-values change with x. By plotting these equations, we can visualize the level curves.