Question

i need to find the function that generates this graph with his level curves

Answer 1

Ans >

bunch of different constants cc together in a level curve plot, which is sometimes called a contour plot.

We return to the above example function f(x,y)=−x2−2y2f(x,y)=−x2−2y2. For some constant cc, the level curve f(x,y)=cf(x,y)=c is the graph of c=−x2−2y2c=−x2−2y2. As long as c<0c<0, this graph is an ellipse, as one can rewrite the equation for the level curve as

x2−c+y2−c/2=1.x2−c+y2−c/2=1.

(If cc is negative, then both denominators are positive.) For example, if c=−1c=−1, the level curve is the graph of x2+2y2=1x2+2y2=1. In the level curve plot of f(x,y)f(x,y) shown below, the smallest ellipse in the center is when c=−1c=−1. Working outward, the level curves are for c=−2,−3,…,−10c=−2,−3,…,−10.

2 1 0 2

The below graph illustrates the relationship between the level curves and the graph of the function. The key point is that a level curve f(x,y)=cf(x,y)=c can be thought of as a horizontal slice of the graph at height z=cz=c. This slice is the intersection of the graph with the plane z=cz=c.

Level curves of an elliptic paraboloid shown with graph. The graph of the function f(x,y)=−x2−2y2f(x,y)=−x2−2y2 is shown is the first panel along with a level curve plot in the second panel. The level curve f(x,y)=cf(x,y)=c is shown in red in the level curve plot, which is the same as the slice of the graph z=f(x,y)z=f(x,y) by the plane z=cz=c. You can change cc by dragging the plane slicing the graph up or down with the mouse. You can also change cc by dragging the red level curve.