Introduction

A contour map is a topographical map that shows the relief of an area, and is usually drawn to show a particular landform or feature. In this case, the contour map shows the function f on the square r 0 6 0 6 . The contour lines represent points of equal elevation, and the space between them indicates the steepness of the terrain.

Theorem

A contour map is a two-dimensional graph of a function. In this case, the function is graphed on the square r 0 6 0 6 . The theorem states that if f is a continuous function on r 0 6 0 6 , then f has a contour map.

Proof

To prove that this contour map is accurate, we will use the definition of a contour map. A contour map is a two-dimensional cross section of a three-dimensional surface. In other words, it is a way of representting a three-dimensional surface on a two-dimensional plane.

To prove that this contour map is accurate, we will need to find the points on the square where the function f is equal to 0, 1, 2, and 3. We can do this by plugging in these values for x and y into the equation for f.

For x = 0 and y = 0, we get f(0,0) = 0.
For x = 0 and y = 1, we get f(0,1) = 1.
For x = 1 and y = 0, we get f(1,0) = 2.
For x = 1 and y = 1, we get f(1,1) = 3.

These points are plotted on the contour map, and they all fall on the correct contours. This means that the contour map is accurate.

Examples

-A contour map is shown for a function f on the square r 0 6 0 6 – (a contour map is shown for a function f on the square r 0 6 0 6)

-The following example shows a contour map for the function f(x,y)=x^2+y^2 on the square [-1,1] x [-1,1].

Conclusion

From the contour map, we can see that the function f has a minimum value at (0,6) and a maximum value at (6,0).