Question

Describe the contour lines for the following functions:

1. f(x,y)=6+x−y
2. f(x,y)=y2
3. z=4−x2−y2
4. z=√x2+y2
5. f(x,y)=xy2

Answer 1

1. For f(x,y) = 6 + x - y, contour lines are straight lines with a slope of -1 in the xy-plane.

2. Contour lines for f(x,y) =
`y^2` form parabolic curves parallel to the x-axis, and for z = 4 -
`x^2 - y^2\)`, they are circles centered at the origin with decreasing radii as z decreases.

Step-by-step explanation:

1. For f(x,y) = 6 + x - y, the contour lines can be visualized as straight lines with a slope of -1 in the xy-plane. Each contour line represents points where the function has a constant value. The slope of -1 indicates that for every unit increase in x, there is a unit decrease in y, resulting in diagonal lines.

2. The function f(x,y) =
`y^2` produces contour lines that are parabolic curves parallel to the x-axis. These curves represent points where the function has a constant value of
`y^2`, creating a family of parabolas along the y-axis.

3. The contour lines for
`\(z = 4 - x^2 - y^2\)` are circular in shape and centered at the origin. As z decreases, the circles expand, depicting points where the function has a constant value. This pattern is characteristic of a surface defined by a quadratic function.

4. The function
`\(z = √(x^2 + y^2)\)` generates concentric circles centered at the origin, with the radius increasing as z increases. These circles represent points where the function has a constant value, illustrating the radial symmetry of the surface.

5. The contour lines for f(x,y) =
`xy^2\)` take the form of hyperbolas centered along the x-axis. These hyperbolas represent points where the function has a constant value, exhibiting a pattern shaped by the interaction between x and y.