2. for the following functions, sketch a contour map with at least five level curves, with evenly spaced…

Explanation:

Step1: Recall the concept of contour - maps

Contour maps are obtained by setting (z = k) (constant) and finding the curves in the (xy) - plane.

(a) (z=x^{2}+y^{2})

Set (z = k), then (k=x^{2}+y^{2}). This is the equation of a circle in the (xy) - plane with radius (r = \sqrt{k}) ((k\geq0)). When (k = 0), the circle is a single point ((0,0)). When (k = 1), (r = 1); when (k = 4), (r = 2); when (k=9), (r = 3); when (k = 16), (r = 4). The contour - curves are concentric circles centered at the origin.

(b) (z=\sqrt{x^{2}+y^{2}})

Set (z = k) ((k\geq0)). Then (k=\sqrt{x^{2}+y^{2}}), or (x^{2}+y^{2}=k^{2}). These are also circles in the (xy) - plane with radius (r = k). For (k = 0), it's the point ((0,0)), for (k = 1), (r = 1), for (k = 2), (r = 2), for (k=3), (r = 3), for (k = 4), (r = 4).

(c) (z = 3x-y)

Set (z=k), then (y=3x - k). These are straight - lines in the (xy) - plane with slope (m = 3) and (y) - intercept (b=-k). When (k = 0), (y = 3x); when (k = 1), (y=3x - 1); when (k=-1), (y = 3x+1); when (k = 2), (y=3x - 2); when (k=-2), (y=3x + 2).

To sketch:

• For (z=x^{2}+y^{2}), draw concentric circles centered at the origin with radii (0,1,2,3,4).
• For (z=\sqrt{x^{2}+y^{2}}), draw concentric circles centered at the origin with radii (0,1,2,3,4).
• For (z = 3x - y), draw straight - lines with slope (3) and (y) - intercepts (0,1, - 1,2,-2).

Answer:

Sketch concentric circles for (a) and (b) and straight - lines for (c) as described above.