Page 25 Name Caleulus III Project 3 Due 10072016 l 4 points

Page 25 Name Caleulus III: Project 3 Due 10/07/2016 l. (4 points) diagrams of surfaces each made with a constant difference between contours. Match each contour with the on pages 3 and 4 as well as an equation from the list below which could generate the surface shown. Write Below are eight contour correct 3-D surface graph the two answers for each contour in the blank provided. e) cos(z + v) ) z = cos(x)cos(y) g) z = expl-rs_ (y-1)2) + expl-z2-(w + 1)2) Contour A Contour B

Solution

Ans-

The contour at height c is given by

This is a contour only for c 0, For c > 0 it is a circle of radius
. For c = 0, it is a single point
(the origin). Thus, the contours at an elevation of c = 1, 2, 3, 4, … are all circles centered at the
origin of radius 1,
, , 2, …. The contour diagram is shown in Figure 12.40. The bowl–shaped
graph of f is shown in Figure
12.41. Notice that the graph of f gets steeper as we move further away
from the origin. This is reflected in the fact that the contours become more closely packed as we
move further from the origin; for example, the contours for c = 6 and c = 8 are closer together than
the contours for c = 2 and c = 4.

For c > 0 this is a circle, just as in the previous example, but here the radius is c instead of
. For
c = 0, it is the origin. Thus, if the level c increases by 1, the radius of the contour increases by 1.
This means the contours are equally spaced concentric circles (see Figure
12.42) which do not
become more closely packed further from the origin. Thus, the graph of f has the same constant
slope as we move away from the origin (see Figure
12.43), making it a cone rather than a bowl.