The parametric equations in these exercises represent a quadric surface for positive values of a , b , and c. Identify the type of surface by eliminating the parameters u and v . Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility. x = a cos u cos v , y = b sin u cos v , z = c sin v
The parametric equations in these exercises represent a quadric surface for positive values of a , b , and c. Identify the type of surface by eliminating the parameters u and v . Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility. x = a cos u cos v , y = b sin u cos v , z = c sin v
The parametric equations in these exercises represent a quadric surface for positive values of a, b, and c. Identify the type of surface by eliminating the parameters u and v. Check your conclusion by choosing specific values for the constants and generating the surface with a graphing utility.
x
=
a
cos
u
cos
v
,
y
=
b
sin
u
cos
v
,
z
=
c
sin
v
Consider the surface defined by the equation 2x2 - 2y?-z = 0.
(a) Choose any one of the variables and give it a fixed value. Then sketch the corre-
sponding trace.
you
know.
(b) Identify the surface. Very briefly explain how
Identify and sketch the surface r +y + - 2r +2 =0.
Project: S is a surface in R° with equation x² + y² – z² = 1. The point P (2,1,2) lies on this hyperboloid. It
turns out there are exactly 2 straight lines L, and L2 which pass through the point P and lie entirely inside the
surface S.
Your job: Find the parametric equations of these 2 lines and then display the surface S together with the lines L1
and L2 using computer graphics software.
Methods: Theory: For finding the equations, I suggest this approach. (You can use a different approach if you know
a better one.) The two lines can be found if you know direction vectors for them. To get the direction vector v =
(a, b, c)for L1 or L2 write down the parametric equations of a line with direction vector v and passing through P.
The condition that the line lies on S gives an equation involving t, a, b and c. Solve this equation to find two
solutions for a,b,c.
Chapter 14 Solutions
Calculus Early Transcendentals, Binder Ready Version
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