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 cosh v , y = b sin u cosh v , z = c sinh 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 cosh v , y = b sin u cosh v , z = c sinh 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
cosh
v
,
y
=
b
sin
u
cosh
v
,
z
=
c
sinh
v
Find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point.
Surfaces: x+ y + 2z = 3,
X = 2
Point:
(2,1,0)
Find the equations for the tangent line. Letz= - 2t.
X =
(Type an expression using t as the variable.)
y =
(Type an expression using t as the variable.)
(Type an expression using t as the variable.)
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.
Use the parameter to write parametric equations representing the given curve.
Hyperbola with center (7, 2), vertices (0, 2) and (14, 2), and foci (-18, 2) and (32, 2).
Ox=7+7tant and y = 2+24 sect
Ox=2+7 sect and y = 7+24 tant
Ox=2+7tant and y = 7+ 24 sect
Ox=7+7 sect and y = 2+24 tant
Chapter 14 Solutions
Calculus Early Transcendentals, Binder Ready Version
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