c Dr Oksana Shatalov, Spring 2014
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Spring 2014 Math 251
Week in Review 2
courtesy: Oksana Shatalov
(covering Sections 11.5-11.6 )
11.5: Quadric Surfaces
Key Points
Ellipsoid
Hyperboloid of one sheet
Hyperboloid of two sheets
Elliptic Cones
Elliptic paraboloid
Hyperbolic paraboloid
Elliptic cylinder
Hyperbolic cylinder
Parabolic cylinder
x2 y 2 z 2
+
+
=1
a2 b2 c2
x2 y 2 z 2
+
−
=1
a2 b2 c2
2
2
y
z2
x
− 2− 2+ 2 =1
a
b
c
x2 y 2
z2
+
= 2
a2 b2
c
2
2
x
y
z
+
=
a2 b2
c
2
2
x
y
z
−
=
a2 b2
c
2
2
x
y
+
=1
a2 b2
2
2
x
y
− 2 =1
2
a
b
y = ax2
• In order to sketch the graph of a surface determine the curves of intersection of the surface with planes
parallel to the coordinate planes. The obtained in this way curves are called traces or cross-sections of
the surface.
• Traces: horizontal ( by planes z = k), yz-traces (by x = 0) and xz-traces (by y = 0).
• To find the equation of a trace substitute the equation of the plane into the equation of the surface.
Examples
1. Find an equation of the trace of the given surface by the given plane and state whether it is
an ellipse, a parabola, or a hyperbola.
(a) y − x2 − 9z 2 = 0,
x=2
(b) 16x2 − y 2 − z 2 = 9,
z=4
c Dr Oksana Shatalov, Spring 2014
2. Sketch each of the following:
r
(a) the surface y =
1 − x2 −
2
z2
16
(b) the “ice cream
cone” bounded by the hemisphere x2 + y 2 + z 2 = 9,
p
2
cone z = x + y 2 .
z ≥ 0, and the
(c) the region bounded by the hyperboloid x2 + y 2 − z 2 = −1 and the plane z = 2.
3. The given equation represents a quadric surface whose orientation is different from that in
the above table. Identify and sketch the surface:
(a) y − x2 − 9z 2 = 0
(b) x2 − 4y 2 − 4z 2 = 0.
4. Identify the surface x2 + y 2 + z + 6x − 2y + 10 = 0 and make a rough sketch that shows its
position and orientation
c Dr Oksana Shatalov, Spring 2014
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11.6: Vector Functions and Space Curves
Key Points
• A tangent vector to the graph of r(t) = hx(t), y(t), z(t)i at point P (x(t0 ), y(t0 ), z(t0 )) is r0 (t0 ).
Examples
6. Find the domain of r(t) =
√
7t, ln |1 − t| , sin(πt) .
7. Find a vector equation for the curve of intersection of the surfaces x = y 2 and z = x in
terms of the parameter y = t.
8. Does the graph of the vector-function r(t) =
1 − t2 t + 1
,
, t lie in the plane x−y+z = −1?
t
t
9. Find the points where the curve r(t) = h1 − t, t2 , t2 i intersects the plane 5x − y + 2z = −1.
c Dr Oksana Shatalov, Spring 2014
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√
10. Show that the the curve given by r(t) = sin t, 2 cos t, 3 sin t lies on both a plane and a
sphere. Then conclude that its graph is a circle and find its radius.
11. In each item, match the equation with one of the graphs below. Explain your reasoning.
√
(a) r(t) = ht, cos(2t), sin(2t)i
(d) r(t) = t, −t, 2 − t2
(b) r(t) = hsin(πt), −t, ti
(c) r(t) = ht cos t, t sin t, ti
c Dr Oksana Shatalov, Spring 2014
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12. Find parametric equations of the line tangent to the graph of r(t) = he−t , t3 , ln ti at the
point t = 1.
13. Find a vector equation of the line tangent to the graph of r(t) =
point (4, 0, 3).
3
t ,4 − t ,−
1+t
2
2
at the
c Dr Oksana Shatalov, Spring 2014
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14. Let
r1 (t) = arctan t, t, −t4
and
sin(2πt)
2
.
r2 (t) = t − t, 2 ln t,
2π
(a) Show that the graphs of the given vector-functions intersect at the origin.
(b) Find their angle of intersection at the origin.