Math 242 The freecalc project team

Transcription

Math 242
Sample Lecture
The freecalc project team
Reference Lectures
https://sourceforge.net/projects/freecalculus/
2016
The freecalc team
Sample Lecture
2016
Outline
1
Surfaces
Quadric Surfaces
The freecalc team
Sample Lecture
2016
Outline
1
Surfaces
Quadric Surfaces
2
Tangent Planes
The freecalc team
Sample Lecture
2016
Please view this lecture in full screen mode.
Scroll down the slides (in full screen mode) to see a
demo of the freecalc graphics and presentation.
The freecalc team
Sample Lecture
2016
Surfaces
A surface S can be given in a number of ways.
The freecalc team
Sample Lecture
2016
Surfaces
Implicit form, as a level surface:
F (x, y , z) = 0
.
The freecalc team
Sample Lecture
2016
Surfaces
F (x, y , z) = 0
.
Explicit form, as a parametric surface:
x = g(u, v )
y = h(u, v )
z = f (u, v )
The freecalc team
Sample Lecture
2016
Surfaces
F (x, y , z) = 0
.
x = g(u, v )
y = h(u, v )
z = f (u, v )
A partial case of the parametric surface is the graph surface:
x = u
z = f (x, y ) =⇒ y = v
z =
The freecalc team
Sample Lecture
2016
Surfaces
F (x, y , z) = 0
.
x = g(u, v )
y = h(u, v )
z = f (u, v )
x = u
z = f (x, y ) =⇒ y = v
z =
The freecalc team
Sample Lecture
2016
Surfaces
F (x, y , z) = 0
.
x = g(u, v )
y = h(u, v )
z = f (u, v )
x = u
z = f (x, y ) =⇒ y = v
z = f (u, v )
The freecalc team
Sample Lecture
2016
Surfaces
F (x, y , z) = 0
.
x = g(u, v )
y = h(u, v )
z = f (u, v )
x = u
z = f (x, y ) =⇒ y = v
z = f (u, v )
A graph surface z = f (x, y ) can be represented as a level surface:
z = f (x, y ) ⇐⇒ F (x, y , z) = 0
The freecalc team
for
Sample Lecture
F (x, y , z) = z − f (x, y ) .
2016
Surfaces
Quadric Surfaces
Quadratic surfaces
The level sets for second degree polynomial functions
f (x, y , z) = Ax 2 + By 2 + Cz 2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J.
are called quadratic surfaces.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Quadratic surfaces
At least one of the second-degree terms is required to be
non-zero.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Quadratic surfaces
At least one of the second-degree terms is required to be
non-zero.
The coefficients are allowed to be zero.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Canonical forms of quadratic surfaces.
Through rigid motions (translations and rotations) a quadratic surface
can be reduced to one of the two canonical forms.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
=
0,
(A, B, C) 6= (0, 0, 0).
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
=
0,
(A, B, C) 6= (0, 0, 0).
Ax 2 + By 2 + Iz = 0,
(A, B) 6= (0, 0), I 6= 0.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
=
0,
(A, B, C) 6= (0, 0, 0). These quadratics posses central symmetry:
Ax 2 + By 2 + Iz = 0,
(A, B) 6= (0, 0), I 6= 0.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
A(−x)2 + B(−y )2 + C(−z)2 + D
= 0,
= 0,
if (x, y , z) belongs to the surface, so does (−x, −y , −z).
Ax 2 + By 2 + Iz = 0,
(A, B) 6= (0, 0), I 6= 0.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
A(−x)2 + B(−y )2 + C(−z)2 + D
= 0,
= 0,
Ax 2 + By 2 + Iz = 0,
(A, B) 6= (0, 0), I 6= 0. These quadratics do not posses central
symmetry.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D
A(−x)2 + B(−y )2 + C(−z)2 + D
= 0,
= 0,
Ax 2 + By 2 + Iz = 0,
(A, B) 6= (0, 0), I 6= 0. These quadratics do not posses central
symmetry.
The canonical forms above are in addition split into sub-forms
depending on the sign of A, B, C, D, I.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Consider the surface C = {(x, y , z)|Ax 2 + By 2 + Cz 2 + D = 0}.
Let A > 0, B > 0, C > 0, D > 0.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D > 0.
Then the surface is the empty set.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D > 0.
Then the surface is the empty set.
Example:
x 2 + 2y 2 + 3z 2 + 4 = 0.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
Rescale so D = −1.
y
x
1 2
2x
z
+ 13 y 2
=
=1−
The freecalc team
z2
4
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
1
.
c2
x
1 2
2x
z
+ 13 y 2
=
=1−
The freecalc team
z2
4
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
z
+ 13 y 2
=
=1−
The freecalc team
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1 2
2x
1
,
a2
1
.
c2
z 2
c
z2
4
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
We illustrate the theory on the
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=
=1−
The freecalc team
z2
4
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=
=1−
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
The level curves are:
Sample Lecture
z2
4.
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= -3
=1−
=
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
Sample Lecture
z2
4.
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
x
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= -3
2
= 1 − z4
= − 54
The freecalc team
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
None for z < 2 and z > 2.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= -2
=1−
=
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= -2
=1−
=0
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Two points for z = ±2.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
x
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= −1
2
= 1 − z4
=
The freecalc team
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
x
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
= −1
2
= 1 − z4
= 34
The freecalc team
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Ellipses for z ∈ (−2, 2).
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=0
=1−
=
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=0
=1−
=1
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=1
=1−
=
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=1
=1−
= 34
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=2
=1−
=
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=2
=1−
=0
The freecalc team
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C > 0, D > 0
Let A > 0, B > 0, C > 0, D < 0.
y
Set A =
1
,
a2
B=
1
,
b2
C=
Surface becomes:
n
2
2
(x, y , z)| xa + yb +
x
1
.
c2
z 2
c
2
example: 12 x 2 + 13 y 2 + z4 = 1.
1 2
2x
z
+ 13 y 2
=
=1−
z2
4
Rewrite: 12 x 2 + 31 y 2 = 1 −
z2
4.
Figure is called ellipsoid.
The freecalc team
Sample Lecture
2016
o
=1 .
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
The level curves z = const are:
y
x
x 2
2
+
√y
2
z
2
The freecalc team
=
=
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
y
x
x 2
2
+
√y
2
z
2
The freecalc team
= 0
=
0
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= 0
=
0
=
(0, 0, 0)
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
1
=
=
1
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
Ellipses for z 6= 0.
y
x
2
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
1
=
=
1
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x
2
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
=
±1
=
1 = (±1)2
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= 2
=
(
2)2 = 4
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
4 x
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= 2
2)2 = 4
=
(
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
4 x
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= 2
=
( ± 2)2 = 4
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
3
=
=
(
3)2 = 9
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
3
=
3)2 = 9
=
(
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
=
±3
=
( ± 3)2 = 9
∈
ellipse
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
=
±3
=
( ± 3)2 = 9 ⇒ the ellipses are stacked along
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
For y = 0:
x 2
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= z2
=
±3
=
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
= z2
= ±z
=
±3
=
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y 1
2
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
x
= z2
= ±z
= ±2z
=
±3
=
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
2
y
4
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
x
= z2
= ±z
= ±2z
=
±3
=
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
y
6
3
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
x
= z2
= ±z
= ±2z
=
±3
=
?
.
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
y
6
3
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
x
= z2
= ±z
= ±2z
=
±3
=
lines .
ellipse
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
z
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z
y
x
A point for z = 0.
y
For y = 0:
x 2
2
x
2
x6
x 2
2
+
√y
2
z
2
⇒ (x, y , z)
The freecalc team
x
= z2
= ±z
= ±2z
=
±3
=
lines .
ellipse
⇒ The figure is a (“two-piece”) cone.
∈
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
z=
2
=
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
z =1
2
=1 = 02 + 1
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
z =1
2
=1 = 02 + 1
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
Ellipses
y
2
x 2
2
+
√y
2
x
z =1
2
=1 = 02 + 1
⇒ (x, y , z) ∈ ellipse
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z= 1
2
=2 = 1 + (
1)2
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
√
2 2
x 2
2
+
√y
2
x
z= 1
2
=2 = 1 + (
1)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
Ellipses
y
√
2 2
x 2
2
+
√y
2
x
z =±1
2
=2 = 1 + ( ± 1)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z= 2
2
=5 = 1 + (
2)2
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
√
2 5
x
x 2
2
+
√y
2
z= 2
2
=5 = 1 + (
2)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
Ellipses
y
√
2 5
x
x 2
2
+
√y
2
z =±2
2
=5 = 1 + ( ± 2)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z= 3
2
=10 = 1 + (
3)2
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
√
2 10
z= 3
2
=10 = 1 + (
3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
Ellipses for all z.
y
x
x 2
2
+
√y
2
√
2 10
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
Ellipses for all z. For y = 0:
x 2
= z2 + 1
2
y
x
x 2
2
+
√y
2
√
2 10
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
⇒ ellipses: stacked along ?
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
x 2
= z2 + 1
2
x 2
− z2 = 1
2
√
2 10
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
x 2
2
+
√y
2
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x −z = 0
2
x 2
2
+
√y
2
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
⇒ ellipses: stacked along hyperbolas.
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=10 = 1 + ( ± 3)2
Figure called: one-sheet hyperboloid.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 1
z
y
x
y
x
√
2 10
x 2
= z2 + 1
2
x 2
− z 2 = 1
2
x
x
= 1
2 −z
2 +z
1
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=10 = 1 + ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z + 2
z
x 2
= z 2 + 12
2
x 2
− z 2 = 12
2
x
x
= 12
2 −z
2 +z
1
x
2
=
x
2 −z
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z =±3
2
=
1
2
x −z = 0
2
+ ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z + 4
z
x 2
= z 2 + 14
2
x 2
− z 2 = 14
2
x
x
= 14
2 −z
2 +z
1
x
4
=
x
2 −z
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z =±3
2
=
1
4
x −z = 0
2
+ ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D < 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z + 8
z
x 2
= z 2 + 18
2
x 2
− z 2 = 18
2
x
x
= 18
2 −z
2 +z
1
x
8
=
x
2 −z
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z =±3
2
=
1
8
x −z = 0
2
+ ( ± 3)2
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z + 0
z
y
x
y
x
x 2
= z2 + 0
2
x 2
− z 2 = 0
2
x
x
= 0
2 −z
2 +z
0
x
=
2 −z
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
x −z = 0
2
z =±3
2
=
0 + ( ± 3)2
Figure called: cone.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
When z = const:
y
x
y
x
x 2
2
+
√y
2
z =
2
=
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
When z = const:
y
x
y
x
x 2
2
+
√y
2
z =
1
2
= 0 = 12 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
When z = const:
y
x
y
x
x 2
2
+
√y
2
z =
1
2
= 0 = 12 − 1
⇒ (x, y , z) = (0, 0,
The freecalc team
1)
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
When z = const: Two pts. for z = ±1.
Ellipses
y
x
x 2
2
+
√y
2
z = ±1
2
= 0 = 12 − 1
⇒ (x, y , z) = (0, 0, ± 1)
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z =
2
=
5
3
16
9
=
5
3
2
−1
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
8
3
x 2
2
+
√y
2
x
z =
2
=
5
3
16
9
=
5
3
2
−1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
8
3
x 2
2
+
√y
2
x
z = ± 53
2
5 2
= 16
−1
9 = ± 3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z =
2
=
7
3
40
9
=
7
3
2
−1
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
4
3
x 2
2
+
√y
2
√
10
x
z =
2
=
7
3
40
9
=
7
3
2
−1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
4
3
x 2
2
+
√y
2
√
10
x
z = ± 73
2
7 2
= 40
−1
9 = ± 3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
x
x 2
2
+
√y
2
z =
3
2
=8=(
3)2 − 1
⇒ (x, y , z) ∈
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
Ellipses
y
x
y
√
4 2
x
x 2
2
+
√y
2
z =
3
2
=8=(
3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
Ellipses for |z| > 1.
y
√
4 2
x
x 2
2
+
√y
2
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
Ellipses for |z| > 1. When y = 0:
x 2
= z2 − 1
2
y
√
4 2
x
x 2
2
+
√y
2
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
x 2
= z2 − 1
2
2
x
z2 − 2
= 1
√
4 2
x
x 2
2
+
√y
2
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
2
+
√y
2
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
2
+
√y
2
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
z − x2 = 0
x 2
2
+
√y
2
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
= 8 = ( ± 3)2 − 1
Figure called: two-sheet hyperboloid.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 1
z
y
x
y
√
4 2
x
x 2
= z2 − 1
2
2
x
z2 − 2 = 1
x
= 1
z − 2 x2 + z
1
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
= 8 = ( ± 3)2 − 1
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z − 2
z
x 2
= z 2 − 12
2
2
z2 − x2 = 12
= 12
z − x2 x2 + z
1
2
=
z − x2
x
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z = ±3
2
=
( ± 3)2 −
1
2
z − x2 = 0
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z − 4
z
x 2
= z 2 − 14
2
2
z2 − x2 = 14
= 14
z − x2 x2 + z
1
4
=
z − x2
x
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z = ±3
2
=
( ± 3)2 −
1
4
z − x2 = 0
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D > 0
Consider
the surface
n
o
y2
x2
1
2
C = (x, y , z)| 4 + 2 = z − 8
z
x 2
= z 2 − 18
2
2
z2 − x2 = 18
= 18
z − x2 x2 + z
1
8
=
z − x2
x
( 2 +z )
y
x
y
x
x +z = 0
2
x 2
2
+
√y
2
z = ±3
2
=
( ± 3)2 −
1
8
z − x2 = 0
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Cz 2 + D = 0, A > 0, B > 0, C < 0, D = 0
Consider
the surface
n
o
y2
x2
2
C = (x, y , z)| 4 + 2 = z − 0
z
y
x
y
x
x 2
= z2 − 0
2
2
x
z2 − 2 = 0
x
= 0
z − 2 x2 + z
0
=
z − x2
( x2 +z )
x +z = 0
2
x 2
2
+
√y
2
z − x2 = 0
z = ±3
2
=
( ± 3)2 − 0
Figure called: cone.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
=z
x
x
The freecalc team
Sample Lecture
2016
o
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
x
x 2
2
+
√y
2
z=
2
=
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
x
x 2
2
+
√y
2
z =0
2
=0
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
x
x 2
2
+
√y
2
z =0
2
=0
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
The level curves z = const are: a
point for z = 0;
x
x 2
2
+
√y
2
z =0
2
=0
(x, y , z) =(0, 0, 0)
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =1
2
=1
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =1
2
=1
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
2
x
x 2
2
+
√y
2
z =1
2
=1
⇒ (x, y , z) ∈ellipse
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =2
2
=2
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =2
2
=2
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
√
8
x
x 2
2
+
√y
2
z =2
2
=2
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =3
2
=3
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0;
x
x 2
2
+
√y
2
z =3
2
=3
⇒ (x, y , z) ∈
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
point for z = 0; ellipses for z > 0.
√
12
x
x 2
2
+
√y
2
z =3
2
=3
The freecalc team
o
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
√y
2
z =3
2
=3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
√y
2
z =3
2
=3
⇒ ellipses: stacked along parabola.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
√y
2
z =3
2
=3
Surface name: paraboloid. If A 6= B:
elliptic paraboloid.
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y2
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
√y
2
z =3
2
=3
The freecalc team
What happens if we decrease B?
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y4
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
z =3
2
=3
2
y
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
2
C = (x, y , z)| x4 + y8
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
√y
8
z =3
2
=3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
n
2
y2
C = (x, y , z)| x4 + 64
y
x
=z
o
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
z =3
2
=3
8
y
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B > 0, I 6= 0
z
Consider
the surface
o
n
2
y2
=z
C = (x, y , z)| x4 + ”∞”
y
x
x 2
For y = 0:
= z
2
z
√
12
x
x
x 2
2
+
z =3
2
=3
”∞”
y
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
A = 13 , |B| =
y
1
4
, I = −1.
x
x
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y =0
y
x2
3
x
parab.
− 0 =z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
y
0
, I = −1.
A = 13 , |B| = 41
x
Set: y =0
x2
3
x
parab.
− 0 =z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = 1
y
x2
3
x
−
( 1)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
y
− 41
, I = −1.
A = 13 , |B| = 41
x
Set: y = 1
x2
3
x
−
( 1)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
y
− 41
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 1
x2
3
x
−
(±1)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = 2
y
x2
3
x
−
( 2)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = 2
y
2
−1 = − 24
x
x2
3
−
( 2)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 2
y
2
−1 = − 24
x
x2
3
−
(±2)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = 3
y
x2
3
x
−
( 3)2
4
parab.
=z
z
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
z
x2
3
−
( 3)2
4
parab.
=z
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
z
x2
3
−
(±3)2
4
parab.
=z
y
o
=z .
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
parab.
=z
Set: x =0
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
0−
y2
4
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
parab.
=z
Set: x =0
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
0
o
=z .
y
0−
y2
4
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
Set: x =
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
( 1)2
3
−
y2
4
parab.
=z
1
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
Set: x =
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
1
3
o
=z .
y
( 1)2
3
−
y2
4
parab.
=z
1
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
parab.
=z
Set: x = ± 1
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
1
3
o
=z .
y
(±1)2
3
−
y2
4
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
Set: x =
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
( 2)2
3
−
y2
4
parab.
=z
2
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
2
− 94 = − 34
z
y
o
=z .
−
(±3)2
4
( 2)2
3
−
y2
4
parab.
=z
Set: x =
4 = 22
3
3
y2
4
x2
3
2
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
2
− 94 = − 34
z
4
y
o
=z .
−
(±3)2
4
parab.
=z
Set: x = ± 2
4 = 22
3
3
y2
x2
3
(±2)2
3
−
y2
4
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
2
− 94 = − 34
z
x2
3
−
(±3)2
4
Set: x =
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
y
o
=z .
( 3)2
3
−
y2
4
parab.
=z
3
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
y
o
=z .
x2
3
−
(±3)2
4
Set: x =
( 3)2
3
−
y2
4
parab.
=z
3
parab.
=z
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
y
o
=z .
x2
3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
−
y2
4
parab.
=z
Set: z =2
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
y
o
=z .
x2
3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
−
y2
4
parab.
=z
Set: z =2
2
− y4 =2
x2
3
y
x
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
(±3)2
4
parab.
=z
Set: x = ± 3
−
y2
4
parab.
=z
Set: z =2
y
x
The freecalc team
−
(±3)2
3
y
o
=z .
x2
3
√x −
3
Sample Lecture
y2
x2
3 − 4 =2
y
√x + y
=2
2
2
3
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =2
y
x
√x −
3
y2
x2
3 − 4 =2
y
√x + y
2
2 =2
3
√x − y
= x 2
2
3
√ +
3
The freecalc team
Sample Lecture
y
2
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =2
y
x
√x −
3
y2
x2
3 − 4 =2
y
√x + y
2
2 =2
3
√x − y
= x 2
2
3
√ +
3
y
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 =2
y
√x + y
2
2 =2
3
√x − y
= x 2
2
3
√ +
3
y
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 =2
y
√x + y
2
2 =2
3
√x − y
= x 2
2
3
√ +
3
y
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
hyperb.
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 =2
y
√x + y
2
2 =2
3
√x − y
= x 2
2
3
√ +
3
y
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
hyperb.
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =1
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 =1
y
√x + y
2
2 =1
3
√x − y
= x 1
2
3
√ +
3
y
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
hyperb.
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z =0
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 =0
y
√x + y
=0
2
2
3
two lines
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z = − 1
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 = − 1
y
√x + y
2
2=−1
3
√x − y
= x −1 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
2
(x, y , z)|x3 −
y2
4
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
o
=z .
x2
3
−
y2
4
parab.
=z
Set: z = − 2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 = − 2
y
√x + y
2
2=−2
3
√x − y
= x −2 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
o
2
2
(x, y , z)|x3 − y4 = z .
Name: hyperbolic
paraboloid.
2
− 94 = − 34
2
z
3
3= 3
x2
3
−
(±3)2
4
Set: x = ± 3
(±3)2
3
y
parab.
=z
−
y2
4
parab.
=z
Set: z = − 2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 = − 2
y
√x + y
2
2=−2
3
√x − y
= x −2 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
o
2
2
(x, y , z)|x3 − y4 = z .
Name: hyperbolic
paraboloid.
2
− 94 = − 34
2
z
3
3= 3
x2
3
−
(±3)2
4
Set: x = ± 3
(±3)2
3
y
parab.
=z
−
y2
4
parab.
=z
Set: z = − 2
x + y = 0
√
2
3
y
x
√x −
3
y2
x2
3 − 4 = − 2
y
√x + y
2
2=−2
3
√x − y
= x −2 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
o
2
2
(x, y , z)|x3 − y4 = z .
Name: hyperbolic
paraboloid.
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
−
y2
4
parab.
=z
Set: z = − 2
x + y = 0
√
2
3
y
What happens if
|B| decreases?
x2
3
x
√x −
3
y2
x2
3 − 4 = − 2
y
√x + y
2
2=−2
3
√x − y
= x −2 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
A = 13 , |B| =
y
x
x
2
− 98 = − 38
z
Surface:
C=
n
o
2
2
(x, y , z)|x3 − y8 = z .
Name: hyperbolic
paraboloid.
2
3 = 33
What happens if
|B| decreases?
−
, I = −1.
Set: y = ± 3
(±3)2
8
parab.
=z
Set: x = ± 3
y
y
x + √
√
=0
3
y8
x2
3
1
8
(±3)2
3
−
y2
8
parab.
=z
Set: z = − 2
y2
x2
3 − 8 = − 2
√x − √y
√x + √y
=−2
3
8
3
8
x
√x − √y
= x −2 y 3
8
√ +√
3
8
y
x − √
√
=0
8
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
1
, I = −1.
A = 13 , |B| = 16
x
Set:
y
=
±3
2
y
x
9 = −3
− 16
16
z
Surface:
C=
n
o
2
y2
(x, y , z)|x3 − 16
=z .
Name: hyperbolic
paraboloid.
2
3 = 33
−
(±3)2
16
parab.
=z
Set: x = ± 3
(±3)2
3
y
−
y2
16
parab.
=z
Set: z = − 2
x + y = 0
√
4
3
y
What happens if
|B| decreases?
x2
3
x
√x −
3
y2
x2
3 − 16 = − 2
y
√x + y
4
4=−2
3
√x − y
= x −2 y 4
3
√ +
3
4
x − y = 0
√
4
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
1
A = 13 , |B| = ∞
= 0, I = −1.
x
Set: y = ± 3
y
0
x2
3
x
z
Surface:
C=
n
o
2
(x, y , z)|x3 − 0 = z .
Name: hyperbolic
paraboloid.
2
3 = 33
parab.
Set: x = ± 3
y
(±3)2
3
− 0 =z
parab.
Set: z = − 2
x2
3 −0=−2
x +0 = 0
√
3
y
What happens if
|B| decreases?
0 =z
−
x
x −0 = 0
√
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Ax 2 + By 2 + Iz = 0, A > 0, B < 0, C = 0, I 6= 0
z
z
, I = −1.
A = 13 , |B| = 41
x
Set: y = ± 3
y
x
Surface:
C=
n
o
2
2
(x, y , z)|x3 − y4 = z .
Name: hyperbolic
paraboloid.
2
− 94 = − 34
2
z
3
3= 3
−
(±3)2
4
parab.
=z
Set: x = ± 3
(±3)2
3
y
−
y2
4
parab.
=z
Set: z = − 2
x + y = 0
√
2
3
y
What happens if
|B| decreases?
x2
3
x
√x −
3
y2
x2
3 − 4 = − 2
y
√x + y
2
2=−2
3
√x − y
= x −2 y 2
3
√ +
3
2
x − y = 0
√
2
3
The freecalc team
Sample Lecture
2016
Surfaces
Quadric Surfaces
Summary: surfaces of form Ax 2 + By 2 + Cz 2 + D = 0
A
B
C
>0
>0
>0
>0
>0
>0
>0
>0
>0
>0
>0
=0
>0
>0
=0
>0
>0
=0
>0
>0
<0
>0
>0
<0
>0
>0
<0
>0
=0
=0
>0
=0
=0
>0
=0
=0
Fill in the rest of the table.
The freecalc team
D
>0
=0
<0
>0
=0
<0
>0
=0
<0
>0
=0
<0
x = x0
empty
y = y0
empty
z = z0
empty
Example
x 2 + 2y 2 + 3z 2 + 4 = 0
Name
empty
ellipse
ellipse
ellipse
x 2 + 2y 2 + 3z 2 − 4 = 0
Ellipsoid
Sample Lecture
2016
Surfaces
Quadric Surfaces
Quadratics Ax 2 + By 2 + Iz = 0 (no central symmetry)
Ax 2 + By 2 + Iz = 0
A
B
x = x0
>0
>0
parabola
>0
=0
>0
<0
Fill in the rest of the table.
The freecalc team
y = y0
parabola
z = z0
ellipse, point, or empty
Sample Lecture
Example
x 2 + 2y 2 + 3z = 0
Name
Elliptic paraboloid
2016
Tangent Planes
Tangent Plane
Consider a surface S in space and a
point P on the surface.
The freecalc team
Sample Lecture
2016
Tangent Planes
Tangent Plane
How should we define the notion of “a
plane tangent to S at P” so that it
matches our geometric intuition?
The freecalc team
Sample Lecture
2016
Tangent Planes
Tangent Plane
Intuitively, it should include all
tangents at P to curves passing
through P and contained in the
surface.
The freecalc team
Sample Lecture
2016
Tangent Planes
Tangent Plane
surface.
The freecalc team
Sample Lecture
2016
Tangent Planes
Tangent Plane
surface.
The freecalc team
Sample Lecture
2016
Tangent Planes
Tangent Plane
surface.
Therefore it should be the plane
The freecalc team
passing through P;
parallel to the directions of all
tangent vectors of curves passing
surface.
Sample Lecture
2016
Tangent Planes
Tangent Plane to a Graph Surface
z
y
Graph surface z = f (x, y ), point
P(x0 , y0 , z0 ) on the surface.
x
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
Call p(x) the curve given by f (x, y ) by
keeping y = y0 constant;
x
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x
The freecalc team
keeping y = y0 constant; call q(y ) the
curve given by f (x, y ) by keeping
x = x0 constant.
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
The freecalc team
Sample Lecture
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
The freecalc team
p0 (x0 )
Sample Lecture
=
(1, 0, fx (x0 , y0 ))
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
The freecalc team
p0 (x0 )
Sample Lecture
=
(1, 0, fx (x0 , y0 ))
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
The freecalc team
p0 (x0 )
q0 (y0 )
Sample Lecture
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
Normal to tangent plane at P: n = p0 (x0 ) × q0 (y0 ) =
.
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
(1, 0, fx (x0 , y0 )) × (0, 1, fy (x0 , y0 ))
The freecalc team
Sample Lecture
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
(1, 0, fx (x0 , y0 )) × (0, 1, fy (x0 , y0 ))
The freecalc team
Sample Lecture
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
(1, 0, fx (x0 , y0 )) × (0, 1, fy (x0 , y0 )) =
The freecalc team
Sample Lecture
.
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
(1, 0, fx (x0 , y0 )) × (0, 1, fy (x0 , y0 )) = (−fx (x0 , y0 ), −fy (x0 , y0 ), 1) .
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
Equation of tangent plane at P(x0 , y0 , f (x0 , y0 )): n · (r − r0 ) = 0
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
− fx (x0 , y0 )(x − x0 ) − fy (x0 , y0 )(y − y0 ) + (z − f (x0 , y0 )) = 0
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
The freecalc team
Sample Lecture
2016
Tangent Planes
z
y
x = x0 constant.
x
p(x)
q(y )
= (x, y0 , f (x, y0 ))
= (x0 , y , f (x0 , y ))
p0 (x0 )
q0 (y0 )
=
=
(1, 0, fx (x0 , y0 ))
(0, 1, fy (x0 , y0 )) .
z = f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) .
The freecalc team
Sample Lecture
2016
Tangent Planes
Let z = f (x, y ) be a function and let (x0 , y0 ) be a point such that f
is differentiable in a small disk near (x0 , y0 ).
The freecalc team
Sample Lecture
2016
Tangent Planes
The graph of f (x, y ) is the surface S = {(x, y , f (x, y ))}.
The freecalc team
Sample Lecture
2016
Tangent Planes
Let (x, y ) be a point in the domain of f . By the vertical line test
there is exactly one point in S whose first two coordinates are x, y .
This is the point (x, y , f (x, y )).
The freecalc team
Sample Lecture
2016
Tangent Planes
Let q(t) = (x(t), y (t), z(t)) be a curve lying in S such that
x(0) = x0 and y (0) = y0 .
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
By the preceding remarks it follows that z(t) = f (x(t), y (t)).
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
Then the tangent vector of q(t) at t = 0 is
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
∂f
dy
∂f
dx
=
1, 0,
+
0, 1,
dt |t=0
∂x |t=0 dt |t=0
∂y |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
∂f
dy
∂f
dx
=
1, 0,
+
0, 1,
dt |t=0
∂x |t=0 dt |t=0
∂y |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
∂f
dy
∂f
dx
=
1, 0,
+
0, 1,
dt |t=0
∂x |t=0 dt |t=0
∂y |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
∂f
dy
∂f
dx
=
1, 0,
+
0, 1,
dt |t=0
∂x |t=0 dt |t=0
∂y |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
x(0) = x0 and y (0) = y0 .
dq
dx dy d
=
,
, (f (x(t), y (t)))
dt |t=0
dt dt dt
|t=0
dx dy ∂f dx
∂f dy
=
,
,
+
dt dt ∂x dt
∂y dt |t=0
∂f
dy
∂f
dx
=
1, 0,
+
0, 1,
dt |t=0
∂x |t=0 dt |t=0
∂y |t=0
The freecalc team
Sample Lecture
2016
Tangent Planes
q(t) = (x(t), y (t), z(t)) - smooth
curve in S = {(x, y , f (x, y ))},
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
The freecalc team
Sample Lecture
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
2016
Tangent Planes
curve in S = {(x, y , f (x, y ))},
z
y
x
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
at (x0 , y0 , z0 ) was defined as the
Recall the tangent space
space passing through (x0 , y0 , z0 ) and spanned by the tangents
of all curves lying in the surface and passing through (x0 , y0 , z0 ).
The freecalc team
Sample Lecture
2016
Tangent Planes
curve in S = {(x, y , f (x, y ))},
z
y
x
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
at (x0 , y0 , z0 ) was defined as the
Recall the tangent space
space passing through (x0 , y0 , z0 ) and spanned by the tangents
The freecalc team
Sample Lecture
2016
Tangent Planes
curve in S = {(x, y , f (x, y ))},
z
y
x
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
Recall the tangent (space) plane at (x0 , y0 , z0 ) was defined as the
(space) passing through (x0 , y0 , z0 ) and spanned by the tangents
Corollary (Justification of tangent plane definition)
The tangent vector to any curve
passing
through
(x0 , y0 , z0 ) is a linear
∂f
∂f
combination of the vectors 1, 0, ∂x and 0, 1, ∂y .
In particular the tangent space at (x0 , y0 , z0 ) is a plane.
The freecalc team
Sample Lecture
2016
Tangent Planes
curve in S = {(x, y , f (x, y ))},
z
y
x
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
passing
through
∂f
∂f
The freecalc team
Sample Lecture
2016
Tangent Planes
curve in S = {(x, y , f (x, y ))},
z
y
x
(x0 , y0 , z0 )- point in S, q(0) =
(x(0), y (0), z(0)) = (x0 , y0 , z0 ).
dq
=
dt |t=0
dx
∂f
1, 0,
dt |t=0
∂x |t=0
.
dy
∂f
+
0, 1,
dt |t=0
∂y |t=0
passing
through
∂f
∂f
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
Can a given level surface be represented as a graph surface?
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
Globally, level surfaces are cannot be represented as graph
surfaces.
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
surfaces.
Example: x 2 + y 2 + z 2 = 1: can’t solve for z globally.
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
surfaces.
2
2
2
Example:
p x + y + z = 1: can’t solve for z globally. Informally,
z = ± 1 − x 2 − y 2 , however this is not a function if we can’t
decide on choice of + or −.
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
surfaces.
2
2
2
Example:
Locally, with additional requirements, a level surface can be
represented as graph surfaces.
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
surfaces.
2
2
2
Example:
NearpP(0, 0, 1), the surface is the graph surface of
z = 1 − x 2 − y 2.
The freecalc team
Sample Lecture
2016
Tangent Planes
Question
surfaces.
2
2
2
Example:
NearpP(0, 0, 1), the surface is the graph surface of
z = 1 − x 2 − y 2.
Near P(1, 0, 0), the surface is not a graph surface w.r.t. to z.
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x, y , z)- function, let P(x0 , y0 , z0 ) - point in the domain of F .
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
We say the level surface is a graph surface around P if there is a
function z = f (x, y ) such that:
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
f is defined on an open disk D around (x0 , y0 );
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
f (x0 , y0 ) = z0 ;
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
f (x0 , y0 ) = z0 ;
F (x, y , f (x, y )) = 0 for all (x, y ) in the disk D.
The freecalc team
Sample Lecture
2016
Tangent Planes
Let F (x0 , y0 , z0 ) = k .
f (x0 , y0 ) = z0 ;
F (x, y , f (x, y )) = 0 for all (x, y ) in the disk D.
If the level surface is a graph surface, we say that the equation
F (x, y , z) = k implicitly defines z = f (x, y ) satisfying the condition
f (x0 , y0 ) = z0 .
The freecalc team
Sample Lecture
2016

Math 242 The freecalc project team

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