Transcript L6 Functions of Several Variable

QUADRIC SURFACES
MATH23
MULTIVARIABLE CALCULUS
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
•
•
•
•
Determine functions of several variables.
Identify the domain of several variables
Discuss quadric surfaces, its equations and graphs.
Differentiate various quadrics, based on equation, and graph
Functions of Several Variables
• A function f of two variables is a rule that assigns to
each ordered pair of real numbers (x,y) in a set D a
unique real number denoted by f(x,y). The set D is
the domain of f and its range is the set of values that
f takes on, that is, {f(x,y)|(x,y) an element of D}.
Example 1. Find the domain of the following functions and evaluate f(3, 2).
Domain of Functions
• Determine the Domain of the following
functions
f ( x, y)  4 x  2 y
2
f ( x, y) 
2
x  y 1
x 1
f ( x, y)  x ln(x  y)
2
Quadric Surfaces
• Is the graph traced by any quadratic or second
degree equation in three variables x, y, z.
• The general equation of a quadric surface is:
Ax2 +Bx2+Cy2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0
Cylinders
• A cylinder is a surface that consists of all lines (called
rulings) that are parallel to a given line and pass
though a given plane curve.
• A curve of two variables traced in three-dimensions
• Examples of which are:
–
–
–
–
Circular Cylinder
Elliptical Cylinder
Parabolic Cylinder
Hyperbolic Cylinder
B. CYLINDERS and SPHERE:
1. x2 + y2 = 4 z

(-2, 0, 0)

(0,-2, 0)
(0, 2, 0)

(2, 0, 0)
CIRCULAR CYLINDER
x
y
2. 4x2 + y2 = 4
z

(-1, 0, 0)
(0,-2, 0)

(1, 0, 0)
(0, 2, 0)

ELLIPTICAL CYLINDER
x
y
3. x2 = y
z
y
x
4. y2 = x
z
y
x
5. z2 = y
z
y
x
6. z2 = y – 1
z

V(0, 1, 0)
x
y
SPHERE:
(x – h)2 + (y – k)2 + (z – i)2 = r2
Ax2 + Ay2 + Az2 + Gx + Hy + Iz = J
z
r = 0 (point)
r = - (no locus)
r = + (sphere)
y
x
EXAMPLE:
Describe the locus of x2 + y2 + z2 + 2x – 4y – 8z + 5 = 0.
Sketch the graph.
SOLUTION:
x2 + 2x + 1 + y2 – 4y + 4 + z2 – 8z + 16 = –5 + 1 + 4 + 16
(x + 1)2 + (y – 2)2 + (z – 4)2 = 16
C(–1, 2, 4) and r = 4
z’

(-1, 2, 8)
z

C(-1, 2, 4)

(3, 2, 4)
x


(-1, 6, 4)
(-1,- 2, 4)
x’
(-5, 2, 4)

(-1, 2, 0)
y
y’
QUADRIC SURFACES
Common Types of Quadric Surfaces:
1. Ellipsoid
2. Hyperboloid of One Sheet
3. Hyperboloid of Two Sheets
4. Elliptic Paraboloid
5. Hyperbolic Paraboloid
6. Elliptic Cone
Ellipsoid
QUADRIC SURFACES
ELLIPSOID
2
2
2
x y z
 2  2 1
2
a b c
y
x
Hyperboloid of One Sheet
QUADRIC SURFACES
HYPERBOLOID OF ONE SHEET
x 2 y 2 z2
 2  2 1
2
a b c
Hyperboloid of Two Sheets
QUADRIC SURFACES
HYPERBOLOID OF TWO SHEETS
z2 x 2 y 2
 2  2 1
2
c a b
Elliptic Paraboloid
QUADRIC SURFACES
ELLIPTIC PARABOLOID
x2 y2
z 2  2
a b
Hyperbolic Paraboloid
QUADRIC SURFACES
HYPERBOLIC PARABOLOID
z’
y2 x2
z 2  2
b a
x’
y’
Elliptic Cone
QUADRIC SURFACES
ELLIPTIC CONE
2
2
x
y
2
z  2 2
a b
EXAMPLES: Sketch the quadric surface.
36x2+9y2+4z2=36
Solution:


1
36x  9y  4z  36
36
x 2 y 2 z2
  1
1 4 9
2
2
2
I. Intercepts :
x
y
z
x
±1
0
0
y
0
±2
0
z
0
0
±3
II. Traces :
i) xy - plane
x2 y2
let z  0 :
 1
1 4
(ellipse)
ii) xz - plane
x 2 z2
let y  0 :
 1
1 9
(ellipse)
iii) yz - plane
y 2 z2
let x  0 :
 1
4 9
(ellipse)
z
y 2 z2
 1
4 9

(0,0,3)
x2 y2
 1
1 4

(-1,0,0)
(0,-2,0)


(0,2,0)
(1,0,0)

(0,0,-3)
x
x 2 z2
 1
1 9
y

2. 16x2+36y2-9z2=144
ii) yz - plane
Solution:
y 2 z2
let x  0 :
 1
4 16
(hyperbola)
iii) xz - plane

1
16x  36y  9z  144
144
x 2 y 2 z2
  1
9 4 16
I. Intercepts :
2
2
2
x
y
z
x
±3
0
0
y
0
±2
0
z
0
0
±4i
II. Traces :
i) xy - plane
x2 y2
let z  0 :
 1
9 4
(ellipse)
x 2 z2
let y  0 :
 1
9 16
(hyperbola)
III. Sections parallel to xy - plane:
let z  4
x 2 y 2 (4)2
x2 y2
 
 1   1  1
9 4
16
9 4
 x2 y2
1
x2 y2
 1
   2 
18 8
9 4
2
(ellipse)
x2 y2
 1
18 8
z
(-4.2,0,4)

x 2 z2
 1
9 16
(0,-2.8,4)
(0,2.8,4)

(-3,0,0)

z=4

(4.2,0,4)
x’
(3,0,0)
z=-4
x
x”

(0,-2,0)

x2 y2
 1
9 4
(-4.2,0,-4)
y
y 2 z2

1
4 16


(0,-2.8,-4)

(4.2,0,-4)
(0,2,0)
y’

(0,2.8,-4)
y”
3. 4z2-4x2-y2=4
ii) yz - plane
Solution:
z2 y 2
let x  0 :
 1
1 4
(hyperbola)
iii) xz - plane


1
4z  4x  y  4
4
z2 x 2 y 2
  1
1 1 4
I. Intercepts :
2
2
2
x
y
z
x
±i
0
0
y
0
±2i
0
z
0
0
±1
II. Traces :
i) xy - plane
x2 y2
let z  0 :    1
1 4
(no trace)
z2 x 2
let y  0 :
 1
1 1
(hyperbola)
III. Sections parallel to xy - plane:
let z  3
(3)2 x 2 y 2
x2 y2
  1 9  1
1
1 4
1 4
 2 y2
1
x2 y2

 1
x   8 
4
8 32

8
(ellipse)
x2 y2
 1
8 32
z
(-2.8,0,3)

z=3
(0,-5.7,3)

(0,5.7,3)

(2.8,0,3)
(0,-5.7,-3)
x
x”

(0,0,-1)


(-2.8,0,-3)
2

2
y
z y
 1
1 4
(0,5.7,-3)
(2.8,0,-3)
y’
(0,0,1)

x’
z=-3
z2  x2  1
y”
Example
Example
Example
Example