If x x1 , x2 , , xn represents a point in a subset A of Rn, and fx is exactly one point in Rm, then

1
If x (x1 , x2 , , xn) represents a point in a
subset A of Rn, and f(x) is exactly one point in
Rm, then we say that f is a function from (a
domain in) Rn to Rm. The function f is called a
scalar-valued function if m 1
vector-valued function if m gt 1
function of a single variable if n 1
function of several variables if n gt 1
f(x) ?8 x is a
scalar-valued function of
a single variable with
domain
x x ? 8 and range
y y ? 0.
f(x1 , x2 , x3) x2/x1 ? 8 x32 is a
scalar-valued function of
three
variables with domain
(x1 , x2 , x3) x1 ? 0 , ? lt x2 lt ? , x3 ?
?8
y ? lt y lt ?.
and range
2
If x (x1 , x2 , , xn) represents a point in a
subset A of Rn, and f(x) is exactly one point in
Rm, then we say that f is a function from (a
domain in) Rn to Rm. The function f is called a
scalar-valued function if m 1
vector-valued function if m gt 1
function of a single variable if n 1
function of several variables if n gt 1
f(x) (3x 5 , x2) is a
vector-valued function of
a single variable with
domain
x ? lt x lt ? and range
(y1 , y2) y1 ? 5 , y2 ? 0.
f(x1 , x2) (x2/x1 , ? 8 x12 , x1x2) is a
vector-valued function of
two
variables with domain
(x1 , x2) 0 lt x1 ? ?8 , ? lt x2 lt ? and
(y1 , y2 , y3) ? lt y1 lt ? , 0 ? y2 lt ?8 , ?
lt y3 lt ?.
range
Definition of the graph of a single-valued
function (page 97)
Definition of level curves and level surfaces
(page 99)
3
Example 2 (page 98) z f(x,y) x y 2
y
The level curves are x y 2 c c 0 c
1 c 1
c 1
c 0
c 1
x
x y 2
x y 1
x y 3
This is the graph of
a plane.
4
Example 2 (page 98) z f(x,y) x y 2 This
is a plane.
z f(x,y)
(0, 0, 2)
(2, 0, 0)
y
(0, 2, 0)
x
5
z f(x,y)
z f(x,y) x y 2 This is a plane.
(0, 0, 2)
(2, 0, 0)
y
(0, 2, 0)
x
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z f(x,y) 3x This is a plane.
z f(x,y)
(1, 0, 3)
y
(1, 0, 3)
x
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Example 3 (page 99) z f(x,y) x2 y2
y
The level curves are x2 y2 c c 0 c 1 c
1 c 2
c 2
c 1
c 0
x
x2 y2 0
x2 y2 1
x2 y2 1
The level curve is empty.
x2 y2 2
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Example 3 (page 99) z f(x,y) x2 y2 Each
level curve resulting from letting z be a
constant c gt 0 is Each level curve resulting
from letting either x or y be a constant c
is This is the graph of
a circle of radius ?c centered at the origin.
a parabola.
a circular paraboloid (pictured in Figure 2.1.7).
Look at the Conic Sections Handout and the
Quadric Surfaces Handout to see how various
two-dimensional graphs and three-dimensional
graphs can be identified.
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Example 4 (page 100) z f(x,y) x2 y2
y
The level curves are x2 y2 c c 0 c 1 c
1
x
x2 y2 0
x2 y2 1
c 1
x2 y2 1
c 0
c 1
This is the graph of
a hyperbolic paraboloid (pictured in Figure
2.1.10).
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x2 4y2 z2 4 Each level curve resulting
from letting z be a constant c, where cgt2, is
an ellipse.
Each level curve resulting from letting z be a
constant c, where c2, is
a point.
Each level curve resulting from letting z be a
constant c, where clt2, is
no points at all.
a hyperbola.
Each level curve resulting from letting x be a
constant c is Each level curve resulting from
letting y be a constant c is
a hyperbola.
a hyperboloid of two sheets.
This is the graph of
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x2/9 y2/16 z2 1 Each level curve resulting
from letting z be a constant clt1 is
an ellipse.
an ellipse.
Each level curve resulting from letting x be a
constant clt3 is
an ellipse.
Each level curve resulting from letting y be a
constant clt4 is
an ellipsoid.
This is the graph of
x2 y2 z2 4 Each level curve resulting from
letting z be a constant c is
a circle.
a hyperbola.
Each level curve resulting from letting x be a
constant c is
a hyperbola.
Each level curve resulting from letting y be a
constant c is
a hyperboloid of one sheet.
This is the graph of
12
x2 y2 z2 0 Each level curve resulting from
letting z be a constant c gt 0 is
a circle.
The level curve resulting from letting z be the
constant 0 is
the point (0 , 0).
Each level curve resulting from letting x be a
constant c gt 0 is
a hyperbola.
The level curve resulting from letting x be the
constant 0 is
two straight perpendicular lines.
Each level curve resulting from letting y be a
constant c gt 0 is
a hyperbola.
The level curve resulting from letting y be the
constant 0 is
two straight perpendicular lines.
right cone.
This is the graph of
13
z y2 In the yz plane, this graph is In R3,
this graph (extended parallel to the x axis) is
a parabola.
a right parabolic cylinder.
x2 y2 25 In the xy plane, this graph is In
R3, this graph (extended parallel to the z axis)
is
a circle (of radius 5 centered at the origin).
a right circular cylinder.
z x2 y2 4x 6y 13
(x 2)2 (y 3)2
Each level curve resulting from letting z be a
constant c gt 0 is
a circle (of radius ?c entered at (2,3)).
This is the graph of
a shifted circular paraboloid.
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Describe each graph in R3. x2 3y2 z2 11 x2
3y2 z2 0 z2 0 x2 y2 9 x2 y2 0 x2
y2 z2 10 x2 y2 z2 1 0 x2 y2 0
ellipsoid
the single point (0 , 0 , 0)
the xy plane
a circular cylinder of radius 3 centered at the
origin
the line which is the z axis
a sphere of radius ?10 centered at the origin
no points at all
two planes (x y 0 and x y 0 )
Level curves help us picture graphs in R3. It is
impossible to picture graphs in Rn for n gt 3, but
level surfaces can be used to give some insight
into graphs in R4.
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Look at each of the following
Example 5 (page 102) w f(x,y,z) x2 y2 z2
Example 6 (page 103) w f(x,y,z) x2 y2 z2