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(8/17/10)
Math 20C. Lecture Examples.
Section 14.4. Linear approximations and tangent planes
†
A function
z
=
f
(
x,y
) of two variables is
linear
if its graph in
xyz
space is a plane. Equations of planes
were found in Section 12.5 by using their normal vectors. Here we will need instead equations given in
the next theorem for planes in terms of the slopes of their cross sections in the
x
 and
y
directions.
Theorem 1(a) (The slopeintercept equation of a plane)
Suppose that the
z
intercept of a plane
is
b
, the slope of its vertical cross sections in the positive
x
direction is
m
1
, and the slope of its vertical
cross sections in the positive
y
direction is
m
2
(Figure 1). Then the plane has the equation
z
=
m
1
x
+
m
2
y
+
b.
(1)
(b) (The pointslope equation of a plane)
Suppose that a plane contains the point
(
x
0
, y
0
, z
0
)
, the
slope of its vertical cross sections in the positive
x
direction is
m
1
, and the slope of its vertical cross
sections in the positive
y
direction is
m
2
(Figure 4). Then the plane has the equation
z
=
z
0
+
m
1
(
x

x
0
) +
m
2
(
y

y
0
)
.
(2)
The slopeintercept equation
The pointslope equation
FIGURE 1
FIGURE 2
†
Lecture notes to accompany Section 14.4 of
Calculus, Early Transcendentals
by Rogawski.
1
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View Full Document Math 20C. Lecture Examples. (8/17/10)
Section 14.4, p. 2
Example 1
Give an equation of the plane with slope

6
in the positive
x
direction,
slope
7
in the positive
y
direction, and
z
intercept 10.
Answer:
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This note was uploaded on 08/31/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.
 Spring '08
 Helton
 Calculus, Equations, Approximation, Linear Approximation

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