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Math 2433 – Week 8
Popper008
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View Full Document 15.4
 The Gradient as a Normal; Tangent Lines and Tangent Planes
Suppose that
f
(
x
,
y
)
is a nonconstant function that is continuously differentiable.
That means
f
is differentiable and its gradient
∇
f
is continuous.
We saw last week that at each point in the
domain
∇
f
(if
≠
0
) points in the direction of the most rapid increase of
f
.
Also, at each point of
the domain, the gradient vector
∇
f
(if
≠
0
) is perpendicular to the level curve of
f
that passes
through that point.
The vector
Is perpendicular to the gradient so it is the
tangent vector
.
The equation of the
tangent line
is:
And the equation of the
normal line
is:
Example:
Write an equation for the tangent line and an equation for the normal line at point
P
.
For
f(x,y,z)
, the equation for the
tangent plane
is
And the
normal line
is
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View Full Document Example:
Find an equation for the tangent plane and
scalar parametric equations for the normal line at the
point
P
.
More examples:
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This note was uploaded on 07/20/2010 for the course MATH 18427 taught by Professor Etgen during the Fall '10 term at University of Houston.
 Fall '10
 Etgen

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