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Sec. 12.6 Tangent Planes and Differentials
In single variable calculus, at a point on a differentiable function, the tangent line to the function at that point
and the function are essentially the same. So it is with a point on a twovariable differentiable function and its
tangent plane at that point. Thus if we can find the tangent plane at a point to a surface we can use it to
estimate the value of the surface near that point.
The gradient at a point on a surface is orthogonal to the tangent plane at that point on the surface. So using the
equation of a plane:
0
0
nrr
we come up with
00
0
,,
0
fx
x
y
y
z
z
for the tangent
plane.(sec12.5)
(Implicit) The tangent plane to the level surface
(, ,) 0
Fxyz
at point
000
(, ,)
Px y z
is the plane that
passes through
P
and has the normal vector
F
(x
0
,y
0
,z
0
). So the equation of the tangent plane is
0
0
0
(,,)
0
xy z
Fxyz x x
Fxyz y y
Fxyz z z
.
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This note was uploaded on 02/11/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.
 Spring '03
 MECothren
 Calculus, Multivariable Calculus

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