Math 251 Section 12.4
Tangent Planes and Differentials
If f ( x , y ) has continuous first partial derivative s at (a,b) then the
graph of z  f ( x , y ) has a tangent plane at (a,b). The normal to
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this tangent plane is n   f x ( a , b ) i  f y ( a , b ) j  k .
Explanation: Find the tangent lines to the curves
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z  f ( x , b ) and z  f ( a , y ) .
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u1 , u 2 be their direction vectors. Find u1 u 2 .
The equation of the tangent plane to z  f ( x , y ) at ( a , b, f ( a , b )) is
 f x ( a , b )( x  a )  f y ( a , b )( y  b )  z  f ( a , b )  0.
Examples: Find an equation of the tangent plane to the surface at the given point.
1.
z  x2  y2
(3, 2,  5)
2.
z  e xy  6
( 2,3,1)
3.
z  ln(1  x 2 y )
(1,1, ln 2 )
Differentials: The tangent plane at
close enough to
( a , b , f ( a , b )) can be used to approximate f ( x , y ) if ( x , y ) is
(a, b) .
The notation of differentials:
L ( x , y )  the point on the tangent plane at ( x , y )
dz  L ( x , y )  f ( a , b )  L ( x , y )  L ( a , b )
z  f ( x, y )  f ( a , b)
dx   x  x  a
dy   y  y  b
The equation of the tangent plane is then
The hope is that
z  dz
dz  f x ( a , b ) dx  f y ( a , b ) dy
, which is true if
f ( x , y ) is continuous and ( x , y ) is close enough to
(a, b) .
Example: Use differentials to estimate the amount of metal in a closed can that is 10cm high and 4cm in
diameter if the wall is .05cm thick.