 
Summary: 14.4 Tangent Planes Linear Approximations
Tangent Planes
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What are tangent planes?
Given the surface ( , )z f x y= and a point 0 0 0( , , )P x y z on the surface.
Given the surface .( , )z f x y=
Question 5/930: Find an equation of the tangent plane to the surface
cos( )z y x y=  at .(2,2,2)
)The equation of the tangent plane at the point 0 0 0( , ,P x y z is
0 0 0 0 0 0( , )( ) ( , )(x yfz z x y x x f x y y y=  + 0)
In other words, an equation of the tangent plane to the graph of a
function ( , )f x y of two variable at the point ( , is, ( , ))a b f a b
( , ) ( , )( ) ( , )( )x yf a b fz a b x a f a b= +
f has
continuous
partial
derivatives
y b + 
Linear Approximations of z=f(x,y)
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