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Chap14_Sec4

# Chap14_Sec4 - PARTIAL DERIVATIVES 14.4 Tangent Planes and...

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14.4 Tangent Planes and Linear Approximations PARTIAL DERIVATIVES In this section, we will learn how to: Approximate functions using tangent planes and linear functions.

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TANGENT PLANES Suppose a surface S has equation z = f ( x , y ), where f has continuous first partial derivatives. Let P ( x 0 , y 0 , z 0 ) be a point on S . Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S . square4 Then, the point P lies on both C 1 and C 2 . square4 Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2 at the point P . Then, the tangent plane to the surface S at the point P is defined to be the plane that contains both tangent lines T 1 and T 2 .
TANGENT PLANES If C is any other curve that lies on the surface S and passes through P , then its tangent line at P also lies in the tangent plane. Therefore, you can think of the tangent plane to S at P as consisting of all possible tangent lines at P to curves that lie on S and pass through P .

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Chap14_Sec4 - PARTIAL DERIVATIVES 14.4 Tangent Planes and...

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