14.4 - 14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS KIAM...

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Unformatted text preview: 14.4: TANGENT PLANES AND LINEAR APPROXIMATIONS KIAM HEONG KWA 1. Tangent Planes Suppose f ( x,y ) is continuously differentiable in the sense that it has continuous partial derivatives. Then an equation of the tangent plane to the surface S defined by the equation z = f ( x,y ) at the point P ( x ,y ,z ) is (1.1) z- z = f x ( x ,y )( x- x ) + f y ( x ,y )( y- y ) . By definition, the tangent plane at the point P is the plane that con- tains all possible tangent lines at P to curves 1 that lie on the surface S and pass through P . We use this to construct (1.1) in the following. Recall that we can represent a curve C as a vector function r ( t ) = h x ( t ) ,y ( t ) ,z ( t ) i 2 . For our purpose, we suppose that r ( t ) = h x ,y ,z i and z ( t ) = f ( x ( t ) ,y ( t )) for all parameter values t close to t for some t , so that C lies on the surface S and passes through the point P . Recall also that r ( t ), the derivative of r ( t ) with respect to the parameter t , is tangent to the curve C at the point whose position vector is r ( t ) 3 . In particular, r ( t ) is tangent to the curve at P . On the other hand, we will learn in the next section that z ( t ) = f x ( x ( t ) ,y ( t )) x ( t ) + f y ( x ( t ) ,y ( t )) y ( t ) . The upshot is: for any space curve C , defined by a vector function r ( t ) = h x ( t ) ,y ( t ) ,z ( t ) i , that lies on the surface S and passes through the point P when t = t for some parameter value t , the vector r ( t ) = h x ( t ) ,y ( t ) ,f x ( x ,y ) x ( t ) + f y ( x ,y ) y ( t ) i is tangent to the curve C at the point P and is thus parallel to the tangent plane at P . Date : October 1, 2010. 1 We assume that all curves are smooth. 2 Review section 13.1 if necessary. 3 Refer to section 13.2 if necessary. 1 2 KIAM HEONG KWA Suppose C 1 is a curve that lies on the surface S and passes through the point P . In addition, we also suppose that C 1 lies on the plane y = y . Then C 1 can be represented by a vector function r 1 ( t ) = h x 1 ( t ) ,y ,z 1 ( t ) i . Now if the curve C 1 passes through the point P when t = t 1 , then we know from the preceding paragraph that the vector r 1 ( t 1 ) = h x 1 ( t 1 ) , ,f x ( x ,y ) x 1 ( t 1 )) i is parallel to the tangent plane at P . Similarly, if C 2 is a curve that lies on the surface S as well as on the plane x and passes through the point P , then its representative vector function r 2 ( t ) = h x ,y 2 ( t ) ,z 2 ( t ) i gives rise to the vector r 2 ( t 2 ) = h ,y 2 ( t ) ,f y ( x ,y ) y 2 ( t 2 )) i that is parellel to the tangent plane at...
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14.4 - 14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS KIAM...

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