Engineering Calculus Notes 292

Engineering Calculus Notes 292 - x = −→ x − −→ x...

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280 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION For a curve in the plane given as the graph of a di±erentiable function f ( x ), the tangent to the graph at the point corresponding to x = x 0 is the line through that point, P ( x 0 ,f ( x 0 )), with slope equal to the derivative f ( x 0 ). Another way to look at this, though, is that the tangent at x = x 0 to the graph of f ( x ) is the graph of the linearization T x 0 f ( x ) of f ( x ) at x = x 0 . We can take this as the de²nition in the case of a general graph: Defnition 3.5.1. The tangent plane at −→ x = −→ x 0 to the graph z = f ( −→ x ) of a diFerentiable function f : R 3 R is the graph of the linearization of f ( −→ x ) at −→ x = −→ x 0 ; that is, it is the locus of the equation z = T −→ x 0 f ( −→ x ) = f ( −→ x 0 ) + d −→ x 0 f ( −→ x ) where −→
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Unformatted text preview: x = −→ x − −→ x . Note that in the de²nition above we are specifying where the tangent plane is being found by the value of the input −→ x ; when we regard the graph as simply a surface in space, we should really think of the plane at ( x,y ) = ( x ,y ) as the tangent plane at the point P ( x ,y ,z ) in space, where z = f ( x ,y ). For example, consider the function f ( x,y ) = x 2 − 3 y 2 2 : the partials are ∂f ∂x = x ∂f ∂y = − 3 y so taking −→ x = p 1 , 1 2 P , we ²nd f p 1 , 1 2 P = 1 8 ∂f ∂x p 1 , 1 2 P = 1 ∂f ∂y p 1 , 1 2 P = − 3 2...
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