Engineering Calculus Notes 294

Engineering Calculus Notes 294 - 282 CHAPTER 3. REAL-VALUED...

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Unformatted text preview: 282 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION which we recognize as a restatement of Equation (3.22) identifying this plane as a graph: 3 1 + (x − 1) − 8 2 = T(1, 1 ) f (x, y ) . y− z= 1 2 2 These formulas have a geometric interpretation. The parameter s = x − 1 represents a displacement of the input from the base input (1, 1 ) parallel 2 to the x-axis—that is, holding y constant (at the base value y = 1 ). The 2 intersection of the graph z = f (x, y ) with this plane y = 1 is the curve 2 1 2 z = f x, which is the graph of the function z= x2 3 −; 2 8 at x = 1, the derivative of this function is dz dx = 1 ∂f ∂x 1, 1 2 =1 9 and the line through the point x = 1, z = 4 in this plane with slope 1 lies in → → →− 1 the plane tangent to the graph of f (x, y ) at (1, 2 ); the vector − 1 = − + k v ı is a direction vector for this line: the line itself is parametrized by x= y= z= 1 +s 1 2 1 8 +s which can be obtained from the parametrization of the full tangent plane by fixing t = 0. (see Figure 3.11.) Similarly, the intersection of the graph z = f (x, y ) with the plane x = 1 is the curve z = f (1, y ) ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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