Engineering Calculus Notes 327

Engineering Calculus Notes 327 - 3.6. EXTREMA 315 11. For...

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Unformatted text preview: 3.6. EXTREMA 315 11. For each surface defined implicitly, decide at each given point whether one can solve locally for (a) z in terms of x and y ; (b) x in terms of y and z ; y in terms of x and z . Find the partials of the function if it exists. (a) x3 z 2 − z 3 xy = 0 at (1, 1, 1) and at (0, 0, 0). (b) xy + z + 3xz 5 = 4 at (1, 0, 1) √ √ (c) x3 + y 3 + z 3 = 10 at (1, 2, 1) and at ( 3 5, 0, 3 5). (d) sin x cos y − cos x sin z = 0 at (π, 0, π ). 2 → − 12. Prove that the gradient vector ∇ f is perpendicular to the level surfaces of f , using the Chain Rule instead of Equation (3.18). 13. Mimic the argument for Theorem 3.5.3 to show that we can solve for any variable whose partial does not vanish at our point. Challenge problem: 14. Suppose P (x0 , y0 , z0 ) is a regular point of the C 1 function f (x, y, z ); → for definiteness, assume ∂f (P ) = 0. Let − be a nonzero vector v ∂z → − perpendicular to ∇ f (P ). → → (a) Show that the projection − = (v1 , v2 ) of − onto the xy -plane is w v a nonzero vector. (b) By the Implicit Function Theorem, the level set L(f, c) of f through P near P can be expressed as the graph z = φ(x, y ) of some C 1 function φ(x, y ). Show that (at least for |t| < ε for some → ε > 0) the curve − (t) = (x0 + v1 t, y0 + v2 t, φ(x0 + v1 t, y0 + v2 t)) p → lies on L(f, c), and that p ′ (0) = − . v (c) This shows that every vector in the plane perpendicular to the gradient is the velocity vector of some curve in L(f, c) as it goes → − through P , at least if ∇ f (P ) has a nonzero z -component. What → − do you need to show this assuming only that ∇ f (P ) is a nonzero vector? 3.6 Extrema Bounded Functions Recall the following definitions from single-variable calculus: ...
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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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