Engineering Calculus Notes 348

Engineering Calculus Notes 348 - 336 CHAPTER 3 REAL-VALUED...

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336 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION We are looking for the extreme values of this function on the curve of intersection of two level surfaces. In principle, we could parametrize the ellipse, but instead we will work directly with the constraints and their gradients: g 1 ( x,y,z ) = x 2 + y 2 −→ g 1 = (2 x, 2 y, 0) g 2 ( x,y,z ) = x + y + z −→ g 2 = (1 , 1 , 1) . Since our curve lies in the intersection of the two level surfaces L ( g 1 , 4) and L ( g 2 , 1), its velocity vector must be perpendicular to both gradients: −→ v · −→ g 1 = 0 −→ v · −→ g 2 = 0 . At a place where the restriction of f to this curve achieves a local (relative) extremum, the velocity must also be perpendicular to the gradient of f
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