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Math32A Topics for second Midterm

Math32A Topics for second Midterm - Uses(important item on...

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Topics for second Midterm, Monday March 12 2012 Mathematics 32A Functions of several variables f(x,y) , f(x,y,z) Partial derivatives: definition, notation ,first derivative, second partial derivatives Equality of mixed partials Gradient of functions Directional derivatives, gradient as direction of fastest increase Tangent plane to level surface , tangent plane to z= f(x,y) Critical points: where all first partials vanish Local max and min occur only at critical points(or on boundary of region) Applications: inequalities e.g. if x, y, and z are non-negative then Cube root of xyz is less than or equal to (1/3) (x+y+z) (arithmetic mean is greater than or equal to “geometric mean”) Test for local max or min at a critical point using second derivatives[two variable functions only] Derivation via looking at second derivative along lines through the point)
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Unformatted text preview: Uses(important item on its own) Second derivative along lines in terms of partial derivative of the function Also, second derivatives along curves in general(get extra term from “acceleration”—involves curvature, cf. homework problem and solution handed out) Lagrange multipliers: method of finding local max or min points of function G of two or three(or more) variables with variable subject to a “Constraint” F=0. Condition: Local max or min subject to the constraint implies that At the point grad F and grad G have the same direction(assume grad F is not 0, then grad G has to be a multiple of grad F). Why this is true! Laplacian in polar coordinates and changing second derivatives in one coordinate system to another....
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