ExamSheet - Vectors and the Geometry of Space Angle between...

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Click to edit Master subtitle style 5/11/09 Distance from point S to the line through P parallel to v : Vectors and the Geometry of Space Angle between two vectors: Projection of u onto v : Triple scalar product of u , v , and w : Distance from point S to the line through P normal to n : Vector-Valued Functions Definition of limit: Arc length from t = a to t = b : Arc length parameter: Unit tangent vector: t Partial Derivatives Definition of limit: Implicit differentiation: F ( x , y ) = 0 F ( x , y,z ) = 0 Directional derivative of f in the direction of unit vector u at P 0: Partial Derivatives, continued Change in f in the direction of u , moving a distance ds : Standard linear approximation of f ( x , y ) at ( x 0, y 0): Error in linear approximation: where M is un upper bound for | fxx |, | fyy |, and | fxy |. Total differential of f moving from ( x 0, y 0) to ( x 0 + dx , y 0 + dy ): Exteme points of f ( x , y ): If ( a , b ) is a critical point of f , then fx(a,b) = fy(a,b) = 0 Second derivative test for local extrema: local max local min saddle point ? Lagrange Multipliers: Finding extreme values of f , subject to one constraint, g ( x , y , z
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