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Math 125 Exam 1 June 26, 2007 50 points possible
1. (a) (3pts) Clearly state the Cauchy-Schwarz Inequality (b) (2pts) Dene the angle between the vectors a and b in Rn . (c) (2pts) Explain why Cauchy-Schwarz is necessary to dene angles in Rn when n >
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Math 380 Exam 2 July 16, 2007 50 points possible yz 2xz xy , , . 2 + z 2 x2 + y 2 + z 2 x2 + y 2 + z 2 +y (a) (4pts) Compute the divergence of F . x2
1. Let F (x, y, z ) =
(b) (2pts) Interpret the divergence of F physically (or geometrically).
2. (a
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Math 380 Exam 3 August 1, 2007 50 points possible 1. If the density of a wire that lies along the planar curve (t) = (3t, t4 ), 0 t 1, is given by the function f (x, y ) = xy , nd the total mass of the wire.
2. Let D be a region appropriate for Gree
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Math 380 Exam 1 June 26, 2008 50 points possible 3 2 0 1. Let T (x) : R3 R3 be dened by the matrix A = 1 3 2 and let S (y) : R3 R 105 be dened by the matrix B = 4 0 2 (a) (1pt) Is S a vector-valued function? Briey explain your answer. (b) (2pts) Fin
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Math 380 Exam 2 July 16, 2008 50 points possible x at the point (1, 2). y
1. Find the second-order Taylor formula for the function f (x, y ) = (There is no need to fully simplify the formula.) 2 4 3 2. Let A = 4 1 3 . 3 3 1 (a) Dene the quadratic fo
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Math 380 Exam 3 July 30, 2008 50 points possible
1. (a) (3pts) State the conclusion to the Change of Variables Theorem for Double Integrals and explain the role of the Jacobian factor in the integrand. (b) (7pts) Determine an appropriate change of v