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Ten problems from an examination in multivariable calculus taught by mr. Haines. The problems cover topics such as jacobian matrices, derivatives, tangent planes, divergence, curl, integrals, taylor polynomials, directional derivatives, and line integrals.
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March 12 2010 Multivariable CalculusMathematics 206 Mr. Haines Examination #
(10)I. Suppose F : with rule and that G : with rule. Use the chain rule to: A. calculate the Jacobian matrix of the function G ο F at the point (2, 1).
B. calculate the derivative of the function G ο F at the point (2, 1).
C. calculate the Jacobian matrix of the function F ο G at 1.
D. calculate the derivative of the function F ο G at 1.
(10) II. Find the equation of the tangent plane at the point (0, -1, 2) to the surface whose equation is.
(10) III. Suppose that F : with rule. A. Calculate div F
B. Calculate curl F
(10) V. Let. A. Calculate the First Taylor polynomial for f at a = (1, 2).
B. Calculate the Second Taylor polynomial for f at a = (1, 2).
(10) VI. Evaluate:.
(10) VII. Suppose and a = (1, 1, 1). Calculate the directional derivative of f at a in the direction parallel to x = (1, 2, 2)
(10) X. Evaluate the line integral where if A) C is the straight line segment from (0,0,0) to (1,1,1).
B) C is the straight line segment from (1,1,1) to (0,0,0).