HARVARD MATH 21A - Final Exam Practice II (14 pages)

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Final Exam Practice II



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Final Exam Practice II

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Pages:
14
School:
Harvard University
Course:
Math 21a - Multivariable Calculus
Multivariable Calculus Documents

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8 5 2010 FINAL EXAM PRACTICE II Maths 21a O Knill Summer 2010 Name Start by printing your name in the above box Try to answer each question on the same page as the question is asked If needed use the back or the next empty page for work Do not detach pages from this exam packet or unstaple the packet Please try to write neatly Answers which are illegible for the grader can not be given credit No notes books calculators computers or other electronic aids are allowed Problems 1 3 do not require any justifications For the rest of the problems you have to show your work Even correct answers without derivation can not be given credit You have 180 minutes time to complete your work 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 Total 150 1 Problem 1 20 points 1 T F The quadratic surface x2 y z 2 5 is a hyperbolic paraboloid Solution Write it as y 5 z 2 x2 to see it better 2 T F There are vectors u and v such that u v u v Solution We have a general identity u v u v sin alpha which contradicts the claim 3 T F R 2 R 5 0 0 r d dr is the area of a disc of radius 5 Solution There seems nothing wrong with that But note that the 0 to 5 integration is paired with the d 4 T F If a vector field F x y satisfies curl F x y Qx Py 0 for all points x y in the plane then F is a gradient field Solution True We have derived this from Green s theorem 5 T F The jerk of a parameterized curve r t hx t y t z t i is parallel to the acceleration if the curve r t is a line Solution The velocity the acceleration and the jerk are all parallel on a line 6 T F The curvature of the curve r t h3 sin t 0 3 cos t i is twice the curvature of the curve s t h6 6 sin t 6 cos t 0i Solution If we scale a curve by a factor 2 its curvature is divided by 2 2 7 T F The curve r t hsin t t2 cos t i for t 0 10 is located on a cylinder Solution Indeed one can check that x t 2 z t 2 1 8 T F If a function f x y has the property that fx x y is zero for all x y then f is the constant function



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