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Exam Ch 13 and 14.1 - 14.4 Fall 09 Takehome

Exam Ch 13 and 14.1 - 14.4 Fall 09 Takehome - M281 Ch 13...

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M281: Ch 13 and 14.1 – 14.4 Exam (TH): Please write your name at the top on the back of the exam. You may work on this take-home portion of the exam with your classmates but no-one else. You must support your conclusions with a sufficient amount of appropriate work. The correct result without the correct work/explanation is worth zero points. 1) Consider a vector function ( 29 ( 29 ( 29 , , v f t g t h t = v . Is it true that d d v v dt dt = v v ? Why or why not? 2) Show that the vector derivative for the helix cos , sin , v a t a t bt = v makes a constant angle with the z -axis. What is the angle if cos , sin , 3 v t t t = v ? 1
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3) Give a parametric representation for the intersection of the cylinder 2 2 2 x y + = and the plane 2 2 x z - = . 4) Suppose that a point moves along the surface of a sphere. Show that its velocity vector is always perpendicular to its position vector. 2
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5) Consider the ellipse ( 29 2cos , 3sin r t t t = v . Where is the curvature maximal? Where is the curvature minimal? Give reasons for your answers.
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