plugin-ReviewProbEx1

# plugin-ReviewProbEx1 - -1 + sin t, 1 + cos t ) for t ∈ [0...

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Math 255 Winter 2010 Review problems for exam 1 The exam covers sections 11.1-11.4, 13.1-13.7, 14.1-14.4. Problem 1 : Compute lim x 0 h cos x - 1 x 2 , sin 2 x x , e x - 1 x i . 1

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Problem 2 : Find the osculating and normal planes of the curve ~ r ( t ) = h e t ,e t sin t,e t cos t i at (1 , 0 , 1). 2
Problem 3 : Identify each of the following surfaces: 1. x 2 + y 2 + 9 z 2 = 36. 2. z 2 = x 2 + y 2 . Let V be the set of points ( x,y,z ) contained between the inside x 2 + y 2 +9 z 2 = 36 and outside z 2 = x 2 + y 2 with z 0. Find a description of V using spherical and cylindrical coordinates. 3

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Problem 4 : (10 pts) Show that for vectors ~x = h x 1 ,x 2 ,x 3 i and ~ y = h y 1 ,y 2 ,y 3 i , | ~x + ~ y || ~x - ~ y | ≤ | ~x | 2 + | ~ y | 2 where | ~x | denotes the magnitude of the vector ~x . 4
Problem 5 : Consider parametric curves (cos t, sin t ) and (

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Unformatted text preview: -1 + sin t, 1 + cos t ) for t ∈ [0 , 2 π ]. 1. Find the equations in x and y representing them and sketch the curves. 2. Each parametric curve represents a motion of an object on each curve. If consider t as the time, will these two objects collide? Explain you answer. 5 Problem 6 : Sketch the graph of r 2 = cos(2 θ ) in polar coordinates. Find the area enclosed by the curve. 6 Problem 7 : Consider the space curve r ( t ) = < sin t,t, cos t > for t ∈ [-π,π ]. 1. Compute the arc length. 2. Compute the curvature. 7...
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## This note was uploaded on 12/31/2011 for the course MATH 255 taught by Professor Jackwaddell during the Fall '08 term at University of Michigan.

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plugin-ReviewProbEx1 - -1 + sin t, 1 + cos t ) for t ∈ [0...

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