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Unformatted text preview: MA\PH 607 Howell 10/14/2011 Homework Handout VII A. Two coordinate systems for the plane are sketched below. On each, plot the points e f & ? @ & & & ! ! " # &" &# ? @ , and . Also, sketch the curve (well as you can) . B. Sketch the curve traced out by each given below B > . (Assume, as appropriate, that and are, respectively, the standard e f e f B C 3 9 Cartesian and polar coordinate systems for two-dimensional Euclidean space, and that e f e f B C D < and are, respectively, the standard Cartesian and spherical ) 9 coordinate systems for three-dimensional Euclidean space. See figures 2.4 and 2.7 on pages 116 and 125 of our text.) 1. x with & & & & & > B > C > > > &" > # # 2. x with & & & & & & > B > C > D > > > > > sin cos 3. x with & & & > > > > > 3 9 # > 1 4. x with & & > > > > & > 3 9 # 1 $ 5. x with 4 & & & > > > % > 3 9 1 > 6. x with 2 & & > <> > > % > ) 9 1 1 $ > C. Find parameterizations for the following curves. Assume Cartesian coordinates. If an orientation is indicated, be sure your parametrization is appropriate for that orientation. 1. The straight line from to where and . + , + , ! ! # $ 2. The straight line from to where and . + , + , # $ ! ! 3. The straight line from to where and . + , + , # $ ' "! 4. The straight line from to where and . + , + , # $ ) ' "! ! 5. The parabola given by . B C # page 2 6. The circle of radius 3 about the origin . 7. The circle of radius 3 about the point . : $ % D. For each of the following two curves with Cartesion parameterizations, with x & & & & & > B > C > > > &" > # # x & & & & & & > B > C > D > > > > > sin cos with , 1. Sketch the curve (again) . 2. Find in terms of , , and, if appropriate, . Also, on your sketch of each curve, . .> x i j k sketch . .> x at different points on the curve. 3. Find , the rate at which the arclength along the curve varies as varies .= .> > use your formula for . Then write out the integral that gives the length of the . .> B parametrized curve (if the integral is simple enough, evaluate it!)....
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- Summer '11