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2220hw8 - where x = y 4 Let c t be a path in R 3 that does...

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Math 2220 Problem Set 8 Spring 2010 These are problems based on sections 3.1 and 3.2 of the book. 1. Consider the curve parametrized by c ( t ) = (2 cos t, 3 sin t ) (a) Verify that this curve is an ellipse by finding an equation in x and y that is satisfied by all points on the curve. (b) At the point on the curve where t = π/ 3 find the velocity vector, the speed, and the tangent line (in parametrized form). (c) Find the points on the curve where the tangent line has slope - 1. 2. Find the tangent line to the curve c ( t ) = (4 cos t, - 3 sin t, 5 t ) at the point where t = π/ 3. 3. (a) Find a parametrization c ( t ) = ( x ( t ) , y ( t )) for the curve defined by the polar equa- tion r = sin 2 θ (a four-leaf rose). Hint: let θ = t . (b) Find the velocity vector and the speed at the point on this curve in the first quadrant
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Unformatted text preview: where x = y . 4. Let c ( t ) be a path in R 3 that does not pass through the origin. If c ( t ) is a point on the path that is closest to the origin, show that the position vector c ( t ) is orthogonal to the velocity vector c ′ ( t ). 5. ±ind the length of the curve c ( t ) = ( t, (2 / 3) t 3 / 2 ), 0 ≤ t ≤ 8. 6. ±ind the point on the curve c ( t ) = (5 sin t, 5 cos t, 12 t ) that is a distance of 26 π units along the curve from the point c (0). 7. The path c ( t ) = ( a cos 3 t, a sin 3 t ) traces out a curve known as an astroid. Sketch this curve and compute its total length. (This is exercise 7 in section 3.2.)...
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