{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2220hw8

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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.)...
View Full Document

{[ snackBarMessage ]}