What is the radius and center of the circle r -8sintheta

Question

1. Convert the following rectangular coordinates into polar coordinates.

(a) (-3.0)

(b) (-2√3, 2)

2. Convert the following polar coordinates into rectangular coordinates.

(a) (3, 5π/4)

(b) (4, 3π)

3. What is the radius and center of the circle r = -8sinθ?

4. Convert rcosθ = 4 to rectangular coordinates.

5. Convert 2xy = x² + y² to polar coordinates.

6. Find the parametric equations for the following lines:

a. The line y = 8x + 2 from x =1 to x =4.

b. The line from the point (-1,3) to {1,7).

7. When do the lines x = 2t, y = 1 +2t and x = 4-5s, y = -9 +2s intersect?

8. Find the parametric equations for the following curves:

(a) y = 1 -x² for positive y.

(b) x² + y² = 14 with particle moving clockwise from (√14, 0) to (√14 , 0)

9. For the parametric curve given by x(t) = t⁴ - cos(t) and y(t) = e^2t - t, determine dy/dx, d²y/dx²

10. Sketch the curve of the following parametric equations:

(a) x= sint, y = t, t ∈ R

(b) x =5t - 3, y = 7t - 2 where 0 ≤ t ≤ 1

(c) x= sin(2t), y = cos(2t) where 0 ≤ t ≤ π/2

11. Let a =< -3,0,4> and b =< -1,2,3 >

(a) Calculate -3a + 4b.

(b) Find a vector of length .6 in the opposite direction of a.

(c) Fid na a unit vector in the direction of b.

(d) Determine proj_b(a) and proj_a(b).