Chapter 12.1, Problem 89E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

1. a. The equations x = −cos t, y = −sin t, for 0 ≤ t ≤ 2πt, generate a circle in the clockwise direction.
2. b. An object following the parametric curve x = 2 cos 2πt, y = 2 sin 2πt circles the origin once every 1 time unit.
3. c. The parametric equations x = cos t, y = t², for t ≥ 0, describe the complete parabola y = x².
4. d. The parametric equations x = cos t, y = sin t, for −π/2 ≤ t ≤ π/2, describe a semicircle.
5. e. There are two points on the curve x = −4 cos t, y = sin t, for 0 ≤ t ≤ 2 π, at which there is a vertical tangent line.