Eliminate the parameter in the parametric equations x=1+sin t, y=3+2 sin t, for 0≤t≤π/2, and describe the curve, indicating its positive orientation. How does this curve differ from the curve x=1+sin t, y=3+2 sin t, for π/2≤t≤π?

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

a. A set of parametric equations for a given curve is always unique.

10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)

10–12. Parametric curves

a. Eliminate the parameter to obtain an equation in x and y.

x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The upper half of the parabola x=y ², originating at (0, 0)

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.

The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.

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

c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.

The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).