Find parametric equations and a parameter interval for the motion of a particle starting at the point (2, 0) and tracing the top half of the circle

$$x ^ { 2 } + y ^ { 2 } = 4$$

four times.

Find parametric equations and a parameter interval for the motion of a particle that starts at the point (-2, 0) in the xy-plane and traces the circle

$$x ^ { 2 } + y ^ { 2 } = 4$$

three times clockwise. (There are many ways to do this.)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 8 cos θ, y = 8 sin θ

If the random variable V has a noncentral $\chi^{2}$ distribution with r degrees of freedom and noncentrality parameter $\gamma,$ use its moment-generating function to find E(V).

Consider the parametric equations x= a cos t, y= a sin t, 0 ≤ t ≤ 2π Eliminate the parameter t in the parametric equations to verify that they are all circles. What is the radius?

Let x = 2 cos t, y = 2 sin t. Find a parameter interval that produces the indicated portion of the graph. The portion in the fourth quadrant, including the points on the axes.

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 6 c0s θ, y = 6 sin θ

Write the equation of the tangent line in Cartesian coordinates for the given parameter t. $x=e^{\sqrt{t}}, \quad y=1-\ln t^{2}, \quad t=1$

Consider the following parametric equations. Eliminate the parameter to obtain an equation in x and y. $x=\sin t, y=2 \sin t+1 ; 0 \leq t \leq \pi / 2$

Find the parametric equation of the line in the x–y plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. (2, 1) and (3, 5)

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $x = e ^ { t }$, $y = 1 - e ^ { 3 t }$, $0 \leq t \leq 1$

Find the arc length of the curve on the indicated interval of the parameter. $x=4 t+3, \quad y=3 t-2, \quad 0 \leq t \leq 2$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 5 sin³ θ, y = 5 cos³ θ

Use a graphing utility to graph the following curves. Be sure to choose an interval for the parameter that generates all features of interest. x=2 cot t, y=1-cos 2t