16. Parametric Equations & Polar Coordinates

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 segment of the parabola y=2x ²−4, where −1≤x≤5

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 horizontal line segment starting at P(8, 2) and ending at Q(−2, 2)

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.

a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.

An ellipse centered at (-2, -3) with major and minor axes of lengths 30 and 20, parallel to the x- and y-axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)

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

d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

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

b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = cos t, y = 8 sin t; t = π/2

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 line that passes through the points P(1, 1) and Q(3, 5), oriented in the direction of increasing x

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 lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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.

A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.

10–12. Parametric curves

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

x = t² + 4, y = -t, for -2 < t < 0; (5, 1)

7–8. Parametric curves and tangent lines

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

x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6