Answer: Finding Parametric EquationsFind parametric

Problem 1E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 2E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 3E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 4E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 5E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 6E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 7E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 8E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 9E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 10E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 11E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 12E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 13E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 14E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 15E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 16E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 17E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 18E Finding Cartesian from Parametric Equations Exercises give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

Problem 19E Finding Parametric Equations Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x2 + y2 = a2 a. once clockwise. b. once counterclockwise. c. twice clockwise. d. twice counterclockwise. (There are many ways to do these, so your answers may not be the same as the ones in the back of the book.)

Problem 20E Finding Parametric Equations Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the ellipse a. once clockwise. b. once counterclockwise. c. twice clockwise. d. twice counterclockwise. (As in Exercise 19, there are many correct answers.)

Problem 21E Finding Parametric Equations In Exercise, find a parametrization for the curve. the line segment with endpoints ( -1, -3) and (4, 1)

Problem 22E Finding Parametric Equations In Exercise, find a parametrization for the curve. the line segment with endpoints (-1, 3) and (3, -2)

Problem 23E Finding Parametric Equations In Exercise, find a parametrization for the curve. the lower half of the parabola x - 1 = y2

Problem 24E Finding Parametric Equations In Exercise, find a parametrization for the curve. the left half of the parabola y = x2 + 2x

Problem 25E Finding Parametric Equations In Exercise, find a parametrization for the curve. the ray (half line) with initial point (2, 3) that passes through the point ( -1, -1)

Problem 26E Finding Parametric Equations In Exercise, find a parametrization for the curve. the ray (half line) with initial point (–1, 2 )that passes through the point (0, 0)

Problem 27E Finding Parametric Equations 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 x2 + y2 = 4 four times.

Problem 28E Finding Parametric Equations Find parametric equations and a parameter interval for the motion of a particle that moves along the graph of in the following way: beginning at (0, 0) it moves to (3, 9), and then travels back and forth from (3, 9) to (–3, 9) infinitely many times.

Problem 29E Finding Parametric Equations Find parametric equations for the semicircle using as parameter the slope t = dy/dx of the tangent to the curve at (x,y)

Problem 30E Finding Parametric Equations Find parametric equations for the circle using as parameter the arc length s measured counterclockwise from the point (a, 0) to the point (x, y).

Problem 31E Finding Parametric Equations Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle ? in the accompanying figure as the parameter.

Problem 32E Finding Parametric Equations Find a parametrization for the curve with terminal point (0, 0) using the angle ? in the accompanying figure as the parameter.

Problem 33E Finding Parametric Equations Find a parametrization for the circle starting at (1, 0) and moving clockwise once around the circle, using the central angle ? in the accompanying figure as the parameter.

Problem 34E Finding Parametric Equations Find a parametrization for the circle starting at (1, 0) and moving counterclockwise to the terminal point (0, 1), using the central angle ? in the accompanying figure as the parameter.

Problem 35E Finding Parametric Equations The witch of Maria Agnesi The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius 1, centered at the point (0, 1), as shown in the accompanying figure. Choose a point A on the line y = 2 and connect it to the origin with a line segment. Call the point where the segment crosses the circle B. Let P be the point where the vertical line through A crosses the horizontal line through B. The witch is the curve traced by P as A moves along the line y = 2. Find parametric equations and a parameter interval for the witch by expressing the coordinates of P in terms of t, the radian measure of the angle that segment O A makes with the positive x-axis. The following equalities (which you may assume) will help.

Problem 36E Finding Parametric Equations Hypocycloid When a circle rolls on the inside of a fixed circle, any point P on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be x2 + y2 = a2, let the radius of the rolling circle be b, and let the initial position of the tracing point P be A (a, 0). Find parametric equations for the hypocycloid, using as the parameter the angle from the positive x-axis to the line joining the circles’ centers. In particular, if b = a/4, as in the accompanying figure, show that the hypocycloid is the asteroid

Problem 37E Finding Parametric Equations As the point N moves along the line y = a in the accompanying figure, P moves in such a way that OP = MN. Find parametric equations for the coordinates of P as functions of the angle t that the line ON makes with the positive y-axis.

Problem 38E Finding Parametric Equations Trochoids A wheel of radius a rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point P on a spoke of the wheel b units from its center. As parameter, use the angle ? through which the wheel turns. The curve is called a trochoid, which is a cycloid when b = a.

Problem 39E Distance Using Parametric Equations Find the point on the parabola closest to the point (2, 1/2). (Hint: Minimize the square of the distance as a function of t.)

Problem 40E Distance Using Parametric Equations Find the point on the ellipse closest to the point (3/ 4, 0). (Hint: Minimize the square of the distance as a function of t.)

Problem 41E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise

Problem 42E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise

Problem 43E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise

Problem 44E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise

Problem 45E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise What happens if you replace 2 with –2 in the equations for x and y? Graph the new equations and find out.

Problem 46E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise A nice curve What happens if you replace 3 with –3 in the equations for x and y? Graph the new equations and find out.

Problem 47E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise

Problem 48E GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercise