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Problem 12.1.13c
11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.
x=−t+6, y=3t−3; −5≤t≤5
Problem 12.1.37
37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.
A circle centered at the origin with radius 4, generated counterclockwise
Problem 12.1.41
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 segment starting at P(0, 0) and ending at Q(2, 8)
Problem 12.1.43
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
Problem 12.1.46
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)
Problem 12.1.96
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
Problem 12.1.95c
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
Problem 12.1.95b
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
Problem 12.1.95a
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
Problem 12.1.94
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.)
Problem 12.1.89d
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.
Problem 12.1.89b
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.
Problem 12.1.78
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 cos t, y = 8 sin t; slope = -1
Problem 12.1.77
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 4 cos t, y = 4 sin t; slope = 1/2
Problem 12.1.73
73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.
x=t ²−1, y=t ³ +t; t=2
Problem 12.1.71a
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 = t + 1/t, y = t − 1/t; t = 1
Problem 12.1.69a
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
Problem 12.1.49
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
Problem 12.1.52
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
Problem 12.1.55
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.
Problem 12.1.67a
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 = 2 + 4t, y = 4 − 8t; t = 2
Problem 12.1.67b
67–72. Derivatives Consider the following parametric curves.
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.
x = 2 + 4t, y = 4 − 8t; t = 2
Problem 12.1.80
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 + √t, y = 2 - 4t; slope = -8
Problem 12.1.82
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
Problem 12.1.84
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
Problem 12.1.86
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 2t sin t - t² cos t, y = 2t cos t + t² sin t; 0 ≤ t ≤ π
Problem 12.1.88
81–88. Arc length Find the arc length of the following curves on the given interval.
x = sin t, y = t - cos t; 0 ≤ t ≤ π/2
Problem 12.1.50
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 left half of the parabola y=x ² +1, originating at (0, 1)
Problem 12.1.51
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)
Problem 12.1.54
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.