Calculus: Early Transcendentals
James Stewart
Educators
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases.
$ x = 1 - t^2 $, $ \quad y = 2t - t^2 $, $ \quad -1 \leqslant t \leqslant 2 $
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases.
$ x = t^3 + t $, $ \quad y = t^2 + 2 $, $ \quad -2 \leqslant t \leqslant 2 $
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases.
$ x = t + \sin t $, $ \quad y = \cos t $, $ \quad -\pi \leqslant t \leqslant \pi $
Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $ t $ increases.
$ x = e^{-a} + t $, $ \quad y = e^a - t $, $ \quad -2 \leqslant t \leqslant 2 $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = 2t - 1 $, $ \quad y = \dfrac{1}{2}t + 1 $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = 3t + 2 $, $ \quad y = 2t + 3 $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = t^2 - 3 $, $ \quad y = t + 2 $, $ \quad -3 \leqslant t \leqslant 3 $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = \sin t $, $ \quad y = 1 - \cos t $, $ \quad 0 \leqslant t \leqslant 2\pi $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = \sqrt{t} $, $ \quad y = 1 - t $
(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
$ x = t^2 $, $ \quad y = t^3 $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \sin\dfrac{1}{2}\theta $, $ \quad y = \cos\dfrac{1}{2}\theta $, $ \quad -\pi \leqslant \theta \leqslant \pi $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \dfrac{1}{2}\cos\theta $, $ \quad y = 2\sin\theta $, $ \quad 0 \leqslant \theta \leqslant \pi $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \sin t $, $ \quad y = \csc t $, $ \quad 0 \leqslant t \leqslant \pi/2 $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = e^t $, $ \quad y = e^{-2t} $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = t^2 $, $ \quad y = \ln t $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \sqrt{t + 1} $, $ \quad y = \sqrt{t - 1} $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \sinh t $, $ \quad y = \cosh t $
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.
$ x = \tan^2 \theta $, $ \quad y = \sec \theta $, $ \quad -\pi/2 < \theta < \pi/2 $
Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.
$ x = 5 + 2\cos \pi t $, $ \; y = 3 + 2 \sin \pi t $, $ \; 1 \leqslant t \leqslant 2 $
Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.
$ x = 2 + \sin t $, $ \; y = 1 + 3\cos t $, $ \; \pi/2 \leqslant t \leqslant 2\pi $
Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.
$ x = 5\sin t $, $ \; y = 2\cos t $, $ \; -\pi \leqslant t \leqslant 5\pi $
Describe the motion of a particle with position $ (x, y) $ as $ t $ varies in a given interval.
$ x = \sin t $, $ \; y = \cos^2 t $, $ \; -2\pi \leqslant t \leqslant 2\pi $
Suppose a curve is a given by the parametric equations $ x = f(t) $, $ y = g(t) $, where the range of $ f $ is $ [1, 4] $ and the range of $ g $ is $ [2, 3] $. What can you say about the curve?
Match the graphs of the parametric equations $ x = f(t) $ and $ y = g(t) $ in (a)-(d) with the parametric curves labeled I-IV. Give reasons for your choices.
Use the graphs of $ x = f(t) $ and $ y = g(t) $ to sketch the parametric curve $ x = f(t) $, $ y = g(t) $. Indicate with arrows the direction in which the cuve is traced as $ t $ increases.
Use the graphs of $ x = f(t) $ and $ y = g(t) $ to sketch the parametric curve $ x = f(t) $, $ y = g(t) $. Indicate with arrows the direction in which the cuve is traced as $ t $ increases.
Use the graphs of $ x = f(t) $ and $ y = g(t) $ to sketch the parametric curve $ x = f(t) $, $ y = g(t) $. Indicate with arrows the direction in which the cuve is traced as $ t $ increases.
Match the parametric equations with the graphs labeled I-VI. Give reasons for your choices. (Do not use a graphing device.)
(a) $ x = t^4 - t + 1 $, $ \; y = t^2 $
(b) $ x = t^2 - 2t $, $ \; y = \sqrt{t} $
(c) $ x = \sin 2t $, $ \; y = \sin(t + \sin 2t) $
(d) $ x = \cos 5t $, $ \; y = \sin 2t $
(e) $ x = t + \sin 4t $, $ y = t^2 + \cos 3t $
(f) $ x = \dfrac{\sin 2t}{4 + t^2} $, $ \; y = \dfrac{\cos 2t}{4 + t^2} $
Graph the curves $ y = x^3 - 4x $ and $ x = y^3 - 4y $ and find their points of intersection correct to one decimal place.
(a) Show that the parametric equations
$ x = x_1 + (x_2 - x_1)t $ $ \quad y = y_1 + (y_2 - y_1)t $
where $ 0 \leqslant t \leqslant 1 $, describe the line segment that joins the points $ P_1(x_1, y_1) $ and $ P_2(x_2, y_2) $.
(b) Find parametric equations to represent the line segment from $ (-2, 7) $ to $ (3, -1) $.
Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices $ A(1, 1) $, $ B(4, 2) $, and $ C(1, 5) $.
Find parametric equations for the parth of a particle that moves along the circle $ x^2 + (y - 1)^2 = 4 $ in the manner described.
(a) Once around clockwise, starting at $ (2, 1) $.
(b) Three times around counterclockwise, starting at $ (2, 1) $.
(c) Halfway around counterclockwise, starting at $ (0, 3) $.
(a) Find parametric equations for the ellipse $ x^2/a^2 + y^2/b^2 = 1 $. [Hint: Modify the equations of the circle in Example 2.]
(b) Use these parametric equations to graph the ellipse when $ a = 3 $ and $ b = 1 $, $ 2 $, $ 4 $, and $ 8 $.
(c) How does the shape of the ellipse change as $ b $ varies?
Compare the curves represented by the parametric equations. How do they differ?
(a) $ x = t^3 $, $ \; y = t^2 $ $ \quad $ (b) $ x = t^6 $, $ \; y = t^4 $
(c) $ x = e^{-3t} $, $\; y = e^{-2x} $
Compare the curves represented by the parametric equations. How do they differ?
(a) $ x = t $, $ \; y = t^{-2} $ $ \quad $ (b) $ x = \cos t $, $ \; y = \sec^2 t $
(c) $ x = e^t $, $\; y = e^{-2t} $
Let $ P $ be a point at a distance $ d $ from the center of a circle of radius $ r $. The curve traced out by $ P $ as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with $ d = r $. Using the same parameter $ \theta $ as for the cycloid, and assuming the line is the $ x $-axis and $ \theta = 0 $ when $ P $ is at one of its lowest points, show that parametric equations of the trochoid are
$$ x = r\theta - d \sin \theta \quad y = r - d \cos \theta $$
Sketch the trochoid for the cases $ d < r $ and $ d > r $.
If $ a $ and $ b $ are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point $ P$ in the figure, using the angle $ \theta $ as the parameter. Then eliminate the parameter and identify the curve.
If $ a $ and $ b $ are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point $ P $ in the figure, using the angle $ \theta $ as the parameter. The line segment $ AB $ is tangent to the larger circle.
A curve, called a witch of Maria Agnesi, consists of all possible positions of the point $ P $ in the figure. Show that the parametric equations for this curve can be written as
$$ x = 2a\cot \theta \quad y = 2a \sin^2 \theta $$
Sketch the curve.
(a) Find parametric equations for the set of all points $ P $ as shown in the figure such taht $ | OP | = | AB | $. (This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube.)
(b) Use the geometric description of the curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.
Suppose that the position of one particle at time $ t $ is given by
$$ x_1 = 3 \sin t \quad y_1 = 1 + \sin t \quad 0 \leqslant t \leqslant 2 \pi $$
and the position of a second particle is given by
$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$
(a) Graph the paths of both particles. How many points of intersection are there?
(b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points.
(c) Describe what happens if the path of the second particle given by
$$ x_2 = 3 + \cos t \quad y_2 = 1 + \sin t \quad 0 \leqslant t \leqslant 2\pi $$
If a projectile is fired with an initial velocity of $ v_0 $ meters per second at an angle $ \alpha $ above the horizontal and air resistance is assumed to be negligible, the its position after $ t $ seconds is given by the parametric equations
$$ x = (v_0 \cos \alpha) t \quad y = (v_0 \sin \alpha)t - \dfrac{1}{2}gt^2 $$
where $ g $ is the acceleration due to gravity $ (9.8 m/s^2) $.
(a) If a gun is fired with $ \alpha - 30^{\circ} $ and $ v_0 = 500\;m/s $, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet?
(b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle $ \alpha $ to see where it hits the ground. Summarize your findings.
(c) Show that the path is parabolic by eliminating the parameter.
Investigate the family of curves defined by the parametric equations $ x = t^2 $, $ y = t^3 - ct $. How does the shape change as $ c $ increases? Illustrate by graphing several members of the family.
The swallowtail catastrophe curves are defined by the parametric equations $ x = 2ct - 4t^3 $, $ y = -ct^2 + 3t^4 $. Graph several of these curves. What features do the curves have in common? How do they change when $ c $ increases?
Graph several members of the family of curves with parametric equations $ x = t + \alpha \cos t $, $ y = t + \alpha \sin t $, where $ \alpha > 0 $. How does the shape change as $ \alpha $ increases? For what values of $ \alpha $ does the curve have a loop?
Graph several members of the family of curves $ x = \sin t + \sin nt $, $ y = \cos t + \cos nt $, where $ n $ is a positive integer. What features do the curves have in common? What happens as $ n $ increases?
The curves with equations $ x = \alpha \sin nt $, $ y = b \cos t $ are called Lissajous figures. Investigate how these curves vary when $ a $, $ b $, and $ n $ vary. (Take $ n $ to be a positive integer.)
Investigate the family of curves defined by the parametric equations $ x = \cos t $, $ y = \sin t - \sin ct $, where $ c > 0 $. Start by letting $ c $ be a positive integer and see what happens to the shape as $ c $ increases. Then explode some of the possibilities that occur when $ c $ is a fraction.
