Explanation Given: r ( t ) = 〈 cos t , cos 2 t , sin t 〉 Formula used: sin x = sin α ⇒ x = n π + ( − 1 ) n α Calculation: For xy -plane, z = 0 . sin t = 0 ⇒ sin t = sin 0 [ ∵ sin x = sin α ⇒ x = n π + ( − 1 ) n α ] ⇒ t = n π + ( − 1 ) n α ⇒ t = n π + ( − 1 ) n × 0 ⇒ t = n π n = 0 , 1 , 2 , 3..... ⇒ t = 0 , π , 2 π , 3 π .... So, different values of t give different point of the intersection of the curve with the xy-plane. That means, the curve intersects the xy-plane infinitely many times. Now, at t = 0 , r ( 0 ) = 〈 cos 0 , cos 2 × 0 , sin 0 〉 = 〈 1 , 1 , 0 〉

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 13.1, Problem 10E

To determine

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.1, Problem 9E

Chapter 13.1, Problem 11E

Students have asked these similar questions

== 1. A separable differential equation can be written in the form hy) = g(a) where h(y) is a function of y only, and g(x) is a function of r only. All of the equations below are separable. Rewrite each of these in the form h(y) = g(x), then find a general solution by integrating both sides. Determine whether the solutions you found are explicit (functions) or implicit (curves but not functions) (a) 1' = — 1/3 (b) y' = = --- Y (c) y = x(1+ y²)

A circle of radius r centered at the point (0,r) in the plane will intersect the y-axis at the origin and the point A=(0,2r), as pictured below. A line passes through the point A and the point C=(11/2,0) on the x-axis. In this problem, we will investigate the coordinates of the intersection point B between the circle and the line, as 1 → ∞ A=(0,2r) B (0,0) (a) The line through A and C has equation: y= 2 117 x+27 (b) The x-coordinate of the point B is 4472 121,2 +4 40 (c) The y-coordinate of the point B is +27 121 44 (d) The limit as r→ ∞ of the x-coordinate of B is 121 (if your answer is oo, write infinity).

1. Show that the vector field F(x, y, z) = (2x sin ye³)ix² cos yj + (3xe³ +5)k satisfies the necessary conditions for a conservative vector field, and find a potential function for F.

Chapter 13 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

Ch. 13.1 - Prob. 1PQ Ch. 13.1 - Prob. 2PQ Ch. 13.1 - Prob. 3PQ Ch. 13.1 - Prob. 4PQ Ch. 13.1 - Prob. 5PQ Ch. 13.1 - Prob. 6PQ Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To show that the path of the curve r ( t ) = 〈 cos t , cos 2 t , sin t 〉 intersects the x y -plane infinitely many times but the underlying space curve intersects the x y -plane only twice.

Solution Summary: The author explains that different values of t give different point of intersection of the curve with the xy-plane.

Concept explainers

Videos

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.

Want to see the full answer?

Chapter 13 Solutions

To show that the path of the curve r ( t ) = 〈 cos t , cos 2 t , sin t 〉 intersects the x y -plane infinitely many times but the underlying space curve intersects the x y -plane only twice.

Solution Summary: The author explains that different values of t give different point of intersection of the curve with the xy-plane.

Concept explainers

Videos

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy -plane only twice.

Want to see the full answer?

Chapter 13 Solutions

To show that the path of the curve r(t)=〈cost,cos2t,sint〉 intersects the xy -plane infinitely many times but the underlying space curve intersects the xy

-plane only twice.