Explanation Given: r ( t ) = 〈 cos 2 t sin t , sin 2 t , cos 2 t cos t 〉 Radius of sphere = 1. Center of sphere = (0,0,0) Formula used: General equation of sphere is ( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 = r 2 , Where a,b,c is center of sphere and r is radius of sphere. sin 2 x + cos 2 x = 1 Calculation: First we will find equation of sphere: ( x − a ) 2 + ( y − b ) 2 + ( z − c ) 2 = r 2 ⇒ ( x − 0 ) 2 + ( y − 0 ) 2 + ( z − 0 ) 2 = ( 1 ) 2 ⇒ x 2 + y 2 + z 2 = 1 r ( t ) = 〈 cos 2 t sin t , sin 2 t , cos 2 t cos t 〉 ⇒ x = cos 2 t sin t ⇒ y = sin 2 t ⇒ z = cos 2 t cos t To show that curve lies on the sphere, we will substitute the coordinates of curve in sphere. If we get value as 0, then we can confirm that the curve lies on the sphere To determine (b) Show that curve intersects the x-axis, the y-axis, and the z-axis.

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ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 13.1, Problem 23E

To determine

(a)

Show that curve r(t)=〈cos2tsint,sin2t,cos2tcost〉 lies on the sphere of radius 1 centered at the origin.

To determine

(b)

Show that curve intersects the x-axis, the y-axis, and the z-axis.

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Chapter 13.1, Problem 22E

Chapter 13.1, Problem 24E

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2. Let C be the parametric curve X = 2 sint + 1 y = 2 cost where t E R. (a) By eliminating t, derive an equation relating and y. (b) Sketch the graph of C on the xy-plane. Indicate the direction as t increases on the graph.

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Chapter 13 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

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

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(a) Show that curve r ( t ) = 〈 cos 2 t sin t , sin 2 t , cos 2 t cos t 〉 lies on the sphere of radius 1 centered at the origin.

Solution Summary: The author explains the equation of sphere: r(t)=langle mathrmcos2t, centered at the origin.

Concept explainers

(a)

Show that curve r(t)=〈cos2tsint,sin2t,cos2tcost〉 lies on the sphere of radius 1 centered at the origin.

(b)

Show that curve intersects the x-axis, the y-axis, and the z-axis.

Want to see the full answer?

Chapter 13 Solutions

(a) Show that curve r ( t ) = 〈 cos 2 t sin t , sin 2 t , cos 2 t cos t 〉 lies on the sphere of radius 1 centered at the origin.

Solution Summary: The author explains the equation of sphere: r(t)=langle mathrmcos2t, centered at the origin.

Concept explainers

(a) Show that curve r(t)=〈cos2tsint,sin2t,cos2tcost〉 lies on the sphere of radius 1 centered at the origin.

(b) Show that curve intersects the x-axis, the y-axis, and the z-axis.

Want to see the full answer?

Chapter 13 Solutions

(a)

Show that curve r(t)=〈cos2tsint,sin2t,cos2tcost〉 lies on the sphere of radius 1 centered at the origin.

(b)

Show that curve intersects the x-axis, the y-axis, and the z-axis.