Explanation Given: The given curve is x ( t ) = cos 3 t , y ( t ) = sin 3 t . Formula used: Curvature = k = | | r ' ( t ) × r ' ' ( t ) | | | | r ' ( t ) | | 3 Calculation: The given curve can be written as r ( t ) = 〈 cos 3 t , sin 3 t 〉 . So, r ' ( t ) = 〈 − 3 cos 2 t sin t , 3 sin 2 t cos t 〉 ‖ r ' ( t ) ‖ = ( − 3 cos 2 t sin t ) 2 + ( 3 sin 2 t cos t ) 2 = 9 cos 4 t sin 2 t + 9 sin 4 t cos 2 t = 9 cos 2 t sin 2 t ( cos 2 t + sin 2 t ) = 9 cos 2 t sin 2 t ( 1 ) = 9 cos 2 t sin 2 t = 3 cos t sin t Now, r ' ' ( t ) = 〈 − 3 cos 3 t + 6 cos t sin 2 t , − 3 sin 3 t + 6 sin t cos 2 t 〉 . Then, find r ' ( t ) × r ' ' ( t ) . r ' ( t ) × r ' ' ( t ) = ( i ^ j ^ k ^ − 3 cos 2 t sin t 3 sin 2 t cos t 0 − 3 cos 3 t + 6 cos t sin 2 t − 3 sin 3 t + 6 sin t cos 2 t 0 ) = ( { [ − 3 cos 2 t sin t ] × [ − 3 sin 3 t + 6 sin t cos 2 t ] To determine (b) The minimum curvature and the point on the curve at which it occurs. To determine (c) For what value of t, curvature is undefined.

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Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Chapter 13.4, Problem 34E

To determine

(a)

To show that the curvature of the given curve is k=1|3sintcost|.

To determine

(b)

The minimum curvature and the point on the curve at which it occurs.

To determine

(c)

For what value of t, curvature is undefined.

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Chapter 13.4, Problem 33E

Chapter 13.4, Problem 35E

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Chapter 13 Solutions

Calculus: Early Transcendentals

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) To show that the curvature of the given curve is k = 1 | 3 sin t cos t | .

Solution Summary: The author explains how the curvature of the given curve can be written as r(t)=langle mathrmcos2t

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(a)

To show that the curvature of the given curve is k=1|3sintcost|.

(b)

The minimum curvature and the point on the curve at which it occurs.

(c)

For what value of t, curvature is undefined.

Want to see the full answer?

Chapter 13 Solutions

(a) To show that the curvature of the given curve is k = 1 | 3 sin t cos t | .

Solution Summary: The author explains how the curvature of the given curve can be written as r(t)=langle mathrmcos2t

Videos

(a) To show that the curvature of the given curve is k=1|3sintcost|.

(b) The minimum curvature and the point on the curve at which it occurs.

(c) For what value of t, curvature is undefined.

Want to see the full answer?

Chapter 13 Solutions

(a)

To show that the curvature of the given curve is k=1|3sintcost|.

(b)

The minimum curvature and the point on the curve at which it occurs.

(c)

For what value of t, curvature is undefined.