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.

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

See similar textbooks

Videos

Question

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.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.4, Problem 33E

Chapter 13.4, Problem 35E

Students have asked these similar questions

Write a formula for the curvature of a plane curve with equation y= f(x)

Find an equation for the circle of curvature of the curve r(t) = ti + (sint)j at the point ('pi/2, 1). (The curve parame- trizes the graph ofy = sin x in the xy-plane.) Need Solution through 15min

Chapter 13 Solutions

EBK 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

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.

(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 | 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.