Explanation Given: Curve = r ( t ) = 〈 e t cos 4 t , e t sin 4 t 〉 s ( t ) = ∫ − ∞ t | | r ( u ) | | d u Formula used: Curvature = k = | | r ' ( t ) × r ' ' ( t ) | | | | r ' ( t ) | | 3 Radius of curvature = r = 1 k Proof: r ' ( t ) = 〈 e t cos 4 t − 4 e t sin 4 t , e t sin 4 t + 4 e t cos 4 t 〉 = 〈 e t ( cos 4 t − 4 sin 4 t ) , e t ( sin 4 t + 4 cos 4 t ) 〉 Then, ‖ r ' ( t ) ‖ = [ e t ( cos 4 t − 4 sin 4 t ) ] 2 + [ e t ( sin 4 t + 4 cos 4 t ) ] 2 = e 2 t ( cos 4 t − 4 sin 4 t ) 2 + e 2 t ( sin 4 t + 4 cos 4 t ) 2 = e 2 t [ ( cos 4 t − 4 sin 4 t ) 2 + ( sin 4 t + 4 cos 4 t ) 2 ] = e 2 t [ cos 2 4 t + 16 sin 2 4 t − 8 cos 4 t sin 4 t + sin 2 4 t + 16 cos 2 4 t + 8 cos 4 t sin 4 t ] = e 2 t [ cos 2 4 t + 16 sin 2 4 t + sin 2 4 t + 16 cos 2 4 t ] = e 2 t [ 17 cos 2 4 t + 17 sin 2 4 t ] = e 2 t [ 17 { cos 2 4 t + sin 2 4 t } ] = e 2 t [ 17 { 1 } ] = 17 e 2 t Therefore, | | r ' ( t ) | | = 17 e t . And, r ' ' ( t ) = 〈 e t cos 4 t − 4 e t sin 4 t − 4 e t sin 4 t − 16 e t cos 4 t , e t sin 4 t + 4 e t cos 4 t + 4 e t cos 4 t − 16 e t sin 4 t 〉 = 〈 − 8 e t sin 4 t − 15 e t cos 4 t , − 15 e t sin 4 t + 8 e t cos 4 t 〉 = 〈 e t ( − 8 sin 4 t − 15 cos 4 t ) , e t ( − 15 sin 4 t + 8 cos 4 t ) 〉 Now, find r ' ( t ) × r ' ' ( t )

Loose-leaf Version For Calculus: Early Transcendentals

4th Edition

ISBN: 9781319055912

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

See similar textbooks

Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magn…

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

Chapter 13.4, Problem 33E

To determine

Show that the radius of curvature of curve 〈etcos4t,etsin4t〉 is proportional to s(t).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.4, Problem 32E

Chapter 13.4, Problem 34E

Students have asked these similar questions

ds 5. Find a solution to this initial value problem: 3t2, s(0) = 5. dt 6. Find a solution to this initial value problem: A' = 0.03A, A(0) = 100.

2) Drive the frequency responses of the following rotor system with Non-Symmetric Stator. The system contains both external and internal damping. Show that the system loses the reciprocity property.

1) Show that the force response of a MDOF system with general damping can be written as: X liax) -Σ = ral iw-s, + {0} iw-s,

Chapter 13 Solutions

Loose-leaf Version For 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.

Show that the radius of curvature of curve 〈 e t cos 4 t , e t sin 4 t 〉 is proportional to s ( t ) .

Solution Summary: The author demonstrates that the radius of curvature of curve langle 'etmathrmcos4t,' s(t)='displaystyle',

Concept explainers

Show that the radius of curvature of curve 〈etcos4t,etsin4t〉 is proportional to s(t).

Want to see the full answer?

Chapter 13 Solutions