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 )

4th Edition

ISBN: 9781319545956

Author: Rogawski

Publisher: MAC HIGHER

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

The spread of an infectious disease is often modeled using the following autonomous differential equation: dI - - BI(N − I) − MI, dt where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of transmission, and μ is the rate at which people recover from infection. Close a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria. b) (5 points) For the equilbria in part a), determine whether each is stable or unstable. c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the dt function by hand.) Identify the equilibria as stable or unstable in the graph. d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.

Find the indefinite integral. Check Answer: 7x 4 + 1x dx

show sketch

Chapter 13 Solutions

CALCULUS W/ACHIEVE PLATFORM

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