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: 9781319309671

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

2) Compute the following anti-derivative. √1x4 dx

Question 3 (5pt): A chemical reaction. In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C : ABC. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: d[C] dt = k[A][B] (where k is a constant positive number). Thus, if the initial concentrations are [A] = = a moles/L and [B] = b moles/L we write x = [C], then we have (E): dx dt = k(ax)(b-x) 1 (a) Write the differential equation (E) with separate variables, i.e. of the form f(x)dx = g(t)dt. (b) Assume first that a b. Show that 1 1 1 1 = (a - x) (b - x) - a) a - x b - x b) (c) Find an antiderivative for the function f(x) = (a-x) (b-x) using the previous question. (d) Solve the differentiel equation (E), i.e. find x as a function of t. Use the fact that the initial concentration of C is 0. (e) Now assume that a = b. Find x(t) assuming that a = b. How does this expression for x(t) simplify if it is known that [C] =…

3) Find the volume of the solid that lies inside both the sphere x² + y² + z² cylinder x²+y² = 1. = 4 and the

Chapter 13 Solutions

CALCULUS: EARLY TRANS 4TH ED W/ ACCESS

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