Explanation Given: c ( t ) = ( e − t cos t , e − t sin t ) for 0 ≤ t < ∞ . Formula Used: The length of a curve c ( t ) from t = a to t = b will be given by: L = ∫ t = a t = b ( d x d t ) 2 + ( d y d t ) 2 d t Calculation: It is given that c ( t ) = ( e − t cos t , e − t sin t ) that means we have x ( t ) = e − t cos t , y ( t ) = e − t sin t Now, differentiate both the equations x ( t ) = e − t cos t , y ( t ) = e − t sin t with respect to t and we get: x ( t ) = e − t cos t , y ( t ) = e − t sin t d x d t = − e − t ( cos t + sin t ) , d y d t = e − t ( cos t − sin t ) So, ( d x d t ) 2 + ( d y d t ) 2 = ( − e − t ( cos t + sin t ) ) 2 + ( e − t ( cos t − sin t ) ) 2 = e − 2 t ( cos t + sin t ) 2 + e − 2 t ( cos t − sin t ) 2 = e − t cos 2 t + sin 2 t + 2 cos t sin t + cos 2 t + sin 2 t − 2 cos t sin t = 2 e − t Now, the slope of the tangent line at any point t will be given by: d y d x = d y d t d x d t = e − t ( cos t − sin t ) − e − t ( cos t + sin t ) = ( cos t − sin t ) − ( cos t + sin t ) The tangent line to c ( t 0 ) is vertical, if the denominator of the above expression is 0...

4th Edition

ISBN: 9781319323394

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 12, Problem 24CRE

To determine

The first positive value of $t_{0}$ such that the tangent line to $c (t_{0})$ is vertical, and to calculate the speed at $t = t_{0}$ .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12, Problem 23CRE

Chapter 12, Problem 25CRE

Students have asked these similar questions

Use the definite integral to find the area between the x-axis and f(x) over the indicated interval. Check first to see if the graph crosses the x-axis in the given interval. f(x)=3e-4; [3,3]

A small company of science writers found that its rate of profit (in thousands of dollars) after t years of operation is given by P'(t) = (7t + 14) (t² + 4t+7) * (a) Find the total profit in the first four years. (b) Find the profit in the sixth year of operation. (c) What is happening to the annual profit over the long run?

Calculus III May I have an expert explained how the terms were simplified into 6(3-x)^2? Thank you,

Chapter 12 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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.

The first positive value of t 0 such that the tangent line to c ( t 0 ) is vertical, and to calculate the speed at t = t 0 .

Solution Summary: The author calculates the first positive value of t 0 such that the tangent line to c is vertical, and the speed.

Concept explainers

The first positive value of $t_{0}$ such that the tangent line to $c (t_{0})$ is vertical, and to calculate the speed at $t = t_{0}$ .

Want to see the full answer?

Chapter 12 Solutions

The first positive value of t 0 such that the tangent line to c ( t 0 ) is vertical, and to calculate the speed at t = t 0 .

Solution Summary: The author calculates the first positive value of t 0 such that the tangent line to c is vertical, and the speed.

Concept explainers

The first positive value of t0 such that the tangent line to c(t0) is vertical, and to calculate the speed at t=t0.

Want to see the full answer?

Chapter 12 Solutions

The first positive value of $t_{0}$ such that the tangent line to $c (t_{0})$ is vertical, and to calculate the speed at $t = t_{0}$ .