Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 11: Q. 60 (page 890)

Use Theorem 11.24 to prove that the curvature of a linear function y = mx + b is zero for every value of x.

Short Answer

Expert verified

It is proved thatthe curvature of a linear functiony = mx + b is zero for every value ofx.

Step by step solution

01

Step 1. Given Information.

The given linear function isy=mx+b.

02

Step 2. Prove.

To prove that the curvature of a linear function y = mx + b is zero for every value of x,we will use the formula for Curvature in the Plane.

The curvature in the plane is defined as letting y = f(x) be a twice-differentiable function. Then the curvature of the graph of fis given by k=f''(x)1+f'x232.

Now,

y=f(x)=mx+bf'x=mf''x=0

Put all the values in the formula,

k=01+m232k=0

Hence proved.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each of the vector-valued functions in Exercises , find the unit tangent vector and the principal unit normal vector at the specified value of t.

rt=cosαt,sinαt,whereα>0,t=π

For each of the vector-valued functions, find the unit tangent vector.

r(t)=(sin2t,cos2t,t)

Evaluate and simplify the indicated quantities in Exercises 35–41.
5(cost,sint)

Let r(t)=x(t),y(t),z(t)be a differentiable vector function on some interval I⊑Rsuch that the derivative of the unit tangent vector T(t0)=0, where t0∈I. Prove that the binormal vectorB(t0)=T(t0)×N(t0)

(a) B(t0)is a unit vector;

(b)B(t0)is orthogonal to both T(t0)and N(t0).

Also, prove that T(t0),N(t0), and B(t0)form a right-handed coordinate system.

Find the unit tangent vector and the principal unit normal vector at the specified value of t.

r(t)=(asinht,bcosht)Whereaandbarepositive,t=0
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free