BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 75E
To determine

Prove that the given relation and explain the difference between given derivatives.

Expert Solution

Explanation of Solution

Given:

The given relation is a(t)=v(t)dvds , where

  a(t)= acceleration.

  v(t)= velocity

  s(t)= displacement

The given derivatives are dvdt and dvds .

Calculation:

Acceleration is the rate of change of velocity with respect to time.

  a(t)=dvdt

For increasing velocity, the sign of dvdt will be positive and for the decreasing velocity, the sign of dvdt will be negative.

Now,

  a(t)=dvdt

Divide both sides by v .

  1va(t)=dvdt1v1va(t)=dvdtdtds

Apply chain rule.

  dfdadadx=df(a)dx

  1va(t)=dvdsa(t)=vdvdsa(t)=v(t)dvds

  dvds is the rate of change of the velocity with respect to displacement.

Whenthe displacement increases, the velocity is increasing, when the displacement decreases, the velocity is decreasing.

Hence a(t)=v(t)dvds proved.

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