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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.

To determine

i) To prove:

 That at some time during the race, the acceleration of the two cars was equal.

2) To state:

The assumptions made to prove i)

Explanation

1) Concept:

If st  is a position function, vt=s't;v  is the velocity function & at=s''t;a  is the acceleration function.

Application of rolls theorem:

 If f  is twice differentiable on R  and has three roots, then f'' has at least one real root.

2) Given:

Car A passed car B twice.

3) Calculation:

a) Let sAt & sBt  be the position functions for car A & car B

Let ft= sAt-sBt

Since car A passes car B twice, there must be three values of t  such that ft=0 ,

Differentiating ft  with respect to t,

f't=s'At-s'Bt

Differentiating f't  with respect to t,

f''t=s''At-s''Bt

Here, f  is a twice differentiable function and has three roots

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