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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

The graph of the acceleration a ( t ) of a car measured in ft/s 2 is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval.

Chapter 4.4, Problem 65E, The graph of the acceleration a(t) of a car measured in ft/s2 is shown. Use the Midpoint Rule to

To determine

To estimate:

The increase in the velocity of the car during the six-second time interval.

Explanation

1) Concept:

Use the Net Change theorem and the Midpoint Rule.

2) Theorem and Rule:

Net Change theorem: The integral of a rate of change is the net change.

abF'(x)dx=Fb-F(a)

The Midpoint Rule:

abfxdxi=1nfxi- x=x fx1-++fxn-

where x=b-an and xi-=12xi-1+xi midpoint of xi-1, xi.

2) Calculation:

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

at=v'(t)

Here t0,6

By Net Change theorem,

06at dt06v'tdt=v6-v(0)

is the change in velocity of the car during the given time interval.

Approximate the value of the integral using the Midpoint Rule with three subintervals: that is with n=3, a=0 and b=6 So width of the interval is

t=b-an

Substitute values

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