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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

Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 am. D is the data throughput, measured in megabits per second. Use the Midpoint Rule to estimate the total amount of data transmitted during that time period.

Chapter 4.4, Problem 66E, Shown is the graph of traffic on an Internet service providers T1 data line from midnight to 8:00

To determine

To estimate:

The total amount of data transmitted during time period midnight to 8:00 A.M.

Explanation

1) Concept:

Use the Net change theorem and midpoint rule.

2) Theorem and Rule:

The 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.

3) Calculation:

Let Mt denote the number of megabits transmitted at the time t(in hours).

Dt  measured in megabits/second is the rate of change of number of megabits transmitted at time t.

Dt=M'(t)

Here,t0,8

By the Net Change theorem,

08Dt dt08M'tdt=M8-M(0) is number of megabits transmitted during the time interval.

Approximate the value of the integral using the Midpoint Rule with n=4, a=0, and b=8

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