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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

When the area under f ( x ) =   x 2 + x  from  x   = 0  to  x =   2 is approximated, the formulas for the sum of n rectangles using left-hand endpoints and right-hand endpoints are

Left-hand endpoints: S L = 14 3 6 n + 4 3 n 2

Right-hand endpoints: S R = 14 n 2 + 18 n + 4 3 n 2

Use these formulas to answer Problems 9-13.

Because f ( x )   =   x 2 +   x is increasing over the interval from x   =   0  to  x =   2 , function values at the right-hand endpoints are maximum values for each subinterval and function values at the left-hand endpoints are minimum values for each subinterval. How would the approximate area using n = 10 and any other point within each subinterval compare with S L ( 10 )  and  S R ( 10 ) ? What would happen to the area result as n if any other point in each subinterval were used?

To determine

The comparison between the approximated area of the increasing function f(x)=x2+x over the interval from x=0 to x=2 using n=10 and any other point in each subinterval with SL(10) and SR(10). Also, calculate the area as n if any other point in each subinterval is used.

Explanation

Given Information:

The provided formulae are SL=1436n+43n2 and SR=14n2+18n+43n2.

Calculation:

Consider the formulae,

SL=1436n+43n2 and SR=14n2+18n+43n2

Recall that to calculate the value of a function f(x) at x=a, substitute a for x in the formula.

First, calculate SL(10).

Substitute 10 for n in SL=1436n+43n2.

SL(10)=143610+43(10)2=143610+4300=1224300=4.08

Thus, the value of SL(10) is 4.08.

Next, calculate SR(10).

Substitute 10 for n in SR=14n2+18n+43n2.

SR=14(10)2+18(10)+43(10)2=1400+180+4300=1584300=5.28

Thus, the value of SR(10) is 5.28.

Hence, the value of SL(10) is 4.08 and the value of SR(10) is 5.28.

Since, the function f(x)=x2+x defined over the interval [0,2] is an increasing function.

Thus, the area calculated at any point in each subinterval is between the values of SL(10) SR(10).

Let A be the area of the function f(x)=x2+x.

Thus, SL(10)ASR(10).

Consider that n.

Recall that for functions f(x) and g(x), limn[f(x)±g(x)]=limnf(x)±limng(x).

First, calculate limnSL.

Substitute SL=1436n+43n2.

limnSL=limn(1436n+43n2)=limn143limn6n+limn43n2=1436limn1n+43limn(1n)2

Use limn1n=0 to simplify further

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Chapter 13 Solutions

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Sect-13.1 P-9ESect-13.1 P-10ESect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.2 P-1CPSect-13.2 P-2CPSect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 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P-2ESect-13.8 P-3ESect-13.8 P-4ESect-13.8 P-5ESect-13.8 P-6ESect-13.8 P-7ESect-13.8 P-8ESect-13.8 P-9ESect-13.8 P-10ESect-13.8 P-11ESect-13.8 P-12ESect-13.8 P-13ESect-13.8 P-14ESect-13.8 P-15ESect-13.8 P-16ESect-13.8 P-17ESect-13.8 P-18ESect-13.8 P-19ESect-13.8 P-20ESect-13.8 P-21ESect-13.8 P-22ESect-13.8 P-23ESect-13.8 P-24ESect-13.8 P-25ESect-13.8 P-26ESect-13.8 P-27ESect-13.8 P-28ESect-13.8 P-29ESect-13.8 P-30ESect-13.8 P-31ESect-13.8 P-32ESect-13.8 P-33ESect-13.8 P-34ECh-13 P-1RECh-13 P-2RECh-13 P-3RECh-13 P-4RECh-13 P-5RECh-13 P-6RECh-13 P-7RECh-13 P-8RECh-13 P-9RECh-13 P-10RECh-13 P-11RECh-13 P-12RECh-13 P-13RECh-13 P-14RECh-13 P-15RECh-13 P-16RECh-13 P-17RECh-13 P-18RECh-13 P-19RECh-13 P-20RECh-13 P-21RECh-13 P-22RECh-13 P-23RECh-13 P-24RECh-13 P-25RECh-13 P-26RECh-13 P-27RECh-13 P-28RECh-13 P-29RECh-13 P-30RECh-13 P-31RECh-13 P-32RECh-13 P-33RECh-13 P-34RECh-13 P-35RECh-13 P-36RECh-13 P-37RECh-13 P-38RECh-13 P-39RECh-13 P-40RECh-13 P-41RECh-13 P-42RECh-13 P-43RECh-13 P-44RECh-13 P-45RECh-13 P-46RECh-13 P-47RECh-13 P-48RECh-13 P-49RECh-13 P-50RECh-13 P-51RECh-13 P-53RECh-13 P-54RECh-13 P-55RECh-13 P-1TCh-13 P-2TCh-13 P-3TCh-13 P-4TCh-13 P-5TCh-13 P-6TCh-13 P-7TCh-13 P-8TCh-13 P-9TCh-13 P-10TCh-13 P-11TCh-13 P-12TCh-13 P-13TCh-13 P-14TCh-13 P-15TCh-13 P-16T

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