   Chapter 13.1, Problem 13E ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### 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 areLeft-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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