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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f(x) = x/(x + 1) on the interval [0, 2] with n = 100. Explain why these estimates show that

0.8946 < 0 2 x x + 1 d x < 0.9081

To determine

To Calculate: The left and right Riemann sums for the function f(x)=x(x+1) on interval [0,2] and n=100 using the calculator.

To Explain: the reason for the expression of estimate as follows:

0.8946<02xx+1dx<0.9081

Explanation

The value of the function f(x)=x(x+1) is a increasing function. The left Riemann Sum gives the lower estimate and right Riemann Sum gives upper estimate Thus, get the expression as follows:

0.8946<02xx+1dx<0.9081

Given:

The function as f(x)=x(x+1)

The region lies between x=0 and x=2. So the limits are a=0 and b=2.

Number of rectangles n=100.

Calculation:

Show the Equation of the function f(x) as follows:

f(x)=x(x+1) (1)

Calculate the value of the function f(x) for different values of x within the interval [0,2].

Substitute 0 for x in Equation (1).

f(0)=0(0+1)=0

Substitute 1.5 for x in Equation (1).

f(1.5)=1.5(1.5+1)=0.6

Substitute 2 for x in Equation (1).

f(2)=2(2+1)=0.666

The calculated values of x shows that the function f(x)=x(x+1) is aincreasing function within the interval [0,2].

The expression to find the left Riemann sum Ln as shown below:

Ln=i=1nf(xi)Δx=f(x0)Δx+f(x1)Δx+...+f(xn1)Δx (2)

Here, the left endpoint height of first rectangle is f(x0), the width is Δx, height of left endpoint of second rectangle is f(x1), and left endpoint height of nth rectangle is f(xn1).

Find the width (Δx) using the relation:

Δx=ban (3)

Here, the upper limit is b, the lower limit is a, and the number of rectangles is n.

Substitute 2 for b, 0 for a and 100 for n in Equation (3).

Δx=20100=0.02

The value of left and right end points within the interval [0,2] are shown below:

The left endpoints are x0=0, x1=0.02, x2=0.04x98=1.96 and x99=1.98.

The right endpoints are x1=0.02, x2=0.04, x3=0.06x99=1.98 and x100=2

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