   Chapter 7.7, Problem 45E

Chapter
Section
Textbook Problem

Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule.

To determine

To sketch:

The graph of a continuous function on [0,2].

To verify:

For the continuous function, trapezoidal rule is more accurate than the midpoint rule.

Explanation

Given information:

The interval is [0,2].

Number of subintervals is 2.

Calculation:

Let consider a function f and draw the graph for the function.

The sketch of the graph of the continuous function is shown in Figure 1.

Show the Trapezoidal rule as follows:

abf(x)dxTn=Δx2[f(x0)+2f(x1)+2f(x2)+...+2f(xn1)+f(xn)]

Here, Tn is trapezoidal approximation, x1,x2,...xn are subintervals, Δx=ban and xi=a+iΔx.

Calculate the length of the subinterval (Δx) using the formula:

Δx=ban

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

Substitute 2 for b, 0 for a, and 2 for n.

Δx=202=1

Hence, the subinterval width is 2 for the limits [0,2].

Apply Trapezoidal rule in Equation (1).

02f(x)dxTn=Δx2[f(x0)+2f(x1)+f(x2)]

Substitute 1 for Δx and 2 for n.

T2=12[f(x0)+2f(x1)+2f(x2)] (2)

Apply the subinterval values in Equation (2).

T2=12[f(x0)+2f(x1)+f(x2)]=12[f(0)+2f(1)+f(2)] (3)

Refer Figure 1

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