   Chapter 7.7, Problem 49E

Chapter
Section
Textbook Problem

# Show that 1 2 ( T n + M n ) = T 2 n .

To determine

To evaluate: To show that

12(Tn+Mn)=T2n

Explanation

Calculation: The trapezoidal rule is the average of the left Riemann sum and the right Riemann sum

Tn=i=0n1Δ2[f(a+iΔ)+f(a+(i+1)Δ)]

Mn=Δf(a+i2Δ)

So,Tn=12Δ[f(x0)+2f(x1)+2f(x2)++2f(xn1)+f(xn)]

And

Mn=Δx[f(x¯1)+f(x¯2)+f(x¯3)++f(x¯n1)+f(x¯n)]x¯i=12(xi1+xi)

Thus we have

T2n=12(12Δ)[f(x0)+2f(x¯1)+2f(x1)+2f(x¯2)+2f(x2)++2f(x¯n1)+

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