Chapter 4: Q. 52 (page 341)

Use Definitions 4.6 and 4.8 to prove that for any function f and interval $[a, b]$ , the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.

Short Answer

Expert verified

It is provedthat for any function f and interval $[a, b]$ , the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.

Step by step solution

Step 1. Given Information

We are given thatfor any function f and interval $[a, b]$ , the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.

Step 2. Proving the statement

The left-sum defined for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} f (x_{k - 1}) Δ x$ .

Where, $Δ x = \frac{b - a}{n}, x_{k} = a + k Δ x$ .

The right sum defined for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} f (x_{k}) Δ x$ .

The average of left sum and right sum is,

$\frac{\sum_{k = 1}^{n} f (x_{k - 1}) Δ x + \sum_{k = 1}^{n} f (x_{k}) Δ x}{2} = \sum_{k = 1}^{n} \frac{f (x_{k - 1}) + f (x_{k})}{2} Δ x$

The trapezoid sum for n rectangles on $[a, b]$ is $\sum_{k = 1}^{n} \frac{f (x_{k - 1}) + f (x_{k})}{2} Δ x$ .

Since f is increasing,

$x_{k - 1} \leq x_{k}$

Hence Proved.

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Use Definitions 4.6 and 4.8 to prove that for any function f and interval $[a, b]$ , the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

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Use Definitions 4.6 and 4.8 to prove that for any function f and interval [a,b], the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving the statement

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Use Definitions 4.6 and 4.8 to prove that for any function f and interval $[a, b]$ , the trapezoid sum with n trapezoids is always the average of the left sum and the right sum with n rectangles.