Q. 5 Sketch an example that shows tha... [FREE SOLUTION]

Chapter 5: Q. 5 (page 490)

Sketch an example that shows that the inequality $L E F T (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� R I G H T (n)$ is not necessarily trueif $f$ is not monotonically increasing on $[a, b]$ .

Short Answer

Expert verified

In the above figure, the region shaded by the single pattern, at the top of each rectangle, represents the difference between the trapezoid sum and the midpoint sum. Hence, if the graph is not monotonically changing, there is no fixed inequality between the left sum and the right sum.

Step by step solution

Step 1. Given information

$L E F T (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� R I G H T (n)$ .

Step 2. Consider a function f which is not monotonically decreasing in the interval a,b.

The integral ${鈭�}_{a}^{b} f (x) d x$ gives the area bounded between the curve and the $x$ -axis in this interval. It is represented by the following figure.

Step 3. Create a graph of the function representing the left sum of the function in the given interval with the shaded region.

Step 4.Create a graph of the function representing the right sum of the function in the given interval with the shaded region.

Step 5. Combine the two graphs to highlight the difference between the two above shaded regions.

In the above figure, the region shaded by the single pattern, at the top of each rectangle, represents the difference between the trapezoid sum and the midpoint sum. It is clear that in some intervals left sum is greater and in some intervals, it is the right sum which has greater value. Hence, if the graph is not monotonically changing, there is no fixed inequality between the left sum and the right sum. Thus, it explains the required statement.

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Sketch an example that shows that the inequality $L E F T (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� R I G H T (n)$ is not necessarily trueif $f$ is not monotonically increasing on $[a, b]$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider a function f which is not monotonically decreasing in the interval a,b.

Step 3. Create a graph of the function representing the left sum of the function in the given interval with the shaded region.

Step 4.Create a graph of the function representing the right sum of the function in the given interval with the shaded region.

Step 5. Combine the two graphs to highlight the difference between the two above shaded regions.

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91影视

Sketch an example that shows that the inequality LEFT(n)鈮�鈭�abfxdx鈮�RIGHT(n)is not necessarily trueif f is not monotonically increasing on [a,b].

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider a function f which is not monotonically decreasing in the interval a,b.

Step 3. Create a graph of the function representing the left sum of the function in the given interval with the shaded region.

Step 4.Create a graph of the function representing the right sum of the function in the given interval with the shaded region.

Step 5. Combine the two graphs to highlight the difference between the two above shaded regions.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Decision Maths

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Sketch an example that shows that the inequality $L E F T (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� R I G H T (n)$ is not necessarily trueif $f$ is not monotonically increasing on $[a, b]$ .