Q. 6 Sketch an example that shows tha... [FREE SOLUTION]

Chapter 5: Q. 6 (page 490)

Sketch an example that shows that the inequality $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ is not necessarily true if $f$ is not concave up on all of $[a, b] .$

Short Answer

Expert verified

Sketch a curve which explains the statement that, "the inequality $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ is not necessarily true".

This sketch explains the required statement.

Step by step solution

Step 1. Given information

$M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ .

Step 2. Consider a function 'f' which is not concave up on all of interval a,b.

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

The objective is to sketch a curve which explains the statement that, "the inequality $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ is not necessarily true.

For the concave up curve, the inequality relating the trapezoidal sum and midpoint sum is given as, $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ .

For the concave down curve, the inequality relating the trapezoidal sum and midpoint sum is given as, $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ .

Thus, for a curve which is both concave up and concave down in interval $[a, b]$ will not necessarily have a inequality to represent the relation.

Step 3. Draw a sketch which is both concave up and concave down in an interval a,b.

Thus, this sketch explains the required statement.

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91影视

Sketch an example that shows that the inequality $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ is not necessarily true if $f$ is not concave up on all of $[a, b] .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider a function 'f' which is not concave up on all of interval a,b.

Step 3. Draw a sketch which is both concave up and concave down in an interval a,b.

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91影视

Sketch an example that shows that the inequality MID(n)鈮�鈭�abfxdx鈮�TRAP(n)is not necessarily true if fis not concave up on all of[a,b].

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider a function 'f' which is not concave up on all of interval a,b.

Step 3. Draw a sketch which is both concave up and concave down in an interval a,b.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

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Theoretical and Mathematical Physics

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Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Sketch an example that shows that the inequality $M I D (n) 鈮� {鈭�}_{a}^{b} f (x) d x 鈮� T R A P (n)$ is not necessarily true if $f$ is not concave up on all of $[a, b] .$