Q. 16 Explain how we get the inequalit... [FREE SOLUTION]

Chapter 4: Q. 16 (page 399)

Explain how we get the inequality
$f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$
in the proof of the Second Fundamental Theorem of Calculus. Make sure you define $m_{h}$ and $M_{h}$ clearly.

Short Answer

Expert verified

Since $m_{h} < f (x) < M_{h}$ , $f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$ . This is how we get the required inequality.

Step by step solution

Step 1. Given information

$f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$ .

Step 2. The objective is to explain how the given inequality is achieved.

The Extreme value Theorem states that if a real-valued function $f$ is continuous in the closed and bounded interval $[a, b]$ , then $f$ must attain a maximum and a minimum, each at least once. That is, there exist numbers $c$ and $d$ in $[a, b]$ such that:

$f (c) 鈮� f (x) 鈮� f (d)$ .

So,

$M_{h} > f (x) > m_{h} m_{k} < f (x) < M_{h}$

where $M_{h}$ is the maximum value on $[a, b]$ and $m_{h}$ is the minimum value on $[a, b]$ .

Hence, on $[x, x + h]$ the area of the rectangles using left and right sum can be written as, $f (m_{h}) h$ and $f (M_{h}) h$ respectively.

Since, $m_{k} < f (x) < M_{h}$ so, $f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$ .

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Explain how we get the inequality
$f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$
in the proof of the Second Fundamental Theorem of Calculus. Make sure you define $m_{h}$ and $M_{h}$ clearly.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to explain how the given inequality is achieved.

One App. One Place for Learning.

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91影视

Explain how we get the inequalityfmhh鈮�鈭�xx+hf(t)dt鈮�fMhhin the proof of the Second Fundamental Theorem of Calculus. Make sure you define mhand Mhclearly.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to explain how the given inequality is achieved.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Explain how we get the inequality
$f (m_{h}) h 鈮� {鈭�}_{x}^{x + h} f (t) d t 鈮� f (M_{h}) h$
in the proof of the Second Fundamental Theorem of Calculus. Make sure you define $m_{h}$ and $M_{h}$ clearly.