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21 What is the area f(x) under the line x) = 6 – x above the interval from 2 to x? What is the derivative of this f(x)?

21 What is the area f(x) under the line x) = 6 – x above the interval from 2 to x? What is the derivative of this f(x)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question 21:** What is the area \( f(x) \) under the line \( u(x) = 6 - x \) above the interval from 2 to \( x \)? What is the derivative of this \( f(x) \)? **Explanation:** In this question, we are asked to compute the area under the line defined by the equation \( u(x) = 6 - x \) over a specific interval, and subsequently find the derivative of the computed area function. 1. **Area Calculation:** - The line \( u(x) = 6 - x \) forms a straight line with a negative slope of -1 and a y-intercept at 6. - We need to find the area under this line from \( x = 2 \) to a general point \( x = x \). - This area can be calculated using definite integration. 2. **Derivative:** - To find the derivative of \( f(x) \), first determine the indefinite integral as part of the area calculation, then differentiate it. No graph or diagram is provided in the image, but if there were, it would typically depict the line \( u(x) = 6 - x \) and the area under this line from 2 to x.
Transcribed Image Text:**Question 21:** What is the area \( f(x) \) under the line \( u(x) = 6 - x \) above the interval from 2 to \( x \)? What is the derivative of this \( f(x) \)? **Explanation:** In this question, we are asked to compute the area under the line defined by the equation \( u(x) = 6 - x \) over a specific interval, and subsequently find the derivative of the computed area function. 1. **Area Calculation:** - The line \( u(x) = 6 - x \) forms a straight line with a negative slope of -1 and a y-intercept at 6. - We need to find the area under this line from \( x = 2 \) to a general point \( x = x \). - This area can be calculated using definite integration. 2. **Derivative:** - To find the derivative of \( f(x) \), first determine the indefinite integral as part of the area calculation, then differentiate it. No graph or diagram is provided in the image, but if there were, it would typically depict the line \( u(x) = 6 - x \) and the area under this line from 2 to x.
Problems 17–22 are about a company whose expense rate \( t(x) = 6 - x \) is decreasing.
Transcribed Image Text:Problems 17–22 are about a company whose expense rate \( t(x) = 6 - x \) is decreasing.
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