-4 -2 -4 288 lower limit is less than 0, so FTC2 cannot be applie re is nothing wrong with the equation. = x-3 is not continuous on the interval [-4, 3] so %3D
-4 -2 -4 288 lower limit is less than 0, so FTC2 cannot be applie re is nothing wrong with the equation. = x-3 is not continuous on the interval [-4, 3] so %3D
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Question: What is wrong with the equation?**
\[
\int_{-4}^{3} x^{-3} \, dx = \left. \frac{x^{-2}}{-2} \right|_{-4}^{3} = -\frac{7}{288}
\]
**Options:**
1. ☐ The lower limit is less than 0, so FTC2 cannot be applied.
2. ☐ There is nothing wrong with the equation.
3. ☐ \( f(x) = x^{-3} \) is not continuous on the interval \([-4, 3]\) so FTC2 cannot be applied.
4. ☐ \( f(x) = x^{-3} \) is not continuous at \( x = -4 \), so FTC2 cannot be applied.
5. ☐ \( f(x) = x^{-3} \) is continuous on the interval \([-4, 3]\) so FTC2 cannot be applied.
---
### Explanation:
The integral in the equation tries to find the antiderivative of \( x^{-3} \) over the interval \([-4, 3]\). However, the function \( f(x) = x^{-3} \) is not defined at \( x = 0 \) and thus is not continuous on the entire interval from \([-4, 3]\). This makes it impossible to apply the Fundamental Theorem of Calculus (part 2).
---
**Note for Educational Use:**
Review the continuity and integrability conditions for applying the Fundamental Theorem of Calculus. In particular, for a function to be integrable on a given interval using this theorem, it must be continuous on that interval. Discontinuous points within the interval, such as \( x = 0 \) in this case, can invalidate the direct application of the theorem.
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Transcribed Image Text:---
**Question: What is wrong with the equation?**
\[
\int_{-4}^{3} x^{-3} \, dx = \left. \frac{x^{-2}}{-2} \right|_{-4}^{3} = -\frac{7}{288}
\]
**Options:**
1. ☐ The lower limit is less than 0, so FTC2 cannot be applied.
2. ☐ There is nothing wrong with the equation.
3. ☐ \( f(x) = x^{-3} \) is not continuous on the interval \([-4, 3]\) so FTC2 cannot be applied.
4. ☐ \( f(x) = x^{-3} \) is not continuous at \( x = -4 \), so FTC2 cannot be applied.
5. ☐ \( f(x) = x^{-3} \) is continuous on the interval \([-4, 3]\) so FTC2 cannot be applied.
---
### Explanation:
The integral in the equation tries to find the antiderivative of \( x^{-3} \) over the interval \([-4, 3]\). However, the function \( f(x) = x^{-3} \) is not defined at \( x = 0 \) and thus is not continuous on the entire interval from \([-4, 3]\). This makes it impossible to apply the Fundamental Theorem of Calculus (part 2).
---
**Note for Educational Use:**
Review the continuity and integrability conditions for applying the Fundamental Theorem of Calculus. In particular, for a function to be integrable on a given interval using this theorem, it must be continuous on that interval. Discontinuous points within the interval, such as \( x = 0 \) in this case, can invalidate the direct application of the theorem.
---
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