Converpes diverpes. Show your

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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### Instruction:
Determine if the following integral converges or diverges. Show your work carefully.

\[ \int_{1}^{\infty} e^{-x} \, dx \]

### Explanation:
To determine whether the integral converges or diverges:

1. **Identify the integral and its limits:**
   The given integral is \[ \int_{1}^{\infty} e^{-x} \, dx \].

2. **Set up the integral:**
   Rewrite the improper integral by introducing a limit.
   \[ \int_{1}^{\infty} e^{-x} \, dx = \lim_{{t \to \infty}} \int_{1}^{t} e^{-x} \, dx \]

3. **Evaluate the finite integral:**
   Find the antiderivative of \( e^{-x} \).

   The antiderivative of \( e^{-x} \) is \(-e^{-x} \).

4. **Apply the Fundamental Theorem of Calculus:**
   \[ \int_{1}^{t} e^{-x} \, dx = \left[ -e^{-x} \right]_{1}^{t} \]
   \[ = -e^{-t} - (-e^{-1}) \]
   \[ = -e^{-t} + e^{-1} \]

5. **Take the limit as \( t \to \infty \):**
   \[ \lim_{{t \to \infty}} (-e^{-t} + e^{-1}) \]

   As \( t \to \infty \), \( e^{-t} \) approaches 0.

   \[ \lim_{{t \to \infty}} (-e^{-t} + e^{-1}) = 0 + e^{-1} = \frac{1}{e} \]

Since \( \frac{1}{e} \) is a finite number, the integral \(\int_{1}^{\infty} e^{-x} \, dx \) converges.

### Conclusion:
The integral \[\int_{1}^{\infty} e^{-x} \, dx\] converges to \[\frac{1}{e}\].
Transcribed Image Text:### Instruction: Determine if the following integral converges or diverges. Show your work carefully. \[ \int_{1}^{\infty} e^{-x} \, dx \] ### Explanation: To determine whether the integral converges or diverges: 1. **Identify the integral and its limits:** The given integral is \[ \int_{1}^{\infty} e^{-x} \, dx \]. 2. **Set up the integral:** Rewrite the improper integral by introducing a limit. \[ \int_{1}^{\infty} e^{-x} \, dx = \lim_{{t \to \infty}} \int_{1}^{t} e^{-x} \, dx \] 3. **Evaluate the finite integral:** Find the antiderivative of \( e^{-x} \). The antiderivative of \( e^{-x} \) is \(-e^{-x} \). 4. **Apply the Fundamental Theorem of Calculus:** \[ \int_{1}^{t} e^{-x} \, dx = \left[ -e^{-x} \right]_{1}^{t} \] \[ = -e^{-t} - (-e^{-1}) \] \[ = -e^{-t} + e^{-1} \] 5. **Take the limit as \( t \to \infty \):** \[ \lim_{{t \to \infty}} (-e^{-t} + e^{-1}) \] As \( t \to \infty \), \( e^{-t} \) approaches 0. \[ \lim_{{t \to \infty}} (-e^{-t} + e^{-1}) = 0 + e^{-1} = \frac{1}{e} \] Since \( \frac{1}{e} \) is a finite number, the integral \(\int_{1}^{\infty} e^{-x} \, dx \) converges. ### Conclusion: The integral \[\int_{1}^{\infty} e^{-x} \, dx\] converges to \[\frac{1}{e}\].
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