Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Educational Website Transcription and Explanation:**
---
### Topic: Inverse Functions and Derivatives
**Function Analysis:**
The function \( f(x) = 15e^{-x} \sqrt{x} \) is shown below. We are examining its behavior for the restricted domain \( x \geq 0.5 \).
**Question 1:** Does the function appear to have an inverse function for this restricted domain? (DO NOT DERIVE IT!!)
- **Why or why not?**
**Question 2:** Estimate \( (f') (1) \): ______
- What then is the approximate derivative \( (f^{-1})'(x) \) at the point \(\left( \frac{15}{e}, 1 \right)\)?
- **Why? What theorem or rule supports your answer?**
**Graph Analysis:**
The graph of the function \( f(x) = 15e^{-x} \sqrt{x} \) is provided. Key features include:
- The graph decreases as \( x \) increases from 0.5.
- The plotted point (0.5, 6.133) is marked, indicating the function value at \( x = 0.5 \).
### Detailed Explanation:
1. **Does the function have an inverse?**
A function has an inverse if it is one-to-one (bijective) in the specified domain. The graph shows a decrease in the value of \( f(x) \) as \( x \) increases from 0.5, suggesting that the function is one-to-one on this domain.
2. **Estimating \( f'(1) \):**
- Examine the graph or utilize the derivative of the function \( f(x) \) to approximate \( f'(1) \).
3. **Derivative of the Inverse:**
- Use the formula for the derivative of the inverse function:
\[
(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}
\]
- Understand that this is supported by the Inverse Function Theorem, which states that if \( f \) is bijective and differentiable, then so is \( f^{-1} \).
By reviewing the graph and applying these concepts, we can analyze and estimate the behavior of both the function and its inverse.
---](https://content.bartleby.com/qna-images/question/8583ef51-cf31-4e05-a3c5-c6f69e55ff88/7c487e8c-95c0-4c60-85aa-171c69bd86f9/nz4k1mb_processed.jpeg)
Transcribed Image Text:**Educational Website Transcription and Explanation:**
---
### Topic: Inverse Functions and Derivatives
**Function Analysis:**
The function \( f(x) = 15e^{-x} \sqrt{x} \) is shown below. We are examining its behavior for the restricted domain \( x \geq 0.5 \).
**Question 1:** Does the function appear to have an inverse function for this restricted domain? (DO NOT DERIVE IT!!)
- **Why or why not?**
**Question 2:** Estimate \( (f') (1) \): ______
- What then is the approximate derivative \( (f^{-1})'(x) \) at the point \(\left( \frac{15}{e}, 1 \right)\)?
- **Why? What theorem or rule supports your answer?**
**Graph Analysis:**
The graph of the function \( f(x) = 15e^{-x} \sqrt{x} \) is provided. Key features include:
- The graph decreases as \( x \) increases from 0.5.
- The plotted point (0.5, 6.133) is marked, indicating the function value at \( x = 0.5 \).
### Detailed Explanation:
1. **Does the function have an inverse?**
A function has an inverse if it is one-to-one (bijective) in the specified domain. The graph shows a decrease in the value of \( f(x) \) as \( x \) increases from 0.5, suggesting that the function is one-to-one on this domain.
2. **Estimating \( f'(1) \):**
- Examine the graph or utilize the derivative of the function \( f(x) \) to approximate \( f'(1) \).
3. **Derivative of the Inverse:**
- Use the formula for the derivative of the inverse function:
\[
(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}
\]
- Understand that this is supported by the Inverse Function Theorem, which states that if \( f \) is bijective and differentiable, then so is \( f^{-1} \).
By reviewing the graph and applying these concepts, we can analyze and estimate the behavior of both the function and its inverse.
---
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