Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Calculating the Derivative Limit
**Objective:**
The limit below represents a derivative \( f'(a) \). Determine \( f(x) \) and \( a \).
\[ \lim_{h \to 0} \frac{\sin\left(\frac{\pi}{2} + h\right) - 1}{h} \]
**Options:**
**Function \( f(x) \) Options:**
1. \( f(x) = \sin(x + y) \)
2. \( f(x) = \sin(x + h) \)
3. \( f(x) = \sin(x) \)
4. \( f(x) = \frac{\sin(x)}{x} \)
5. \( f(x) = \sin(x) - 1 \)
**Point \( a \) Options:**
1. \( a = -\frac{\pi}{2} \)
2. \( a = -1 \)
3. \( a = 0 \)
4. \( a = 1 \)
5. \( a = \frac{\pi}{2} \)
**Explanation of the Limit:**
The given limit expression is a fundamental representation of the definition of the derivative:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
### Steps to Determine \( f(x) \) and \( a \):
1. Compare the given limit with the standard definition to identify \( f(a + h) \) and \( f(a) \):
\[ f(a + h) = \sin\left(\frac{\pi}{2} + h\right) \]
\[ f(a) = 1 \]
2. Recall that:
\[ \sin\left(\frac{\pi}{2} + x\right) = \cos(x) \]
3. Since \( f(a) = 1 \) and comparing with \( f(a) = \sin(a) \), we find:
\[ \sin(a) = 1 \]
This indicates:
\[ a = \frac{\pi}{2} \]
4. Therefore, the original function must be:
\[ f(x) = \sin(x) \]
### Conclusion:
- **Function \( f(x) \):** \( \sin(x) \)
- **Point](https://content.bartleby.com/qna-images/question/2d16e493-8014-4530-b66f-8d290a1d8344/c050660c-9fc3-4a27-9a75-5a9e079d8bb2/we3z4jr.png)
Transcribed Image Text:### Calculating the Derivative Limit
**Objective:**
The limit below represents a derivative \( f'(a) \). Determine \( f(x) \) and \( a \).
\[ \lim_{h \to 0} \frac{\sin\left(\frac{\pi}{2} + h\right) - 1}{h} \]
**Options:**
**Function \( f(x) \) Options:**
1. \( f(x) = \sin(x + y) \)
2. \( f(x) = \sin(x + h) \)
3. \( f(x) = \sin(x) \)
4. \( f(x) = \frac{\sin(x)}{x} \)
5. \( f(x) = \sin(x) - 1 \)
**Point \( a \) Options:**
1. \( a = -\frac{\pi}{2} \)
2. \( a = -1 \)
3. \( a = 0 \)
4. \( a = 1 \)
5. \( a = \frac{\pi}{2} \)
**Explanation of the Limit:**
The given limit expression is a fundamental representation of the definition of the derivative:
\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]
### Steps to Determine \( f(x) \) and \( a \):
1. Compare the given limit with the standard definition to identify \( f(a + h) \) and \( f(a) \):
\[ f(a + h) = \sin\left(\frac{\pi}{2} + h\right) \]
\[ f(a) = 1 \]
2. Recall that:
\[ \sin\left(\frac{\pi}{2} + x\right) = \cos(x) \]
3. Since \( f(a) = 1 \) and comparing with \( f(a) = \sin(a) \), we find:
\[ \sin(a) = 1 \]
This indicates:
\[ a = \frac{\pi}{2} \]
4. Therefore, the original function must be:
\[ f(x) = \sin(x) \]
### Conclusion:
- **Function \( f(x) \):** \( \sin(x) \)
- **Point
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