The limit given below represents the derivative of some function f at some number a. What f(x) and the number a? lim h→0 Provide your answer below: f(x)= =, a= [9(2+h)+(-6-3h) +10] - 148 h
The limit given below represents the derivative of some function f at some number a. What f(x) and the number a? lim h→0 Provide your answer below: f(x)= =, a= [9(2+h)+(-6-3h) +10] - 148 h
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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![### Calculus Problem on Derivatives
The limit given below represents the derivative of some function \( f \) at some number \( a \). What are \( f(x) \) and the number \( a \)?
\[
\lim_{{h \to 0}} \frac{{[9(2 + h)^4 + (-6 - 3h) + 10] - 148}}{h}
\]
Provide your answer below:
\[ f(x) = \_\_, \quad a = \_\_ \]
#### Explanation:
This limit is in the form that represents the definition of the derivative of a function \( f \) at a specific point \( a \). To find the function \( f(x) \) and the point \( a \), we need to identify the function inside the limit and the value of \( a \).
1. **Identify the function \( f \)**:
The expression inside the limit provides clues to what \( f(x) \) should be.
2. **Rewrite the expression**:
Simplify the term inside the limit to find \( f(x) \) explicitly.
3. **Determine the value of \( a \)**:
Compare the simplified expression with the standard form of the difference quotient to identify the point \( a \).
Please show your work and reasoning below when determining \( f(x) \) and \( a \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3c9b7cf8-bc31-43fb-8f2d-f6ce11207b4a%2F6f5a484d-3944-4b2c-97f0-581b13e25aa2%2Fa77wtc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problem on Derivatives
The limit given below represents the derivative of some function \( f \) at some number \( a \). What are \( f(x) \) and the number \( a \)?
\[
\lim_{{h \to 0}} \frac{{[9(2 + h)^4 + (-6 - 3h) + 10] - 148}}{h}
\]
Provide your answer below:
\[ f(x) = \_\_, \quad a = \_\_ \]
#### Explanation:
This limit is in the form that represents the definition of the derivative of a function \( f \) at a specific point \( a \). To find the function \( f(x) \) and the point \( a \), we need to identify the function inside the limit and the value of \( a \).
1. **Identify the function \( f \)**:
The expression inside the limit provides clues to what \( f(x) \) should be.
2. **Rewrite the expression**:
Simplify the term inside the limit to find \( f(x) \) explicitly.
3. **Determine the value of \( a \)**:
Compare the simplified expression with the standard form of the difference quotient to identify the point \( a \).
Please show your work and reasoning below when determining \( f(x) \) and \( a \).
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