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
![**Identifying the Original Function and Evaluating Its Derivative at a Point**
The task is to identify the original function \( f(x) \) and determine the value of \( c \) to evaluate \( f'(c) \). We start by exploring the given limit:
\[
\lim_{{x \to 2}} \frac{(3x - 9x^2) + 42}{x + 2}
\]
**Steps to Solve:**
1. **Simplify the Expression:**
- Combine and rearrange terms in the numerator, \((3x - 9x^2) + 42\).
2. **Identify the Original Function \( f(x) \):**
- Determine whether the expression can be factored or is a derivative of another function, leading us to \( f(x) \).
3. **Determine the Value of \( c \):**
- Analyze when evaluating the limit gives an indeterminate form to find \( c \), for derivative calculation.
4. **Calculate the Derivative of \( f(x) \) and Evaluate at \( c \).**
This process involves algebraic manipulation and understanding of calculus principles such as limits and derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0da95ed-4961-453b-bb3e-75b23e1037ca%2F8fd7ec80-ca24-4ce9-8aca-b95974b82021%2Fsw697nb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Identifying the Original Function and Evaluating Its Derivative at a Point**
The task is to identify the original function \( f(x) \) and determine the value of \( c \) to evaluate \( f'(c) \). We start by exploring the given limit:
\[
\lim_{{x \to 2}} \frac{(3x - 9x^2) + 42}{x + 2}
\]
**Steps to Solve:**
1. **Simplify the Expression:**
- Combine and rearrange terms in the numerator, \((3x - 9x^2) + 42\).
2. **Identify the Original Function \( f(x) \):**
- Determine whether the expression can be factored or is a derivative of another function, leading us to \( f(x) \).
3. **Determine the Value of \( c \):**
- Analyze when evaluating the limit gives an indeterminate form to find \( c \), for derivative calculation.
4. **Calculate the Derivative of \( f(x) \) and Evaluate at \( c \).**
This process involves algebraic manipulation and understanding of calculus principles such as limits and derivatives.
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