Identify the original function f(x) and what value of c to evaluate f'(c). (3x-9x²)+(42) x+2 lim x→−2 Edit View Insert Format Tools Table

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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**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.
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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