d f'(3) using f'(a) = lim 2x² + 3x5 f(a+h)-f(x) h for the following fun

Calculus: Early Transcendentals
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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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**Problem 1: Calculating the Derivative at a Point**

1. **Objective**: Find \( f'(3) \) using the definition of the derivative.

2. **Definition of the Derivative**: 

   The derivative of a function \( f \) at a point \( a \) is given by:
   \[
   f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
   \]

3. **Given Function**: 

   \( f(x) = 2x^2 + 3x - 5 \)

**Solution Approach**:

To find \( f'(3) \), we will substitute \( a = 3 \) into the derivative definition and evaluate the limit. We need to compute:
\[
f(3+h) = 2(3+h)^2 + 3(3+h) - 5
\]

After expanding and simplifying, we then evaluate the limit as \( h \) approaches 0 to find the derivative at \( x = 3 \).
Transcribed Image Text:**Problem 1: Calculating the Derivative at a Point** 1. **Objective**: Find \( f'(3) \) using the definition of the derivative. 2. **Definition of the Derivative**: The derivative of a function \( f \) at a point \( a \) is given by: \[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] 3. **Given Function**: \( f(x) = 2x^2 + 3x - 5 \) **Solution Approach**: To find \( f'(3) \), we will substitute \( a = 3 \) into the derivative definition and evaluate the limit. We need to compute: \[ f(3+h) = 2(3+h)^2 + 3(3+h) - 5 \] After expanding and simplifying, we then evaluate the limit as \( h \) approaches 0 to find the derivative at \( x = 3 \).
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