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
![**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 \( x = 3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43e506db-8dc4-4921-8a99-ad8ac8b3c405%2Fcceca3b8-8228-46ff-a20f-5a30b8b802ed%2Fho6j1i_processed.png&w=3840&q=75)
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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