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
![**Graphing and Analyzing Functions**
**Objective:**
- Graph the function \( f(x) \)
- Determine \( f'(x) \)
- Then, graph \( f'(x) \)
**Given Function:**
\[ f(x) = 1 + x + x^2 \]
**Instructions:**
1. Begin by graphing the given function \( f(x) = 1 + x + x^2 \).
2. Calculate the derivative of the function to find \( f'(x) \).
3. Plot the graph of \( f'(x) \).
**Steps to Follow:**
1. **Graph \( f(x) = 1 + x + x^2 \)**
- Identify key points such as the vertex and intercepts.
- Plot several values to establish the shape of the parabola.
2. **Find \( f'(x) \)**
- Apply appropriate differentiation rules to find the first derivative of the function.
3. **Graph \( f'(x) \)**
- Using the derivative obtained, identify key characteristics like slope at given points, intercepts, and general shape.
- Plot this derivative to visualize how \( f'(x) \) behaves relative to the original function \( f(x) \).
**Note:** Pay close attention to shifts in the graph and changes in curvature, inflection points, and how the derivative provides insights into the increasing and decreasing nature of the original function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2F1c8571f3-93da-4c3b-b5c7-5dde81820cef%2F9pd8som_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graphing and Analyzing Functions**
**Objective:**
- Graph the function \( f(x) \)
- Determine \( f'(x) \)
- Then, graph \( f'(x) \)
**Given Function:**
\[ f(x) = 1 + x + x^2 \]
**Instructions:**
1. Begin by graphing the given function \( f(x) = 1 + x + x^2 \).
2. Calculate the derivative of the function to find \( f'(x) \).
3. Plot the graph of \( f'(x) \).
**Steps to Follow:**
1. **Graph \( f(x) = 1 + x + x^2 \)**
- Identify key points such as the vertex and intercepts.
- Plot several values to establish the shape of the parabola.
2. **Find \( f'(x) \)**
- Apply appropriate differentiation rules to find the first derivative of the function.
3. **Graph \( f'(x) \)**
- Using the derivative obtained, identify key characteristics like slope at given points, intercepts, and general shape.
- Plot this derivative to visualize how \( f'(x) \) behaves relative to the original function \( f(x) \).
**Note:** Pay close attention to shifts in the graph and changes in curvature, inflection points, and how the derivative provides insights into the increasing and decreasing nature of the original function.
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