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...
Related questions
Question
![### Understanding Antiderivatives: A Worksheet
**Problem Statement:**
Consider the function \( f(x) = 8x^3 - 6x^2 + 3x - 8 \).
An antiderivative of \( f(x) \) is \( F(x) = Ax^4 + Bx^3 + Cx^2 + Dx \).
**Tasks:**
1. **Identify the Constants:**
Determine the values of the coefficients \( A \), \( B \), \( C \), and \( D \) in the antiderivative \( F(x) \).
- **Where \( A \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( B \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( C \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( D \) is:**
\[ \_\_\_\_\_\_\_\_ \]
---
### Explanation:
**Function and Antiderivative Relationship:**
To find the antiderivative \( F(x) \) of the function \( f(x) \), we need to integrate \( f(x) \):
\[ f(x) = 8x^3 - 6x^2 + 3x - 8 \]
Integrate term by term:
1. The integral of \( 8x^3 \):
\[ \int 8x^3 \, dx = 2x^4 \]
2. The integral of \( 6x^2 \):
\[ \int 6x^2 \, dx = 2x^3 \]
3. The integral of \( 3x \):
\[ \int 3x \, dx = x^2 \]
4. The integral of \( -8 \):
\[ \int -8 \, dx = -8x \]
Combine these integrals:
\[ F(x) = 2x^4 - 2x^3 + x^2 - 8x + C \]
**Values of \( A \), \( B \), \( C \), and \( D \):**
- \( A = 2 \)
- \( B = -2 \)
- \( C = 1 \)
- \( D =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F20f1e57d-415e-41ed-9e7b-a3cc3f0444d0%2Fmks46go_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Antiderivatives: A Worksheet
**Problem Statement:**
Consider the function \( f(x) = 8x^3 - 6x^2 + 3x - 8 \).
An antiderivative of \( f(x) \) is \( F(x) = Ax^4 + Bx^3 + Cx^2 + Dx \).
**Tasks:**
1. **Identify the Constants:**
Determine the values of the coefficients \( A \), \( B \), \( C \), and \( D \) in the antiderivative \( F(x) \).
- **Where \( A \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( B \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( C \) is:**
\[ \_\_\_\_\_\_\_\_ \]
- **And \( D \) is:**
\[ \_\_\_\_\_\_\_\_ \]
---
### Explanation:
**Function and Antiderivative Relationship:**
To find the antiderivative \( F(x) \) of the function \( f(x) \), we need to integrate \( f(x) \):
\[ f(x) = 8x^3 - 6x^2 + 3x - 8 \]
Integrate term by term:
1. The integral of \( 8x^3 \):
\[ \int 8x^3 \, dx = 2x^4 \]
2. The integral of \( 6x^2 \):
\[ \int 6x^2 \, dx = 2x^3 \]
3. The integral of \( 3x \):
\[ \int 3x \, dx = x^2 \]
4. The integral of \( -8 \):
\[ \int -8 \, dx = -8x \]
Combine these integrals:
\[ F(x) = 2x^4 - 2x^3 + x^2 - 8x + C \]
**Values of \( A \), \( B \), \( C \), and \( D \):**
- \( A = 2 \)
- \( B = -2 \)
- \( C = 1 \)
- \( D =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning