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
![**Limit and Derivatives: Concept and Calculation**
In this section, we will explore the concept of limits and derivatives through a specific example.
### Example Problem:
\[ \lim_{h \to 0} \left( \frac{f(1 + h) - f(1)}{h} \right) \]
where \( f(x) = 9x^2 + x - 4 \)
### Step-by-Step Solution:
1. **Function Definition:**
\[ f(x) = 9x^2 + x - 4 \]
2. **Compute \( f(1 + h) \):**
Substitute \( 1 + h \) for \( x \) in the function \( f(x) \):
\[ f(1 + h) = 9(1 + h)^2 + (1 + h) - 4 \]
3. **Expand \( f(1 + h) \):**
\[ f(1 + h) = 9(1 + 2h + h^2) + 1 + h - 4 \]
\[ = 9 + 18h + 9h^2 + 1 + h - 4 \]
\[ = 9h^2 + 19h + 6 \]
4. **Compute \( f(1) \):**
\[ f(1) = 9(1)^2 + 1 - 4 \]
\[ = 9 + 1 - 4 \]
\[ = 6 \]
5. **Formulating the Difference:**
\[ f(1 + h) - f(1) = (9h^2 + 19h + 6) - 6 \]
\[ = 9h^2 + 19h \]
6. **Substitute into Limit Expression:**
\[ \lim_{h \to 0} \left( \frac{9h^2 + 19h}{h} \right) \]
7. **Simplify the Fraction:**
\[ = \lim_{h \to 0} \left( 9h + 19 \right) \]
8. **Taking the Limit:**
\[ \lim_{h \to 0} (9h + 19) = 9(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F492efe9f-0a9a-45cd-99ee-2fd493a23754%2F3f3277c2-14ca-490b-ad9a-3fd3308819f0%2Fuydhkh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Limit and Derivatives: Concept and Calculation**
In this section, we will explore the concept of limits and derivatives through a specific example.
### Example Problem:
\[ \lim_{h \to 0} \left( \frac{f(1 + h) - f(1)}{h} \right) \]
where \( f(x) = 9x^2 + x - 4 \)
### Step-by-Step Solution:
1. **Function Definition:**
\[ f(x) = 9x^2 + x - 4 \]
2. **Compute \( f(1 + h) \):**
Substitute \( 1 + h \) for \( x \) in the function \( f(x) \):
\[ f(1 + h) = 9(1 + h)^2 + (1 + h) - 4 \]
3. **Expand \( f(1 + h) \):**
\[ f(1 + h) = 9(1 + 2h + h^2) + 1 + h - 4 \]
\[ = 9 + 18h + 9h^2 + 1 + h - 4 \]
\[ = 9h^2 + 19h + 6 \]
4. **Compute \( f(1) \):**
\[ f(1) = 9(1)^2 + 1 - 4 \]
\[ = 9 + 1 - 4 \]
\[ = 6 \]
5. **Formulating the Difference:**
\[ f(1 + h) - f(1) = (9h^2 + 19h + 6) - 6 \]
\[ = 9h^2 + 19h \]
6. **Substitute into Limit Expression:**
\[ \lim_{h \to 0} \left( \frac{9h^2 + 19h}{h} \right) \]
7. **Simplify the Fraction:**
\[ = \lim_{h \to 0} \left( 9h + 19 \right) \]
8. **Taking the Limit:**
\[ \lim_{h \to 0} (9h + 19) = 9(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 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