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
![**Understanding the Difference Quotient**
In this example, we aim to compute the difference between the function \(f(x)\) evaluated at \(c+h\) and at \(c\), specifically at the point \(c=3\).
The function given is:
\[ f(x) = 6x^2 \]
We are to evaluate \( f(c+h) - f(c) \) at \( c = 3 \).
### Steps to Compute:
1. **Substitute \(x = c + h\) and \(x = c\) into the function \(f(x)\).**
\[
f(c+h) = 6(c+h)^2
\]
\[
f(c) = 6c^2
\]
2. **Evaluate these expressions specifically at \(c = 3\):**
\[
f(3+h) = 6(3+h)^2
\]
\[
f(3) = 6 \cdot 3^2 = 6 \cdot 9 = 54
\]
3. **Calculate the difference:**
\[
f(3+h) - f(3) = 6(3+h)^2 - 54
\]
### Conclusion
This expression represents the change in the function’s value as \(x\) changes from \(c\) to \(c+h\).
**Note:**
- The expression inside the box signifies that it should be simplified further using \(h\) as the variable.
- There are no additional graphs or diagrams in this image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F708b5043-edcc-41eb-8461-039a390d64a5%2Fb1534ee1-5cca-4910-8425-6429c75b453d%2F567ucwe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Difference Quotient**
In this example, we aim to compute the difference between the function \(f(x)\) evaluated at \(c+h\) and at \(c\), specifically at the point \(c=3\).
The function given is:
\[ f(x) = 6x^2 \]
We are to evaluate \( f(c+h) - f(c) \) at \( c = 3 \).
### Steps to Compute:
1. **Substitute \(x = c + h\) and \(x = c\) into the function \(f(x)\).**
\[
f(c+h) = 6(c+h)^2
\]
\[
f(c) = 6c^2
\]
2. **Evaluate these expressions specifically at \(c = 3\):**
\[
f(3+h) = 6(3+h)^2
\]
\[
f(3) = 6 \cdot 3^2 = 6 \cdot 9 = 54
\]
3. **Calculate the difference:**
\[
f(3+h) - f(3) = 6(3+h)^2 - 54
\]
### Conclusion
This expression represents the change in the function’s value as \(x\) changes from \(c\) to \(c+h\).
**Note:**
- The expression inside the box signifies that it should be simplified further using \(h\) as the variable.
- There are no additional graphs or diagrams in this image.
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