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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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![### Simplifying Functions: A Step-by-Step Guide
**Problem Statement:**
Find and simplify the following for the function. (Assume \( h \neq 0 \)).
Given:
\[ f(x) = 9 \sqrt{x} \]
We need to simplify:
\[ \frac{f(a + h) - f(a)}{h} \]
**Solution Steps:**
1. **Substitute the function \( f(x) \) with \( f(a + h) \) and \( f(a) \):**
Given \( f(x) = 9 \sqrt{x} \), then:
\[
f(a + h) = 9 \sqrt{a + h}
\]
and
\[
f(a) = 9 \sqrt{a}
\]
2. **Apply these into the difference quotient formula:**
\[
\frac{f(a + h) - f(a)}{h} = \frac{9 \sqrt{a + h} - 9 \sqrt{a}}{h}
\]
Simplifying this expression can help to find the limiting value as \( h \) approaches 0.
3. **Simplify the numerator:**
Notice that the common factor of 9 can be factored out:
\[
\frac{9 (\sqrt{a + h} - \sqrt{a})}{h}
\]
4. **Rationalize the numerator if necessary, using techniques like multiplying and dividing by the conjugate, to further simplify the expression.**
5. **As \( h \) approaches 0, the simplified form of the limit will give the derivative.**
**Need Help?**
- **Read It:** Access detailed written explanations and examples.
- **Watch It:** View instructional videos for visual and auditory learning.
- **Talk to a Tutor:** Get personalized assistance from a tutor to improve your understanding.
This problem is an introduction to the concept of finding the derivative of a function using the difference quotient, which is fundamental in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad356ace-94cd-4e8a-aa7f-bdf17308114e%2F985e1b4c-ca49-42e6-8572-5e0e153ed7ed%2Faia3v58.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Functions: A Step-by-Step Guide
**Problem Statement:**
Find and simplify the following for the function. (Assume \( h \neq 0 \)).
Given:
\[ f(x) = 9 \sqrt{x} \]
We need to simplify:
\[ \frac{f(a + h) - f(a)}{h} \]
**Solution Steps:**
1. **Substitute the function \( f(x) \) with \( f(a + h) \) and \( f(a) \):**
Given \( f(x) = 9 \sqrt{x} \), then:
\[
f(a + h) = 9 \sqrt{a + h}
\]
and
\[
f(a) = 9 \sqrt{a}
\]
2. **Apply these into the difference quotient formula:**
\[
\frac{f(a + h) - f(a)}{h} = \frac{9 \sqrt{a + h} - 9 \sqrt{a}}{h}
\]
Simplifying this expression can help to find the limiting value as \( h \) approaches 0.
3. **Simplify the numerator:**
Notice that the common factor of 9 can be factored out:
\[
\frac{9 (\sqrt{a + h} - \sqrt{a})}{h}
\]
4. **Rationalize the numerator if necessary, using techniques like multiplying and dividing by the conjugate, to further simplify the expression.**
5. **As \( h \) approaches 0, the simplified form of the limit will give the derivative.**
**Need Help?**
- **Read It:** Access detailed written explanations and examples.
- **Watch It:** View instructional videos for visual and auditory learning.
- **Talk to a Tutor:** Get personalized assistance from a tutor to improve your understanding.
This problem is an introduction to the concept of finding the derivative of a function using the difference quotient, which is fundamental in calculus.
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