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
![**Title: Understanding Limits and Functions**
**Objective:**
In this lesson, we will learn how to find a function \( f(x) \) and evaluate the limit of an expression involving that function.
**Problem Statement:**
Given the following limit expression, find the function \( f(x) \) and determine the value of the limit as \( h \) approaches 0.
\[ \lim_{{h \to 0}} \frac{[3(2+h)^2 + 2] - 14}{h} \]
**Step-by-Step Solution:**
1. **Identify the function \( f(x) \):**
To find \( f(x) \), we compare the expression inside the limit to the standard form of a difference quotient \( \frac{f(x+h) - f(x)}{h} \).
* Expression inside the limit: \( 3(2+h)^2 + 2 \)
* Constant term being subtracted: \( 14 \)
By comparing the expression \( 3(2+h)^2 + 2 \) with \( 14 \), it is clear that \( f(2) = 14 \).
2. **Determine the function \( f(x) \):**
To find \( f(x) \), inspect the form of \( 3(2+h)^2 + 2 \) and recognize that it fits the pattern for a quadratic function evaluated at \( x = 2 + h \).
Let's set \( x = 2 + h \). Then:
\[
f(x) = 3x^2 + 2
\]
* Verification:
For \( x = 2 \):
\[
f(2) = 3(2^2) + 2 = 3 \cdot 4 + 2 = 12 + 2 = 14
\]
This confirms that our function choice is correct.
3. **Evaluate the limit:**
Substitute \( f(x) = 3x^2 + 2 \) back into the limit expression:
\[
\lim_{{h \to 0}} \frac{f(2+h) - f(2)}{h} = \lim_{{h \to 0}} \frac{[3(2+h)^2 + 2] - 14}{h}
\]
4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2F3e6d8b56-834e-413f-a31a-c3c4d19a0cd0%2Fvj11omq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Limits and Functions**
**Objective:**
In this lesson, we will learn how to find a function \( f(x) \) and evaluate the limit of an expression involving that function.
**Problem Statement:**
Given the following limit expression, find the function \( f(x) \) and determine the value of the limit as \( h \) approaches 0.
\[ \lim_{{h \to 0}} \frac{[3(2+h)^2 + 2] - 14}{h} \]
**Step-by-Step Solution:**
1. **Identify the function \( f(x) \):**
To find \( f(x) \), we compare the expression inside the limit to the standard form of a difference quotient \( \frac{f(x+h) - f(x)}{h} \).
* Expression inside the limit: \( 3(2+h)^2 + 2 \)
* Constant term being subtracted: \( 14 \)
By comparing the expression \( 3(2+h)^2 + 2 \) with \( 14 \), it is clear that \( f(2) = 14 \).
2. **Determine the function \( f(x) \):**
To find \( f(x) \), inspect the form of \( 3(2+h)^2 + 2 \) and recognize that it fits the pattern for a quadratic function evaluated at \( x = 2 + h \).
Let's set \( x = 2 + h \). Then:
\[
f(x) = 3x^2 + 2
\]
* Verification:
For \( x = 2 \):
\[
f(2) = 3(2^2) + 2 = 3 \cdot 4 + 2 = 12 + 2 = 14
\]
This confirms that our function choice is correct.
3. **Evaluate the limit:**
Substitute \( f(x) = 3x^2 + 2 \) back into the limit expression:
\[
\lim_{{h \to 0}} \frac{f(2+h) - f(2)}{h} = \lim_{{h \to 0}} \frac{[3(2+h)^2 + 2] - 14}{h}
\]
4.
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