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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![### Analyzing and Sketching Polynomial Functions
In this tutorial, we will analyze and sketch the graph of a given polynomial function. We will find any intercepts, relative extrema, points of inflection, and asymptotes for the function:
\[ y = 3x^4 + 4x^3 \]
#### Steps for Analysis:
1. **Intercepts:**
- **Y-intercept:** Found by evaluating the function at \( x = 0 \).
- **X-intercepts:** Found by solving \( y = 0 \).
2. **Relative Extrema:**
- Use the first derivative test to find critical points.
- Determine whether each critical point is a local minimum or maximum by analyzing the sign changes of the first derivative.
3. **Points of Inflection:**
- Identify points where the second derivative changes sign.
- Verify by evaluating the second derivative at those points to confirm if they are points of inflection.
4. **Asymptotes:**
- For polynomial functions, vertical asymptotes do not exist.
- Check for horizontal asymptotes by analyzing the end behavior of the function as \( x \) approaches infinity.
5. **Sketching the Graph:**
- Combine all the information gathered to plot the intercepts, critical points, and points of inflection.
- Show the overall shape considering the end behavior indicated by the leading term (in this case, \( 3x^4 \)).
#### Detailed Calculation:
1. **Finding the Intercepts:**
- Y-intercept at \( x = 0 \):
\[
y(0) = 3(0)^4 + 4(0)^3 = 0
\]
So, the y-intercept is at (0,0).
- X-intercepts by solving \( 3x^4 + 4x^3 = 0 \):
\[
x^3(3x + 4) = 0
\]
This gives \( x = 0 \) and \( x = -\frac{4}{3} \).
Therefore, the x-intercepts are at (0,0) and \((-4/3, 0)\).
2. **Finding Relative Extrema:**
- First derivative: \( y' = \frac{d}{dx}(3x^4 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6e1a3f1-b538-4464-8269-7761edea80d0%2F084c4ee0-173b-4430-af4c-ed592e4820b9%2Fmo6xm4_processed.png&w=3840&q=75)
Transcribed Image Text:### Analyzing and Sketching Polynomial Functions
In this tutorial, we will analyze and sketch the graph of a given polynomial function. We will find any intercepts, relative extrema, points of inflection, and asymptotes for the function:
\[ y = 3x^4 + 4x^3 \]
#### Steps for Analysis:
1. **Intercepts:**
- **Y-intercept:** Found by evaluating the function at \( x = 0 \).
- **X-intercepts:** Found by solving \( y = 0 \).
2. **Relative Extrema:**
- Use the first derivative test to find critical points.
- Determine whether each critical point is a local minimum or maximum by analyzing the sign changes of the first derivative.
3. **Points of Inflection:**
- Identify points where the second derivative changes sign.
- Verify by evaluating the second derivative at those points to confirm if they are points of inflection.
4. **Asymptotes:**
- For polynomial functions, vertical asymptotes do not exist.
- Check for horizontal asymptotes by analyzing the end behavior of the function as \( x \) approaches infinity.
5. **Sketching the Graph:**
- Combine all the information gathered to plot the intercepts, critical points, and points of inflection.
- Show the overall shape considering the end behavior indicated by the leading term (in this case, \( 3x^4 \)).
#### Detailed Calculation:
1. **Finding the Intercepts:**
- Y-intercept at \( x = 0 \):
\[
y(0) = 3(0)^4 + 4(0)^3 = 0
\]
So, the y-intercept is at (0,0).
- X-intercepts by solving \( 3x^4 + 4x^3 = 0 \):
\[
x^3(3x + 4) = 0
\]
This gives \( x = 0 \) and \( x = -\frac{4}{3} \).
Therefore, the x-intercepts are at (0,0) and \((-4/3, 0)\).
2. **Finding Relative Extrema:**
- First derivative: \( y' = \frac{d}{dx}(3x^4 +
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