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
Need first derivative analysis. Y and Y’’ where given. Y’ is written with arrow next to it.
![### Mathematical Transcriptions:
**Function:**
\[ y = \frac{-x^3}{(x+2)(x-2)} \]
**First Derivative:**
\[ y' = \frac{-x^4 + 12x^2}{(x-2)^2(x+2)^2} \]
**Second Derivative:**
\[ y'' = \frac{-6x(x^2 - 2x + 4)}{(x+2)^3(x-1)^3} \]
**Notes:**
- The analysis starts with the first derivative.
- An arrow points to the first derivative with the note “need first derivative analysis.”
### Explanation for Educational Purpose:
1. **Original Function:**
- This is a rational function where the numerator is a cubic polynomial, \(-x^3\), and the denominator is a product of two linear factors, \((x+2)(x-2)\).
2. **First Derivative:**
- The first derivative \( y' \) is obtained through differentiation and is also a rational function. It shows the rate of change of the original function.
- The numerator is a quartic polynomial, \(-x^4 + 12x^2\).
- The denominator consists of the factors \((x-2)^2\) and \((x+2)^2\), indicating possible points of inflection or vertical asymptotes.
3. **Second Derivative:**
- The second derivative \( y'' \) is used to determine the concavity and points of inflection of the function.
- Its numerator features a product of a linear term \(-6x\) and a quadratic trinomial \((x^2 - 2x + 4)\).
- Its denominator, \((x+2)^3(x-1)^3\), implies more complex behavior near these points.
4. **Analysis Notes:**
- Before proceeding with further analysis like concavity or points of inflection using \( y'' \), the behavior of \( y' \) needs to be understood, which typically includes finding critical points and analyzing intervals for increasing or decreasing behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3341284f-d761-40a1-94fb-8b0932aec828%2F406d3ace-1814-4f8d-ab96-285a23fab146%2Fx46in5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Transcriptions:
**Function:**
\[ y = \frac{-x^3}{(x+2)(x-2)} \]
**First Derivative:**
\[ y' = \frac{-x^4 + 12x^2}{(x-2)^2(x+2)^2} \]
**Second Derivative:**
\[ y'' = \frac{-6x(x^2 - 2x + 4)}{(x+2)^3(x-1)^3} \]
**Notes:**
- The analysis starts with the first derivative.
- An arrow points to the first derivative with the note “need first derivative analysis.”
### Explanation for Educational Purpose:
1. **Original Function:**
- This is a rational function where the numerator is a cubic polynomial, \(-x^3\), and the denominator is a product of two linear factors, \((x+2)(x-2)\).
2. **First Derivative:**
- The first derivative \( y' \) is obtained through differentiation and is also a rational function. It shows the rate of change of the original function.
- The numerator is a quartic polynomial, \(-x^4 + 12x^2\).
- The denominator consists of the factors \((x-2)^2\) and \((x+2)^2\), indicating possible points of inflection or vertical asymptotes.
3. **Second Derivative:**
- The second derivative \( y'' \) is used to determine the concavity and points of inflection of the function.
- Its numerator features a product of a linear term \(-6x\) and a quadratic trinomial \((x^2 - 2x + 4)\).
- Its denominator, \((x+2)^3(x-1)^3\), implies more complex behavior near these points.
4. **Analysis Notes:**
- Before proceeding with further analysis like concavity or points of inflection using \( y'' \), the behavior of \( y' \) needs to be understood, which typically includes finding critical points and analyzing intervals for increasing or decreasing behavior.
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