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
Find dy/dx
![**Mathematical Expression:**
In this section, we will analyze the given mathematical expression. The expression is written as:
\[ y = \left(1 + \tan^{-1}(x)\right)^3 \]
**Explanation:**
1. **Equation Components:**
- \( y \) represents the dependent variable.
- The expression on the right side of the equation is \( \left(1 + \tan^{-1}(x)\right)^3 \).
2. **Inverse Tangent Function (\(\tan^{-1}(x)\)):**
- \(\tan^{-1}(x)\) is the inverse tangent function, also known as arctan. It returns the angle whose tangent is \(x\).
- The range of \(\tan^{-1}(x)\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
3. **Addition:**
- The inverse tangent function \(\tan^{-1}(x)\) is added to 1.
4. **Exponentiation:**
- The resulting sum \(\left(1 + \tan^{-1}(x)\right)\) is then raised to the power of 3.
**Graphical Representation:**
To understand this function's behavior, we can plot \( y = \left(1 + \tan^{-1}(x)\right)^3 \). We would observe the following characteristics:
- **Asymptotic Behavior:** Since \(\tan^{-1}(x)\) approaches \(\frac{\pi}{2}\) as \(x\) approaches \(\infty\) and approaches \(-\frac{\pi}{2}\) as \(x\) approaches \(-\infty\), the function will show smoothing out effect towards the extremities.
- **Increasing Function:** The function increases as \(x\) increases due to the positivity of the exponent.
This expression finds its applications in various mathematical analyses and helps in understanding the intricacies of transformations and functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2Fde080474-d43e-4a87-96ba-cb607b76231c%2F52fhrng_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Expression:**
In this section, we will analyze the given mathematical expression. The expression is written as:
\[ y = \left(1 + \tan^{-1}(x)\right)^3 \]
**Explanation:**
1. **Equation Components:**
- \( y \) represents the dependent variable.
- The expression on the right side of the equation is \( \left(1 + \tan^{-1}(x)\right)^3 \).
2. **Inverse Tangent Function (\(\tan^{-1}(x)\)):**
- \(\tan^{-1}(x)\) is the inverse tangent function, also known as arctan. It returns the angle whose tangent is \(x\).
- The range of \(\tan^{-1}(x)\) is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians.
3. **Addition:**
- The inverse tangent function \(\tan^{-1}(x)\) is added to 1.
4. **Exponentiation:**
- The resulting sum \(\left(1 + \tan^{-1}(x)\right)\) is then raised to the power of 3.
**Graphical Representation:**
To understand this function's behavior, we can plot \( y = \left(1 + \tan^{-1}(x)\right)^3 \). We would observe the following characteristics:
- **Asymptotic Behavior:** Since \(\tan^{-1}(x)\) approaches \(\frac{\pi}{2}\) as \(x\) approaches \(\infty\) and approaches \(-\frac{\pi}{2}\) as \(x\) approaches \(-\infty\), the function will show smoothing out effect towards the extremities.
- **Increasing Function:** The function increases as \(x\) increases due to the positivity of the exponent.
This expression finds its applications in various mathematical analyses and helps in understanding the intricacies of transformations and functions.
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