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
Instructions say to find the first derivative.
![### Mathematical Function Representation
The given image shows a mathematical function written in standard algebraic notation. The function \( f(x) \) is defined as follows:
\[ f(x) = x^a \sec(ax) \]
Here is a breakdown of the components of the function:
1. **\( f(x) \)**: This denotes a function \( f \) with \( x \) as the independent variable.
2. **\( x^a \)**: This term represents \( x \) raised to the power of \( a \), where \( a \) is a constant exponent.
3. **\( \sec(ax) \)**: This term involves the secant function, \( \sec \), which is the reciprocal of the cosine function. It is written as \( \sec(ax) \), indicating that the argument of the secant function is \( ax \), where \( a \) is a constant coefficient.
### Explanation of Components
- **Exponentiation**: The part \( x^a \) describes exponential growth or decay depending on the value of \( a \). If \( a \) is positive, the term \( x^a \) increases as \( x \) increases. If \( a \) is negative, \( x^a \) decreases as \( x \) increases.
- **Secant Function**: The secant of an angle (or argument) is the reciprocal of its cosine. That is:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
Applied in the given function with \( ax \) as the argument, it accounts for periodic behavior that has asymptotic points where the cosine function is zero (since division by zero is undefined).
### Implications and Usage
The combination of a power function \( x^a \) with a secant function \( \sec(ax) \) can model various phenomena in physics and engineering where both growth and periodic behavior are observed. The function is also used in mathematical analysis for solving differential equations and evaluating integrals involving trigonometric identities.
### Visualization
While the image does not contain any graphs or diagrams, visualizing this function would reveal:
- **Oscillations**: Due to the \( \sec(ax) \) term, the function will have vertical asymptotes where \( \cos(ax) \) is zero.
- **Growth/Decay Behavior**: Depending on the value of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3e7eb8b-0b75-4c6c-a4cc-30a54bf299fc%2F4157018a-4763-4bc1-821e-0e2124af90a7%2Fw8j2jrq_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Function Representation
The given image shows a mathematical function written in standard algebraic notation. The function \( f(x) \) is defined as follows:
\[ f(x) = x^a \sec(ax) \]
Here is a breakdown of the components of the function:
1. **\( f(x) \)**: This denotes a function \( f \) with \( x \) as the independent variable.
2. **\( x^a \)**: This term represents \( x \) raised to the power of \( a \), where \( a \) is a constant exponent.
3. **\( \sec(ax) \)**: This term involves the secant function, \( \sec \), which is the reciprocal of the cosine function. It is written as \( \sec(ax) \), indicating that the argument of the secant function is \( ax \), where \( a \) is a constant coefficient.
### Explanation of Components
- **Exponentiation**: The part \( x^a \) describes exponential growth or decay depending on the value of \( a \). If \( a \) is positive, the term \( x^a \) increases as \( x \) increases. If \( a \) is negative, \( x^a \) decreases as \( x \) increases.
- **Secant Function**: The secant of an angle (or argument) is the reciprocal of its cosine. That is:
\[
\sec(\theta) = \frac{1}{\cos(\theta)}
\]
Applied in the given function with \( ax \) as the argument, it accounts for periodic behavior that has asymptotic points where the cosine function is zero (since division by zero is undefined).
### Implications and Usage
The combination of a power function \( x^a \) with a secant function \( \sec(ax) \) can model various phenomena in physics and engineering where both growth and periodic behavior are observed. The function is also used in mathematical analysis for solving differential equations and evaluating integrals involving trigonometric identities.
### Visualization
While the image does not contain any graphs or diagrams, visualizing this function would reveal:
- **Oscillations**: Due to the \( \sec(ax) \) term, the function will have vertical asymptotes where \( \cos(ax) \) is zero.
- **Growth/Decay Behavior**: Depending on the value of
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