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 f prime of x
![This mathematical function is expressed as follows:
\[ f(x) = \sqrt[3]{\frac{e^{4x} - 1}{1 + 4^x}} \]
Here’s the breakdown of the components of the function:
1. **Exponentiation and the Natural Exponential Function:**
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
- \( e^{4x} \) represents the exponential function where the exponent is \( 4x \).
2. **Subtraction:**
- The expression inside the numerator, \( e^{4x} - 1 \), subtracts 1 from the exponential expression.
3. **Division:**
- This numeral is then divided by \( 1 + 4^x \) where \( 4^x \) is \( 4 \) raised to the power of \( x \).
4. **Cube Root:**
- Finally, the entire fraction is under a cube root, denoted by \( \sqrt[3]{\cdot} \).
This function describes a relationship between \( x \) and \( f(x) \) which involves exponential growth, normalization by an additional exponential term, and the transformation by taking the cube root of the resulting fraction. This type of function can model certain types of growth curves and transformations in advanced mathematical studies such as calculus and differential equations.
In terms of graphing, this function could exhibit rapid changes depending on the values of \( x \), especially since it involves exponential terms. However, it would ultimately be tempered by the cube root operation which moderates the growth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe7817cb-5274-4434-90ec-13ab480c973d%2F12e22079-ba77-448b-a54c-386eef04f62a%2Fcc53ww_processed.png&w=3840&q=75)
Transcribed Image Text:This mathematical function is expressed as follows:
\[ f(x) = \sqrt[3]{\frac{e^{4x} - 1}{1 + 4^x}} \]
Here’s the breakdown of the components of the function:
1. **Exponentiation and the Natural Exponential Function:**
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
- \( e^{4x} \) represents the exponential function where the exponent is \( 4x \).
2. **Subtraction:**
- The expression inside the numerator, \( e^{4x} - 1 \), subtracts 1 from the exponential expression.
3. **Division:**
- This numeral is then divided by \( 1 + 4^x \) where \( 4^x \) is \( 4 \) raised to the power of \( x \).
4. **Cube Root:**
- Finally, the entire fraction is under a cube root, denoted by \( \sqrt[3]{\cdot} \).
This function describes a relationship between \( x \) and \( f(x) \) which involves exponential growth, normalization by an additional exponential term, and the transformation by taking the cube root of the resulting fraction. This type of function can model certain types of growth curves and transformations in advanced mathematical studies such as calculus and differential equations.
In terms of graphing, this function could exhibit rapid changes depending on the values of \( x \), especially since it involves exponential terms. However, it would ultimately be tempered by the cube root operation which moderates the growth.
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