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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find it's derivative and solve for its critical points
![**Problem 46:**
The given function is:
\[ y = e^{2x} - e^x \]
In this expression, \(e\) is the base of the natural logarithm. This equation represents a difference between two exponential functions with bases \(e\), where the exponents are \(2x\) and \(x\) respectively.
**Explanation:**
- The term \(e^{2x}\) grows rapidly as \(x\) increases because the exponent is multiplied by 2.
- The term \(e^x\) also grows exponentially, but at a slower rate compared to \(e^{2x}\).
- The function \(y = e^{2x} - e^x\) will demonstrate behavior based on the rates of growth of these exponential components. At larger values of \(x\), \(e^{2x}\) will dominate the expression.
This function might be further explored by analyzing its derivatives, finding critical points, or evaluating its limits at specific values of \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90d2137b-83f2-4c15-8300-1a033b76054d%2F109266f4-b734-4b21-acd8-b5752ab0cb1b%2Fn9ftgli_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 46:**
The given function is:
\[ y = e^{2x} - e^x \]
In this expression, \(e\) is the base of the natural logarithm. This equation represents a difference between two exponential functions with bases \(e\), where the exponents are \(2x\) and \(x\) respectively.
**Explanation:**
- The term \(e^{2x}\) grows rapidly as \(x\) increases because the exponent is multiplied by 2.
- The term \(e^x\) also grows exponentially, but at a slower rate compared to \(e^{2x}\).
- The function \(y = e^{2x} - e^x\) will demonstrate behavior based on the rates of growth of these exponential components. At larger values of \(x\), \(e^{2x}\) will dominate the expression.
This function might be further explored by analyzing its derivatives, finding critical points, or evaluating its limits at specific values of \(x\).
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