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 the derivative
![The mathematical equation shown is:
\[ y = \sqrt{2 - x} \cdot \ln(x) \]
Here's a breakdown of the components:
- \( y \) represents the dependent variable or output of the function.
- \( \sqrt{2 - x} \) is the square root of the expression \( (2 - x) \). This part is only defined for values of \( x \) such that \( 2 - x \geq 0 \), or \( x \leq 2 \).
- \( \ln(x) \) is the natural logarithm of \( x \), which is defined for \( x > 0 \).
- The expression uses multiplication (\( \cdot \)) between \( \sqrt{2 - x} \) and \( \ln(x) \).
For the function to be defined, \( x \) must satisfy both \( x \leq 2 \) and \( x > 0 \). Therefore, the function is valid for \( 0 < x \leq 2 \).
Graphs and diagrams could illustrate how the function behaves over this interval, showcasing any critical points, intercepts, or asymptotic behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa32a331a-44ec-4e8d-a40f-5fbd27452f88%2F8d8c8635-600f-47e9-8c62-cce2c918f30e%2Frbyty4a_processed.png&w=3840&q=75)
Transcribed Image Text:The mathematical equation shown is:
\[ y = \sqrt{2 - x} \cdot \ln(x) \]
Here's a breakdown of the components:
- \( y \) represents the dependent variable or output of the function.
- \( \sqrt{2 - x} \) is the square root of the expression \( (2 - x) \). This part is only defined for values of \( x \) such that \( 2 - x \geq 0 \), or \( x \leq 2 \).
- \( \ln(x) \) is the natural logarithm of \( x \), which is defined for \( x > 0 \).
- The expression uses multiplication (\( \cdot \)) between \( \sqrt{2 - x} \) and \( \ln(x) \).
For the function to be defined, \( x \) must satisfy both \( x \leq 2 \) and \( x > 0 \). Therefore, the function is valid for \( 0 < x \leq 2 \).
Graphs and diagrams could illustrate how the function behaves over this interval, showcasing any critical points, intercepts, or asymptotic behavior.
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