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
![**Equation:**
\[ y = (\sec x)^{\log_2 x} \]
**Explanation:**
The given equation describes a mathematical function where the output value \( y \) is defined as the secant of \( x \), raised to the power of the logarithm base 2 of \( x \).
**Terms:**
- \( \sec x \): It stands for the secant of the angle \( x \), which is the reciprocal of the cosine function, \(\sec x = \frac{1}{\cos x}\).
- \( \log_2 x \): This is the logarithm of \( x \) with base 2.
This function intertwines trigonometric and logarithmic components, making it an example of a composite function that can exhibit complex behavior dependent on the properties of both the secant function and the logarithm function.
**Graphical Analysis (if applicable):**
- The behavior of the secant function, \(\sec x\), is undefined at odd multiples of \(\frac{\pi}{2}\) since \(\cos x = 0\) at these points.
- The domain of \( \log_2 x \) is \( x > 0 \).
Combining these behaviors, the function \( y = (\sec x)^{\log_2 x} \) will have discontinuities or undefined points, particularly where \( x \) corresponds to values leading to undefined secant values.
When graphing, look for:
- Intervals where \( \sec x \) is defined: \( x \neq (2k+1) \frac{\pi}{2} \), where \( k \) is any integer.
- Positive \( x \) values due to the logarithm’s domain requirements.
By examining these intervals and properties, you can analyze the function's behavior more comprehensively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc931b694-aaff-47c2-895e-73bc1d52993a%2F50d82be3-41a6-4642-b8a1-7d4162d8fcc0%2F6w3pdjm_processed.png&w=3840&q=75)
Transcribed Image Text:**Equation:**
\[ y = (\sec x)^{\log_2 x} \]
**Explanation:**
The given equation describes a mathematical function where the output value \( y \) is defined as the secant of \( x \), raised to the power of the logarithm base 2 of \( x \).
**Terms:**
- \( \sec x \): It stands for the secant of the angle \( x \), which is the reciprocal of the cosine function, \(\sec x = \frac{1}{\cos x}\).
- \( \log_2 x \): This is the logarithm of \( x \) with base 2.
This function intertwines trigonometric and logarithmic components, making it an example of a composite function that can exhibit complex behavior dependent on the properties of both the secant function and the logarithm function.
**Graphical Analysis (if applicable):**
- The behavior of the secant function, \(\sec x\), is undefined at odd multiples of \(\frac{\pi}{2}\) since \(\cos x = 0\) at these points.
- The domain of \( \log_2 x \) is \( x > 0 \).
Combining these behaviors, the function \( y = (\sec x)^{\log_2 x} \) will have discontinuities or undefined points, particularly where \( x \) corresponds to values leading to undefined secant values.
When graphing, look for:
- Intervals where \( \sec x \) is defined: \( x \neq (2k+1) \frac{\pi}{2} \), where \( k \) is any integer.
- Positive \( x \) values due to the logarithm’s domain requirements.
By examining these intervals and properties, you can analyze the function's behavior more comprehensively.
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