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f(x) = 3 sec( x (0, 4п) 2

f(x) = 3 sec( x (0, 4п) 2

Calculus: Early Transcendentals
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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Describe the concavity. Write out the answer in interval notation.

The function is defined as follows: \[ f(x) = 3 \sec \left( x - \frac{\pi}{2} \right) \] The domain of the function is specified as \( (0, 4\pi) \). ### Explanation: This function represents a secant function, which is the reciprocal of the cosine function, scaled by a factor of 3. The secant function is defined as \(\sec(x) = \frac{1}{\cos(x)}\). In this context, the expression \(x - \frac{\pi}{2}\) indicates a horizontal shift of \(\frac{\pi}{2}\) units to the right. The domain of the function, \( (0, 4\pi) \), means the function is considered over the interval from \(0\) to \(4\pi\), not including either endpoint. ### Graphical Considerations: - **Amplitude**: The amplitude is scaled by a factor of 3. - **Period**: The period of the basic secant function is \(2\pi\), but due to the horizontal shift, the interval will encompass two full periods within \( (0, 4\pi) \). - **Vertical Asymptotes**: The secant function will have vertical asymptotes where the cosine of the shifted angle is zero. These occur regularly along the x-axis. This function graphically represents vertical elongations and compressions typical to a secant curve, repeating over the specified domain.
Transcribed Image Text:The function is defined as follows: \[ f(x) = 3 \sec \left( x - \frac{\pi}{2} \right) \] The domain of the function is specified as \( (0, 4\pi) \). ### Explanation: This function represents a secant function, which is the reciprocal of the cosine function, scaled by a factor of 3. The secant function is defined as \(\sec(x) = \frac{1}{\cos(x)}\). In this context, the expression \(x - \frac{\pi}{2}\) indicates a horizontal shift of \(\frac{\pi}{2}\) units to the right. The domain of the function, \( (0, 4\pi) \), means the function is considered over the interval from \(0\) to \(4\pi\), not including either endpoint. ### Graphical Considerations: - **Amplitude**: The amplitude is scaled by a factor of 3. - **Period**: The period of the basic secant function is \(2\pi\), but due to the horizontal shift, the interval will encompass two full periods within \( (0, 4\pi) \). - **Vertical Asymptotes**: The secant function will have vertical asymptotes where the cosine of the shifted angle is zero. These occur regularly along the x-axis. This function graphically represents vertical elongations and compressions typical to a secant curve, repeating over the specified domain.
Expert Solution
Step 1

Consider the provided function,

fx=3secx-π2,0,4π

Describe the concavity.

If f'x>0 then fx concave upwards.

If f'x<0 then fx concave downwards.

Now, find the derivative.

f'x=3secx-π2tanx-π2=-3cosecxcotx

 

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