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kate middleton william public appearance balmoral - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 A linguist studying language evolution uses computational models to analyze the frequency of word usage, modeled by the equation \( f(x) = a \sin(bx + c) + d \). If \( f(0) = 2 \), \( f(\frac{\pi}{2b}) = 4 \), and the average value of the function over one period is 3, find the constants \( a \), \( b \), \( c \), and \( d \).

📌 The given function is \( f(x) = a \sin(bx + c) + d \).

📌 Using \( f(0) = 2 \):

📌 \[ f(0) = a \sin(c) + d = 2 \]

📌 Using \( f\left(\frac{\pi}{2b}\right) = 4 \):

📌 \[ f\left(\frac{\pi}{2b}\right) = a \sin\left(b \cdot \frac{\pi}{2b} + c\right) + d = a \sin\left(\frac{\pi}{2} + c\right) + d = 4 \]

📌 Since \(\sin\left(\frac{\pi}{2} + c\right) = \cos(c)\), we have:

📌 \[ a \cos(c) + d = 4 \]

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