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The linear approximation at \( x=0 \) to \( \sin (6 x) \) is \( A+B x \) where \( A \) is: and where \( B \) is:

Ask by Lee Weaver. in the United States
Mar 13,2025

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\( A = 0 \) and \( B = 6 \).

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Bonus Knowledge

To find the linear approximation of \( \sin(6x) \) at \( x=0 \), we can use the Taylor series expansion. The first step is to evaluate \( \sin(6x) \) and its derivative at \( x=0 \). 1. \( A = \sin(6 \cdot 0) = \sin(0) = 0 \). 2. For \( B \), we calculate the derivative \( \frac{d}{dx}\sin(6x) = 6\cos(6x) \). At \( x=0 \), \( B = 6\cos(6 \cdot 0) = 6\cos(0) = 6 \). So, the linear approximation is \( 0 + 6x \), where \( A = 0 \) and \( B = 6 \).

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