Question

the linear approximation at ( x = 0 ) to ( sin(7x) ) is ( a + bx ) where ( a ) is: and where ( b ) is: question help: video message instructor submit question jump to answer

Explanation:

Step1: Recall the formula for linear approximation

The linear approximation of a function \(y = f(x)\) at \(x = a\) is \(L(x)=f(a)+f^{\prime}(a)(x - a)\). Here \(a = 0\) and \(f(x)=\sin(7x)\).

Step2: Calculate \(f(0)\)

We know that if \(f(x)=\sin(7x)\), then \(f(0)=\sin(7\times0)=\sin(0) = 0\). So \(A=f(0)=0\).

Step3: Calculate the derivative of \(f(x)\)

Using the chain - rule, if \(y=\sin(u)\) and \(u = 7x\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Since \(\frac{d}{du}(\sin(u))=\cos(u)\) and \(\frac{d}{dx}(7x)=7\), we have \(f^{\prime}(x)=7\cos(7x)\).

Step4: Calculate \(f^{\prime}(0)\)

Substitute \(x = 0\) into \(f^{\prime}(x)\). Then \(f^{\prime}(0)=7\cos(7\times0)=7\cos(0)\). Since \(\cos(0)=1\), we get \(f^{\prime}(0)=7\). So \(B = f^{\prime}(0)=7\).

Answer:

\(A = 0\) and \(B=7\)