Question

let ( y = 4x^{2}+4x + 4 ). if ( delta x=0.3 ) at ( x = 4 ), use linear approximation to estimate ( delta y )( delta yapprox )
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Explanation:

Step1: Find the derivative of the function

The function is \(y = 4x^{2}+4x + 4\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(y^\prime=\frac{dy}{dx}=8x + 4\).

Step2: Evaluate the derivative at \(x = 4\)

Substitute \(x = 4\) into \(y^\prime\). Then \(y^\prime(4)=8\times4 + 4=32 + 4=36\).

Step3: Use the linear - approximation formula \(\Delta y\approx dy=y^\prime(x)\Delta x\)

We know that \(y^\prime(4) = 36\) and \(\Delta x=0.3\). So \(\Delta y\approx y^\prime(4)\times\Delta x\).
Substitute the values: \(\Delta y\approx36\times0.3\).

Answer:

\(10.8\)