let ( y = 2x^{3}+4x + 2 ). if ( delta x = 0.3 ) at ( x = 1 ), use linear approximation to estimate ( delta y…

let ( y = 2x^{3}+4x + 2 ). if ( delta x = 0.3 ) at ( x = 1 ), use linear approximation to estimate ( delta y ). ( delta y approx )

let ( y = 2x^{3}+4x + 2 ). if ( delta x = 0.3 ) at ( x = 1 ), use linear approximation to estimate ( delta y ). ( delta y approx )

Answer

Explanation:

Step1: Find the derivative of (y)

The derivative of (y = 2x^{3}+4x + 2) with respect to (x) is (y^\prime=\frac{dy}{dx}=6x^{2}+4) (using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}) and (\frac{d}{dx}(ax)=a)).

Step2: Use the linear approximation formula

The linear approximation formula is (\Delta y\approx dy=y^\prime(x)\Delta x). Given (x = 1) and (\Delta x=0.3), substitute (x = 1) into (y^\prime): (y^\prime(1)=6(1)^{2}+4=6 + 4=10). Then (\Delta y\approx y^\prime(1)\cdot\Delta x).

Step3: Calculate (\Delta y)

Substitute (y^\prime(1) = 10) and (\Delta x=0.3) into (\Delta y\approx y^\prime(x)\Delta x), we get (\Delta y\approx10\times0.3 = 3).

Answer:

(3)