15. -/3 points details my notes scalc9 2.9.024. compute δy and dy for the given values of x and dx = δx. y =…

Explanation:

Step1: Calculate $y(x)$ and $y(x + \Delta x)$

First, find $y(x)$ when $x = 3$. $y(3)=3\times3 - 3^{2}=9 - 9=0$. Then find $x+\Delta x=3+( - 0.4)=2.6$. $y(2.6)=3\times2.6-2.6^{2}=7.8 - 6.76 = 1.04$.

Step2: Compute $\Delta y$

$\Delta y=y(x+\Delta x)-y(x)$. $\Delta y=y(2.6)-y(3)=1.04 - 0=1.04$.

Step3: Find the derivative of $y$

Differentiate $y = 3x - x^{2}$ with respect to $x$. $y^\prime=\frac{d}{dx}(3x - x^{2})=3 - 2x$.

Step4: Compute $dy$

$dy=y^\prime(x)dx$. When $x = 3$ and $dx=\Delta x=-0.4$, $y^\prime(3)=3-2\times3=-3$. $dy=y^\prime(3)dx=(-3)\times(-0.4)=1.2$.

Answer:

$\Delta y = 1.04$ $dy = 1.2$