what is the differential, dy, if y = 8 - x^2, x = - 2, and dx = 1.3?

Explanation:

Step1: Differentiate y with respect to x

Given $y = 8 - x^{2}$, using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ and $\frac{d}{dx}(c)=0$ (where c is a constant), we have $\frac{dy}{dx}=-2x$.

Step2: Use the total differential formula

The total differential formula is $dy=\frac{dy}{dx}dx$. We know that $\frac{dy}{dx}=-2x$ and $dx = 1.3$, and $x=-2$. Substitute $x = - 2$ into $\frac{dy}{dx}$: $\frac{dy}{dx}\big|_{x=-2}=-2\times(-2)=4$.

Step3: Calculate dy

Since $dy=\frac{dy}{dx}dx$, substitute $\frac{dy}{dx}=4$ and $dx = 1.3$ into the formula. Then $dy=4\times1.3 = 5.2$.

Answer:

$5.2$