13. 1/2 points details my notes scalc9 2.9.020. (a) find the differential dy. y = cos(πx) dy = -sin(πx)πdx…

Explanation:

Step1: Recall the differential formula

We know that if $y = f(x)$, then $dy=f^{\prime}(x)dx$. Given $y = \cos(\pi x)$, by the chain - rule, if $u=\pi x$, $y=\cos(u)$ and $\frac{dy}{du}=-\sin(u)$, $\frac{du}{dx}=\pi$. So $\frac{dy}{dx}=-\sin(\pi x)\cdot\pi$, and $dy = -\pi\sin(\pi x)dx$.

Step2: Substitute given values

We are given $x=\frac{1}{3}$ and $dx = 0.1$. First, find $\sin(\pi x)$ when $x = \frac{1}{3}$. Since $\sin(\pi\cdot\frac{1}{3})=\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$. Then substitute into the $dy$ formula: $dy=-\pi\sin(\pi x)dx$. Substitute $x = \frac{1}{3}$ and $dx=0.1$ into it, we get $dy=-\pi\cdot\frac{\sqrt{3}}{2}\cdot0.1$.

Step3: Calculate the value

$dy=-\pi\cdot\frac{\sqrt{3}}{2}\cdot0.1\approx - 3.14159\times\frac{1.73205}{2}\times0.1\approx - 0.272$.

Answer:

$-0.272$