a graphing calculator is recommended. let y = 6x sin(x). (a) find y(x). y(x)=6 sin(x)+6x cos(x) nice work…

Explanation:

Step1: Recall tangent - line formula

The equation of a tangent line to a curve $y = f(x)$ at the point $(x_0,y_0)$ is given by $y - y_0=y'(x_0)(x - x_0)$. We know that $x_0=\frac{\pi}{2}$, $y_0 = 3\pi$, and $y'(x)=6\sin(x)+6x\cos(x)$.

Step2: Evaluate $y'(x)$ at $x_0$

Substitute $x = \frac{\pi}{2}$ into $y'(x)$: $y'\left(\frac{\pi}{2}\right)=6\sin\left(\frac{\pi}{2}\right)+6\times\frac{\pi}{2}\cos\left(\frac{\pi}{2}\right)$. Since $\sin\left(\frac{\pi}{2}\right)=1$ and $\cos\left(\frac{\pi}{2}\right)=0$, we have $y'\left(\frac{\pi}{2}\right)=6\times1+6\times\frac{\pi}{2}\times0 = 6$.

Step3: Find the equation of the tangent line

Substitute $x_0=\frac{\pi}{2}$, $y_0 = 3\pi$, and $y'\left(\frac{\pi}{2}\right)=6$ into the tangent - line formula $y - y_0=y'(x_0)(x - x_0)$: $y-3\pi=6\left(x-\frac{\pi}{2}\right)$. Expand the right - hand side: $y-3\pi=6x - 3\pi$. Add $3\pi$ to both sides of the equation: $y = 6x$.

Answer:

$y = 6x$