a graphing calculator is recommended. let y = 8x sin(x). (a) find an equation of the tangent line to the…

Explanation:

Step1: Differentiate using product - rule

The product - rule states that if $y = uv$, where $u = 8x$ and $v=\sin(x)$, then $y^\prime=u^\prime v + uv^\prime$. Since $u^\prime = 8$ and $v^\prime=\cos(x)$, we have $y^\prime=8\sin(x)+8x\cos(x)$.

Step2: Find the slope at $x = \frac{\pi}{2}$

Substitute $x=\frac{\pi}{2}$ into $y^\prime$. $y^\prime\left(\frac{\pi}{2}\right)=8\sin\left(\frac{\pi}{2}\right)+8\times\frac{\pi}{2}\cos\left(\frac{\pi}{2}\right)$. Since $\sin\left(\frac{\pi}{2}\right) = 1$ and $\cos\left(\frac{\pi}{2}\right)=0$, the slope $m = 8$.

Step3: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. Here, $x_1=\frac{\pi}{2}$, $y_1 = 4\pi$ and $m = 8$. So $y-4\pi=8\left(x-\frac{\pi}{2}\right)$.

Step4: Simplify the equation

Expand the right - hand side: $y-4\pi=8x - 4\pi$. Then $y = 8x$.

Answer:

$y = 8x$