Question

a graphing calculator is recommended. let y = 12x sin(x). (a) find y(x). y(x)= find an equation of the tangent line to the curve y at the point (π/2, 6π). y = (b) illustrate part (a) by graphing the curve y and the tangent line on the same screen.

Explanation:

Step1: Apply product - rule for differentiation

The product - rule states that if $y = u\cdot v$, where $u = 12x$ and $v=\sin(x)$, then $y'=u'v + uv'$. The derivative of $u = 12x$ is $u'=12$, and the derivative of $v=\sin(x)$ is $v'=\cos(x)$. So, $y'(x)=12\sin(x)+12x\cos(x)$.

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

Substitute $x=\frac{\pi}{2}$ into $y'(x)$. We know that $\sin(\frac{\pi}{2}) = 1$ and $\cos(\frac{\pi}{2})=0$. So, $y'(\frac{\pi}{2})=12\sin(\frac{\pi}{2})+12\cdot\frac{\pi}{2}\cos(\frac{\pi}{2})=12\times1 + 6\pi\times0=12$.

Step3: Use the point - slope form of a line

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(\frac{\pi}{2},6\pi)$ and $m = 12$. Substituting these values, we get $y-6\pi=12(x - \frac{\pi}{2})$.

Step4: Simplify the equation of the tangent line

Expand the right - hand side: $y-6\pi=12x-6\pi$. Then, $y = 12x$.

Answer:

$y'(x)=12\sin(x)+12x\cos(x)$
$y = 12x$