Single Variable Calculus, Chapter 3, 3.4, Section 3.4, Problem 25

a.) Determine the equation of the tangent line to the curve $y = 2x \sin x$ at the point $\displaystyle \left( \frac{\pi}{2}, \pi \right)$


$
\begin{equation}
\begin{aligned}

\qquad y' =& (2x) \frac{d}{dx} (\sin x) + (\sin x) 2 \frac{d}{dx} (x)
&&
\\
\\
\qquad y' =& 2x \cos x + 2 \sin x
&&
\\
\\

\end{aligned}
\end{equation}
$




Let $y' = m_T$ (slope of the tangent line)


$
\begin{equation}
\begin{aligned}


y' = m_T =& 2 \left( \frac{\pi}{2} \right) \cos \left( \frac{\pi }{2} \right) + 2 \sin \left( \frac{\pi}{2} \right)
&& \text{Substitute value of $x$}
\\
\\
m_T =& 2
&&

\end{aligned}
\end{equation}
$


Using Point Slope Form substitute the values of $x, y$ and $m_T$


$
\begin{equation}
\begin{aligned}

y - y_1 =& m (x - x_1)
&&
\\
\\
y - \pi =& 2 \left( x - \frac{\pi}{2} \right)
&&
\\
\\
y - \pi =& 2x - \frac{2 \pi}{2}
&&
\\
\\
y =& 2x - \pi + \pi
&&
\\
\\
y =& 2x
&& \text{Equation of the tangent line at $\large \left( \frac{\pi}{2}, \pi \right)$}

\end{aligned}
\end{equation}
$



b.) Graph the curve and the tangent line in part (a) on the same screen