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