What is the equation of the tangent line of  f(x)=sinpix  at  x=3 ?

Dec 30, 2015

$y = \pi \left(3 - x\right)$

Explanation:

$\textcolor{b l u e}{\text{Graph of } f \left(x\right)}$
graph{sin(pix) [-10, 10, -5, 5]}

To construct a straight line, one needs either two points, or a point and the gradient. In this case, the latter would be more appropriate.

$f \left(3\right) = 0$

We know that the line passes through the point $\left(3 , 0\right)$.

$f ' \left(x\right) = \pi \cos \pi x$

$f ' \left(3\right) = - \pi$

We know that the gradient of the line is $- \pi$.

$- \pi = \frac{y - 0}{x - 3}$

Equation of tangent line:

$y = \pi \left(3 - x\right)$

$\textcolor{b l u e}{\text{Graph of tangent line}}$
graph{pi(3-x) [-10, 10, -5, 5]}