I am stumped with this question.
Find the line tangent $f(t)=3\sin(2t)+5$ at the point where $t=\pi$.
You must first find the derivative of $f$ at $\pi$. Next, find the equation of the tangent line using the slope$=f′(π)$ and the point $P(\pi)$. Yep, you need to find $y$.
Best Answer
The problem specifies the point $(t_1,y_1)=\left(\pi,f(\pi)\right)$ where $f(t)=3\sin(2t)+5$ and $y=f(t)$. The equation of a tangent line is given by
$$y-y_1=m(t-t_1).\tag{1}$$
You need to find $f(\pi)$ and $m=f'(\pi)$ and then substitute them into $(1)$.
Step 1: Find the derivative of $f$ at $\pi$.
$$f(t)=3\sin(2t)+5$$ $$f'(t)=6\cos(2t)$$ $$f'(\pi)=6\cos(2\pi)=6$$
Step 2: Find $f(\pi)$.
$$f(\pi)=3\sin(2\pi)+5=5$$
Therefore, the point is $(t_1,y_1)=(\pi,5)$ and the slope of the tangent line is $m=f'(\pi)=6$. Substitute these values into $(1)$ to find the equation of the tangent line.