Find the equation of the tangent line to the curve y=2tan(x) at the point (pi/4,2). The equation of this tangent line can be written in the form y=mx+b. Find m and b

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to remember the form of equation of tangent line to the graph of a function at a certain point such that:

`y - f(x_0) = f'(x_0)(x - x_0)`

You need to determine each term of the equation such that:

`f(x_0)|_x_0=pi/4 = 2tan(pi/4) = ` 2 (remember that tan `pi/4 = 1` )

`f'(x) = 2*(tan x)' =gt f'(x) = 2*(1/(cos^2 x))`

`f'(x_0)|_x_0=pi/4 = 2/(cos^2 (pi/4))`

`f'(x_0)|_x_0=pi/4 = 2/(2/4) =gt f'(x_0)|_x_0=pi/4 = 4`

You need to substitute 2 for `f(x_0)`  and 4 for `f'(x_0)`  in equation of tangent line such that:

`y - 2= 4(x - pi/4)`

You need to use the slope intercept form of the equation of tangent line, thus, you should isolate y to the left side such that:

`y = 4x - pi + 2`

Hence, evaluating the equation of tangent line to the graph of `y=2tan x,`  at `(pi/4,2)`  yields `y = 4x - (pi-2).`