find the equation of the tangent line to the curve ( y = 2\tan x ) at the point ( (pi/4,2) ). the equation…

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

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

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of (y = 2\tan x) is (y'=2\sec^{2}x).

Step2: Evaluate the derivative at (x = \frac{\pi}{4})

When (x=\frac{\pi}{4}), (\sec x=\sqrt{2}), so (y'|_{x = \frac{\pi}{4}}=2\times(\sqrt{2})^{2}=4). Thus, (m = 4).

Step3: Use the point - slope form to find (b)

The equation of the line is (y - y_{0}=m(x - x_{0})), substituting (x_{0}=\frac{\pi}{4}), (y_{0} = 2) and (m = 4): (y-2=4(x-\frac{\pi}{4})). Simplify to (y=4x + 2-\pi). So (b=2-\pi).

Answer:

(m = 4), (b=2-\pi)