Question

What is the equation of the line tangent to `f(x)=2 tan(3pix +4)` at `x=1`?

Answer 1

Tangent Line

`y=6 pi(sec^2 4)x -6 pi* sec^2 4 + 2*tan 4`

Explanation:

The given
`f(x)=2 tan (3 pi x+4)` at `x=1`

`f(1)=2 tan(3 pi *1+4)`
`f(1)=2 tan 4`

The point of Tangency `x_1=1` Abscissa
`y_1= 2 tan 4= 2.31564` Ordinate

The Slope
`f' (x) =2 sec^2(3 pi x+4)* 3 pi`
`f' (x) = 6 pi* sec^2 (3 pi x +4)`
slope `m=6 pi sec^2 (3 pi * 1 +4)`
slope `m=6 pi sec^2 4`

Tangent Line

`y-y_1=m*(x-x_1)`

`y-2 tan 4 = 6 pi sec^2 4 (x - 1)`

`y= 6 pi (sec^2 4)x -6 pi sec^2 4 + 2 tan 4`