Question

What is the equation of the line tangent to `f(x)=x + cos (x)` at `x=0`?

Answer 1

`x-y+1=0`. Interestingly, this is the bounding tangent for the whole periodic wave, from above. The parallel one below is x-y-1=0. See Socratic graphs.

Explanation:

At x = 0, f = 1. So, the point of contact P of the tangent is (0, 1)

Slope of the tangent is f' =1+sin x=1, at P.

Now, the equation to the tangent at P(0, 1) is

`y-1=1(x-0)`, giving

`x-y+1=0`

graph{(x+cosx-y)(x-y+1.05)(x^2+(y-1)^2-.01)=0 [-5, 5, -2.5, 2.5]}
graph{(x+cosx-y)(x-y+1.05)(x-y-1)(x^2+(y-1)^2-.01)=0 [-30, 30 -15, 15]}