Question

Find all functions f(x) that have the property that the tangent lines to the graphs of f(x)pass through the point (x+2,0).

Answer 1

`y=Ae^(-2x)`

Explanation:

Given that the functions f(x) that have the property that the tangent lines to the graphs of f(x)pass through the point (x+2,0).

Let (x,y) be any arbitrary point of contract

The tangent line passes through two points (x,y) and (x+2,0)

Slope of tangent line = f'(x) = change in y/change in x=
`(-y)/(2)`

i.e. we have

`(dy)/(dx) =(-y)/(2)`

Separate the variables

`(dy)/(y) =-2x\\lny =-2x+c`

Raise to power e

`y=Ae^(-2x)`

Thus the functions would have the above form for various values of A