Question

Tangent lines?

Show that the graphs of the two equations `y = x` and `y = 1 / x` have tangent lines that are perpendicular to each other at the point of intersection.

Answer 1

See below

Explanation:

Here we have two functions of `x`:

`y= f_1(x) =x`

`y=f_2(x) =1/x`

To prove that their tangents are perpendicular at their points of intersection we need to find the slopes of `f_1` `(m_1)` and `f_2` `(m_2)` at intersection and show that `m_1xxm_2 = -1` in each case.

`f_1(x)` is a straight line through the origin with slope of `1`

`:. m_1 = 1 forall x`

To determine the slope of `f_2` at any point we need to find `f'_2(x)`

`f'_2(x) = d/dx(1/x) = -1/x^2`

To determine the points of intersection between `f_1(x)` and `f_2(x)`

`f_1(x) = f_2(x) -> x= 1/x`

`x^2 =1 -> x=+-1`

Therefore at the intersections of `f_1` and `f_2`, `m_2 = -1/(+-1)^2=-1`

`:. m_2 =-1` at both intersections

Hence, `m_1xxm_2 = 1 xx-1 = -1` at both intersections.

Therefore `f_1` and `f_2` have tangent lines that are perpendicular to each other at their points of intersection.