Math Derivative applications Contact between curves

Contact between curves

A contact between curves (point of contact) is a common point of two function graphs, where both functions have the same tangent (same slope).

Contact between curves


Two functions have a contact between curves, if:
$f(x_C)=g(x_C)$ and


  1. Take derivatives
  2. Equate function equations: $f(x_C)=g(x_C)$
  3. Check slopes
  4. Specify touch point


Determine the point of contact of the function $f(x)=x^2$ and $g(x)=-x^2+4x-2$.

  1. Take derivatives


  2. Equate function equations

    First condition: Both functions must have a common point.

    There is a quadratic equation that can be solved, for example, with the PQ formula.
    $x_{C_{1,2}} = -\frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$
    $x_{C_{1,2}} = 1 \pm\sqrt{1-1}$
  3. Check slope

    Second Condition: Both functions must have the same slope at the point.
    => The functions touch at the position $x_C=1$
  4. Specify point of contact

    The point of contact should be specified: Therefore, calculate the y-coordinate with one of the original functions.

    => Point of contact: $C(\color{red}{1}|\color{blue}{1})$