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).

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Remember

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

Method

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

Example

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

1. Take derivatives

$f(x)=x^2$
$f'(x)=2x$

$g(x)=-x^2+4x-2$
$g'(x)=-2x+4$
2. Equate function equations

First condition: Both functions must have a common point.
$f(x_C)=g(x_C)$
$x^2=-x^2+4x-2\quad|-x^2$
$-2x^2+4x-2=0\quad|:(-2)$
$x^2-2x+1=0$

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

Second Condition: Both functions must have the same slope at the point.
$f'(x_C)=g'(x_C)$
$f'(\color{red}{1})=g'(\color{red}{1})$
$2\cdot\color{red}{1}=-2\cdot\color{red}{1}+4$
$2=2$
=> 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.

$f(\color{red}{1})=\color{red}{1}^2=\color{blue}{1}$
=> Point of contact: $C(\color{red}{1}|\color{blue}{1})$