Example: Find the acute angles at an intersection of two functions, a parabola and line.

Steps:

First find where the two functions intersect. Set the two equal to each other.

Simplify, set equal to 0.

Find values of x by factoring.

Plug in these values of x into the parabola equation to find the corresponding values of y.

Find the derivative of parabola, y'. Plug in the values of x found earlier. These give the slopes of the functions at x1, x2...

Find the tangent vectors for each function:

For a line: take the slope of the line from the line function. ex. for a slope of three, the tangent vector is <1,3> (1 unit over, 3 units up)

For a parabola: need tangent vectors at both points of intersection.

Find the value of y' for each x (plug in the found x values into the derivative of the parabola)

For each, this is the slope.

Turn the slope into a tangent vector like the example for a line.

From here, find the angles between the tangent vectors, using the theta dot product formula (next page).