Determine the instantaneous slope of f(x) = x^2 - 5x + 4 for all x, without using the derivative or limit.
Hint: tangent line intersects f(x) at one point.
Suppose that the equation of the tangent is y = ax+b.
Its slope, (and therefore the slope of the curve), will be a.
The line y = ax + b intersects the curve y = x^2 - 5x + 4 at points with x coordinate(s) given by
x^2 - 5x + 4 = ax + b,
x^2 - ( 5 + a)x + (4 - b) = 0.
If the line is a tangent to the curve then the single point will be x = (5 + a)/2, (that's from the formula for the solution of a quadratic with the b^2 - 4ac part equal to zero).
From that, a = 2x - 5, and that's the slope.