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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.

 Oct 4, 2022
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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,

or 

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.

 Oct 4, 2022

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