Tangent Line Problem

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1.  Given a function f, and a point P on this function, we are asked to find an
equation of the tangent to the graph at P.

2.  Why did scientists of the early 17th century think that defining the tangent
line at an arbitrary point P on a curve and deducing its slope from the
formula y=f(x) a question worthy of their time and energy?

Some Reasons:

a)  In optics, the angle at which a ray of light strikes the surface of a
lens is defined in terms of the tangent to the surface.

b)  In physics, the direction of a body's motion at any point of its path
is along the tangent to the path.

c)  In geometry, the angle between two intersecting curves is the angle
between their tangents at the point of intersection.

3.  How will we find the equation of this tangent line?  We will use a method
developed by Pierre De Fermat in 1629.

a)  We start with what we can calculate, namely the slope of a secant
through P and a point Q nearby on the curve.

b)  We find the limiting value of the secant slope (if it exists) as Q
approaches P along the curve.

c)  We take this number to be the slope of the curve at P and define the
tangent to the curve at P to be the line through P with this slope.

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