Tangent Line Problem - Descartes vs Fermat

Tangent Line\

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Is it possible to find the tangent line at any point x=a?

Method Method

Example 1 - Find the slope and then write an equation of the tangent line to the function y = x2 at the

point (1,1) using Descartes' Method. '2-• i - n- M _xc u " 1L -~T- ~ O

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Example 2 - Find the slope slope and then write an equation of the tangent line to the function y = x2

at the point (1,1) using Fermat's Method. . ^- [ -

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*- ) " /Example 3 - Find an equation for the tangent line of y = V3x passing through the point (3,3) using

Descartes' Method.

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Example 4 - Find an equation for the tangent line of y — V3x passing throu^rTthe"poInt(3,3) using

Fermat's Method.

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slope (t)

Q Q'Example 5 - Write an equation of the tangent line to the ctFrve y = V4x at the point A = (1,2) using

either method.

Descartes heavily relied on standard algebraic manipulation which made his method limited to simple

algebraic curves.

Fermat's approach could be used on a wide variety of curves because of the use of a limiting process.

Sources:

http://math.kennesaw.edu/~jdoto/13.pdf

http://users.etown.edU/s/sanchisgr/HistoryOfMathematics/Calculusl/Worksheets/W3.pdf

Bauldry, W. (2009). Introduction to real analysis: an educational approach, (pp. 42-44). Hoboken, NJ: John Wiley & Sons, Inc.