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FCS # y = tan_(fcs)( x; -1 ) = tan ( x -tan(x- tan ( x - ...#. How do you prove that, piecewise, at y = 1, the tangent is inclined at #33^o 41' 24'' # to the x-axis, nearly?

Answer 1

Paisley Johnson

See explanation and the wholesome Socratic graph.

The FCS generator is #y =tan ( x - y )#. Inversely,

#x = y + kpi + tan^(-1)y, k = 0, +-1, +-2, +-3, ..#.

At #y = 1, x = kpi + pi/4 +1, k = 0, +-1, +-2, +-3, ..= 1.76539,....#.

#(dx)/(dy) = 1 + 1/( 1 + y^2 )

#y' = 1/(dx)/(dy). And so,

#y' = ( 1 + y^2 )/( 2 + y^2 ) = 1 - 1/( 2 + y^2 ) in [ 1/2, 1).#

#= 2/3#, at y = 1, and so,

#the inclination of he tangent to the x-axis is

#tan^(-1)( 2/3 ) = 33.69^o = 33^o 41' 24''#, nearly. See graph for this

tangent #(y-1) = 2/3 (x-1.7654 )# graph{(y-tan(x-y))((x-1.7654)^2+(y-1)^2-.01)((y-1) -2/3 (x-1.7654 ))=0[-10 6 -3 3]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.