The discussion focuses on the definition of epsilon (ε) in the context of the Chain Rule as presented in James Stewart's "Calculus: Early Transcendentals." It establishes that ε represents the difference between the difference quotient and the derivative, defined mathematically as ε = (Δy/Δx) - f'(a). The limit of ε approaches zero as Δx approaches zero, leading to the conclusion that ε can be treated as a continuous function of Δx, despite Δx not being in the domain when it equals zero. The participants express confusion regarding the definition of ε at this point.
PREREQUISITESStudents of calculus, educators teaching differential calculus, and anyone seeking a deeper understanding of the Chain Rule and the role of epsilon in mathematical proofs.
If y=f(x) and x changes from a to a + \Deltax , we defined the increment of y as
\Deltay = f(a + \Deltax) - f(a)
Accordin to definition of a derivative ,we have
lim \frac{\Delta y}{\Delta x} = f'(a)
so if we denote by \epsilon the difference between Difference Qutient and the derivative we obtain
lim \epsilon = ( lim \frac{\Delta y}{\Delta x} - f'(a) ) = 0
But \epsilon = \frac{\Delta y}{\Delta x} - f'(a) \Rightarrow \Delta y = f'(a) \Delta x + \epsilon \Delta x
If we Define \epsilon to be 0 when \Delta x=0.then \epsilon becomes a continuous function of \Delta x