determine the function 𝜀 that occurs in the theorem below.

Question

If f is differentiable at c then Δy = f(c + Δx) − f(c) satisfies Δy = (f '(c) + 𝜀)Δx where the function 𝜀 has the property that lim Δx→0 𝜀(Δx) = 0.

1. f(x) = x² − x + 6

a) 𝜀 = Δx(2x − 1 + Δx)

b) 𝜀 = Δx

c) 𝜀 = (Δx)²

d) 𝜀 = 2x − 1

Can someone walk me through this problem so I can understand how to do it. My teacher did not do an example or anything similar to this.

Answer 1

f(x+Δx)=x²+2xΔx+(Δx)²-(x+Δx)+6

f(x+Δx)-f(x)=2xΔax+(Δx)²-Δx=Δy

Δy/Δx=2x-1+Δx

but the Δx->0 and therefore ε=2x-1