Using the derivative f(x) evaluated at x0, we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative f(x) evaluated at x0, we have: ΔF0Δx≈f(x0)(in the limitΔx→0)∴ΔF0=f(x0)Δx We can then estimate the value of the function F1=F(x1) at the next point, x1=x0+Δx, as illustrated by the black arrow in Figure A2.3.1 \[F1=F(x1)=F(x+Δx)≈F0+ΔF0≈F0+f(x0)Δx…