Compare the values of ∆y and dy if y = f (x) = x^{3}+x^{2}−x+2 and x changes
(a) from 2 to 2.06 and
(b) from 2 to 2.01
Do you remember the definitions? Recall that
$\displaystyle \Delta y=y_2-y_1=f(x_2)-f(x_1)$
$\displaystyle \Delta x=x_2-x_1$
$\displaystyle dy=f'(x)dx$
and
$\displaystyle \Delta x =dx$
In problem a) you have
$\displaystyle x=2 \quad \Delta x =0.06$
$\displaystyle \Delta y =f(2.06)-f(2)$
$\displaystyle dy = f'(2)\cdot (0.06)$