Determine dy & Δy  if y = cos ( x² + 1) - x as x changes from x = 2 to x = 2.03 .

 Solution

Firstly let's deetrmine actual the change in y, Δy .

Δy = cos (( 2.03)²  + 1) - 2.03 - (cos (2²  + 1) - 2) = 0.083581127

Now let's obtain the formula for dy.

dy = (-2 x sin ( x² + 1) -1) dx

Next, the change in x from x = 2 to x = 2.03 is Δx = 0.03 and therefore we then suppose that dx ≈ Δx = 0.03 .  It gives an approximate change in y of,

            dy = (-2 ( 2) sin (2²  + 1) -1)(0.03) = 0.085070913

We can illustrates that actually we do have that Δy ≈ dy provided we keep ?x small.