Determine dy & Δy if y = cos ( x^{2 }+ 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 (2^{2 } + 1) - 2) = 0.083581127
Now let's obtain the formula for dy.
dy = (-2 x sin ( x^{2} + 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 (2^{2 } + 1) -1)(0.03) = 0.085070913
We can illustrates that actually we do have that Δy ≈ dy provided we keep ?x small.