Can you compare the values of dy and Δy for the function

Question

Function		x-Value		Differential of x
f(x) = 0.4x2		x = 6		Δx = dx = −0.1

dy=?

Δy=?

Answer 1

delta y = f( 6) - f(5.9) = 14.4 - 13.924 = 0.473

The forwards difference quotient btw is deltaY / deltaX = 0.476/0.1 = 4.76

f'(x) = 0.8x ---> f ' (6) = 4.8

Answer 2

y = f(x) = 0.4x²

Δy = f(x+Δx) - f(x) = f(5.9) - f(6) = -0.476

dy = f'(x)dx = f(6)(-0.1) = 0.8(6)(-0.1) = -0.48

Note that, since Δx is close to 0, dy is approximately equal to Δy.