Question

Find Δf and df for the function f for the indicated values of x, Δx and compare:

f(x) = x³ – 2x², x = 2, Δx = dx = 0.5

Answer 1

y = f(x) = x³ – 2x²

dy = (3x² – 4x)dx

dy (when x = 2 and dx = 0.5) = [3(2²) – 4(2)](0.5)

= (12 – 8)(0.5)

= 4(0.5)

= 2

(i.e.,) df = 2

Now ∆f = f(x + ∆x) – f(x)

Here x = 2 and ∆x = 0.5

f(x) = x³ – 2x²

So f(x + ∆x) = f(2 + 0.5)

= f(2.5) = (2.5)³ – (2.5)² 

= (2.5)² [2.5 – 2]

= 6.25(0.5)

= 3.125

f(x) = f(2) = 2³ – 2(2²)

= 8 – 8

= 0

So ∆f = 3.125 – 0

= 3.125