Question

Use differentials to approximate the change in the volume of a cube when the side is decreased from 8 to 7.99 cm. (in cm3)

Answer 1

-1.92 cm³

Explanation:

The formula for the volume of a cube in terms of side length is ...

V = s³

Then the first-order approximation of volume for a small change ∆s in side length is ...

V(s +∆s) ≈ V(s) + V'(s)·∆s

and the change in volume is ...

∆V ≈ V(s +∆s) -V(s) = V'(s)·∆s

The derivative V'(s) is ...

V'(s) = 3s²

so the change in volume for s = 8 cm and ∆s = -0.01 cm is ...

∆V ≈ 3s²·∆s = 3(8 cm)²(-0.01 cm) = -1.92 cm³