First slide

Differentials; errors and approximations

Question

Find the approximate change in the volume V of a cube of side ‘x’ meters caused by increasing the side by 3%

Moderate

A

$0.09 x^{3} m^{3}$
B

$9 x^{3} m^{3}$
C

$0.9 x^{3} m^{3}$
D

$0.1 x^{3} m^{3}$

Solution

$\begin{aligned} Note that V = x^{3} \\ \begin{aligned} d V & = (\frac{d V}{d x}) Δ x \\ = (3 x^{2}) Δ x \\ = (3 x^{2}) (0.03 x) \\ = 0.09 x^{3} m^{3} \end{aligned} \end{aligned}$
$Thus, the approximate change in volume is 0.09 x^{3} m^{3}$

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