First slide

Differentials; errors and approximations

Question

$Use differential to approximate (27.3)^{1 / 3}$

Moderate

A

3
B

2.011
C

3.1112
D

3.0111

Solution

$\begin{aligned} Let y = x^{1 / 3} \\ \frac{d y}{d x} = \frac{1}{3} x^{- 2 / 3} \\ = \frac{1}{3 x^{2 / 3}} \end{aligned}$
$\begin{matrix} y + Δ y = (x + Δ x)^{1 / 3} \\ Δ y = (x + Δ x)^{1 / 3} - x^{1 / 3} \\ = \frac{d y}{d x} \cdot Δ x \end{matrix}$
$\begin{aligned} (x + Δ x)^{1 / 3} = x^{1 / 3} + \frac{1}{3 x^{2 / 3}} Δ x \\ = (27)^{1 / 3} = + \frac{1}{3 (27) 2 / 3} (0.3) \\ = 3 + \frac{1}{90} \\ = 3 + \frac{0.1111}{10} \\ = 3.0111 \end{aligned}$

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