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Find $\sqrt[3]{27.2}$

3.1

3.11

3.007

3.012

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A square's side is measured as 3 cm with an error of 0.1 cm. The error in its area will be

$0 \cdot 2 {cm}^{2}$

$0.4 {cm}^{2}$

$0 \cdot 6 {cm}^{2}$

$0.8 {cm}^{2}$

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Concepts Covered - 1

Approximations and Errors using Derivatives

Approximations and Errors Using Derivatives

Let the function, $y = f (x)$ , be a function of $x$
As we have derived derivatives earlier,

$\frac{d y}{d x} = lim_{Δ x \to 0} \frac{Δ y}{Δ x} = lim_{Δ x \to 0} \frac{f (x + Δ x) - f (x)}{Δ x}$

$Δ x$ is small change in x and corresponding change in y is $Δ y$
As in the figure, point $Q$ moves closer to point $P$ on the curve, then $d y$ is a good approximation of $Δ y$ .

$\begin{aligned} \frac{d y}{d x} = \frac{Δ y}{Δ x} = \frac{f (x + Δ x) - f (x)}{Δ x} \\ ∴ & f (x + Δ x) = f (x) + Δ x \cdot \frac{d y}{d x} \end{aligned}$
ERROR
Absolute Error
$Δ x$ or $d x$ is called absolute error in $x$ .
Relative Error
$\frac{Δ x}{x}$ or $\frac{d x}{x}$ is called the relative error in $x$ .
Percentage Error
$\frac{Δ x}{x} \cdot 100$ or $\frac{d x}{x} \cdot 100$ is called the percentage error in $x$ .