Approximations and Errors using Derivatives - Practice Questions & MCQ

Approximations and Errors Using Derivatives

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

$\frac{dy}{dx}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$

$\mathrm{\Delta }\mathrm{x}$ is small change in x and corresponding change in y is $\mathrm{\Delta }\mathrm{y}$
As in the figure, point $Q$ moves closer to point $P$ on the curve, then $dy$ is a good approximation of $\mathrm{\Delta }y$.

$\begin{array}{rl}& \frac{dy}{dx}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}\\ \therefore & \mathbf{f}(\mathbf{x}+\mathrm{\Delta }\mathbf{x})=\mathbf{f}(\mathbf{x})+\mathrm{\Delta }\mathbf{x}\cdot \frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\end{array}$
ERROR
Absolute Error
$\mathrm{\Delta }\mathrm{x}$ or $dx$ is called absolute error in $x$.
Relative Error
$\frac{\mathrm{\Delta }\mathrm{x}}{\mathrm{x}}$ or $\frac{dx}{\mathrm{x}}$ is called the relative error in $x$.
Percentage Error
$\frac{\mathrm{\Delta }\mathrm{x}}{\mathrm{x}}\cdot 100$ or $\frac{dx}{\mathrm{x}}\cdot 100$ is called the percentage error in $x$.