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{d y}{d x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}
$

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

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