# Errors and Approximations

(C)   ERRORS AND APPROXIMATIONS

We can use differentials to calculate small changes in the dependent variable of a function corresponding to small changes in the independent variable. The theory behind it is quite simple: From the chapter on differentiation, we know that

$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \frac{{dy}}{{dx}} = f'\left( x \right)$

For small $$\Delta x,$$ we can therefore approximate $$\Delta y\,{\rm{as}}\,f'\left( x \right)\Delta x.$$  This is all there is to it!

Suppose we have to calculate (4.016)2.

We let $$y = {x^2} \cdot \,{x_0} = 4\,\,\,{\rm{and}}\,\,{y_0} = 169$$

\begin{align} \qquad\quad y' & = 2x,\,\,\Delta x = 0.016\\\\ \Rightarrow \qquad \Delta y & = f'\left( x \right) \cdot \Delta x\\\\ \qquad\qquad & = {\left. {2x} \right|_{{x_0} = 4}} \times 0.016\\\\ \qquad\qquad & = {\rm{ }}8{\rm{ }} \times {\rm{ }}0.016\\\\ \qquad\qquad & = {\rm{ }}0.128\\\\ \quad\Rightarrow \qquad \;\; y & = {y_0} + \Delta y = 16.128\end{align}

Example – 35

Find the value of $${\left( {8.01} \right)^{4/3}} + {\left( {8.01} \right)^2}$$

Solution: Let $$y = f\left( x \right) = {x^{4/3}} + {x^2}$$

Let $${x_0} = 8\,\,\,{\rm{so}}\,\,{\rm{that}}\,\,{y_0} = 16 + 64 = 80$$

\begin{align} \qquad\;\;\Delta x & = 0.01\\\\ \Rightarrow \qquad \Delta y & = {\left. {f'\left( x \right)} \right|_{x = {x_0}}} \times \Delta x\\\\ \qquad\qquad & \;{\left. { = \left( {\frac{4}{3}{x^{1/3}} + 2x} \right)} \right|_{{x_0} = 8}} \times \Delta x\\\\ \qquad\qquad & \; = \left( {\frac{8}{3} + 16} \right) \times 0.01\\\\ \qquad\qquad & \; = \frac{{0.56}}{3}\\\\ \qquad\qquad & \; = 0.1867\\\\ \Rightarrow \qquad \;\, {y_0} & \; = {y_0} + \Delta y\\\\ \qquad\qquad & \; = 80.1867\end{align}

TRY YOURSELF - V

Q. 1   Use the appropriate mean value theorem to show that the square roots of two successvie natural numbers greater thatn Ndiffer by less than \begin{align}\frac{1}{{2N}}\end{align}.

Q. 2   Use LVMT to show that

(a) \begin{align}\frac{{b - a}}{b} < \ln \frac{b}{a} < \frac{{b - a}}{a} \qquad {\text{when}}\; a < a < b\end{align}

(b) \begin{align}{\tan ^{ - 1}}{x_2} - {\tan ^{ - 1}}{x_1} < {x_2} - {x_1} \qquad {\text{when}}\; {x_2} > {x_1}\end{align}

Q. 3   If $$2a + 3b + 6c = 0,$$ can we say that $$a{x^2} + bx + c = 0$$ will have at least one real root in (0, 1)

Q. 4   Find the approximate value of \begin{align}f\left( x \right) = {\left( {\frac{{2 - x}}{{2 + x}}} \right)^{1/5}} \qquad {\text{at}}\;x = 0.15\end{align}

Q. 5   If $$f\left( x \right)\;{\rm{and}}\;g\left( x \right)$$ are differentiable functions for $$0 \le x \le 1$$ such that $$f\left( 0 \right) = 2,g\left( 0 \right) = 0,\,\,f\left( 1 \right) = 6,\,\,g\left( 1 \right) = 2,$$ show that there exists c satisfying $$0 < c < 1\;{\rm{and}}\;f'\left( c \right) = 2g'\left( c \right)$$