# 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 *N*^{2 }differ 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 \(0 < c < 1\;{\rm{and}}\;f'\left( c \right) = 2g'\left( c \right)\)

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