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Differentials, Errors and Approximations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentials, Errors and Approximations4 Marks
Question
Using differentials, find the approximate values of the following:
$(1.999)^5$
✓

Answer

Consider the function $\text{y}=\text{f} (\text{x})=\text{x}^5$

Let:

$\text{x}=5$

$\text{x}+\triangle \text{x}=1.999$

Then,

$\triangle\text{x}=0.001$

For $\text{x}=2$

$\text{y}=2^5=32$

Let:

$\text{dx}=\triangle \text{x}=-0.001$

Now, $\text{y}=\text{x}^5$

$\Rightarrow\frac {\text{dy}}{\text{dx}}=5\text{x}^4$

$\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}=2}=80$

$\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=80\times(-0.001)=-0.08$

$\Rightarrow\triangle\text{y} =-0.08$

$\therefore1.999^5=\text{y}+\triangle\text{y}=31.92$

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