Question

Using differentials, find the approximate value of each of the following up to 3 places of decimal ?
$\sqrt{25.3}$

$\sqrt{49.5}$

$\sqrt{0.6}$

$(0.009{)}^{\frac{1}{3}}$

$(0.999{)}^{\frac{1}{10}}$

$(15{)}^{\frac{1}{4}}$

$(26{)}^{\frac{1}{3}}$

$(255{)}^{\frac{1}{4}}$

$(82{)}^{\frac{1}{4}}$

$(401{)}^{\frac{1}{2}}$

$(0.0037{)}^{\frac{1}{2}}$

$(81.5{)}^{\frac{1}{4}}$

$(3.968{)}^{\frac{3}{2}}$

$(32.15{)}^{\frac{1}{5}}$

Answer 1

$\sqrt{25.3}$
Consider $f\left(x\right)=\sqrt{x}.$ Let x = 25 and $\Delta x=0.3$
$f^{\prime }\left(x\right)\simeq \frac{1}{2\sqrt{x}}$
Now, $f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)$
$\Rightarrow \sqrt{x+\Delta x}=\Delta x+\frac{1}{2\sqrt{x}}\times \Delta x$
$\Rightarrow \sqrt{25.3}\simeq \sqrt{25}+\frac{0.3}{2\sqrt{25}}=5+\frac{0.3}{10}=5.03\Rightarrow \sqrt{25.3}\simeq 5.03$
Note The approximate value is not equal to the exact value.

$\sqrt{49.5}$
Consider $f\left(x\right)=\sqrt{x}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$
Let x = 49 and $Deltax=0.5$
Now $f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)$
$\therefore \sqrt{x+\Delta x}\simeq \sqrt{x}+\frac{1}{2\sqrt{x}}\times \Delta x$
$\Rightarrow \sqrt{49.5}\simeq \sqrt{49}+\frac{0.5}{2\sqrt{49}}=7+\frac{0.5}{14}.=7+0.036\Rightarrow \sqrt{49.5}\simeq 7.036$

$\sqrt{0.6}$
Consider $f\left(x\right)=\sqrt{x}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{2\sqrt{x}}$
Let $x=0.64and\Delta x=-0.04$
$Now,f(x+\Delta x)=f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow \sqrt{x+\Delta x}=\sqrt{x}+\frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \sqrt{0.64-0.04}=\sqrt{0.64}+\frac{(-0.04)}{2\sqrt{0.64}}=0.8-\frac{0.04}{2\times 0.8}=0.8-0.025=0.775\Rightarrow \sqrt{0.6}\simeq 0.775$

$(0.009{)}^{\frac{1}{3}}$
Consider $f\left(x\right)=x^{\frac{1}{3}}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{3}{x}^{\frac{-2}{3}}$
Let $x=0.008and\Delta x=0.001$
Now, $f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{1}{3}}\simeq x^{\frac{1}{3}}+\frac{1}{3x^{\frac{2}{3}}}\times \Delta x\Rightarrow (0.008+0.001{)}^{\frac{1}{3}}\simeq (0.008{)}^{\frac{1}{3}}+\frac{0.001}{3(0.008{)}^{\frac{2}{3}}}=0.2+\frac{0.001}{3\left\{\right(0.2{)}^3{\}}^{\frac{2}{3}}}=0.2+\frac{0.001}{3\times (0.2{)}^2}=0.208\Rightarrow (0.009{)}^{\frac{1}{3}}\simeq 0.208$

$(0.999{)}^{\frac{1}{10}}Consider,f(x)=x^{\frac{1}{10}},Letx=1and\Delta x=-0.001f^{\prime }(x)=\frac{1}{10}{x}^{\frac{-9}{10}}Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow (x+\Delta x{)}^{\frac{1}{10}}\simeq x^{\frac{1}{10}}+\frac{1}{10x^{\frac{9}{10}}}\times \Delta x\Rightarrow (1-0.001{)}^{\frac{1}{10}}\simeq (1{)}^{\frac{1}{10}}+\frac{(-0.001)}{10(1{)}^{\frac{9}{10}}}=1-\frac{0.001}{10\times 1}=1-0.0001=0.9999\therefore (0.999{)}^{\frac{1}{10}}=0.9999$

$(15{)}^{\frac{1}{4}}$
Consider $f\left(x\right)=x^{\frac{1}{4}}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{4}{x}^{-\frac{3}{4}}$
Let $x=16and\Delta x=-1$
Now, $f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)$
$\Rightarrow (x+\Delta x{)}^{\frac{1}{4}}\simeq x^{\frac{1}{4}}+\frac{\Delta x}{4x^{\frac{3}{4}}}\Rightarrow (16-1{)}^{\frac{1}{4}}\simeq (16{)}^{\frac{1}{4}}+\frac{(-1)}{4(16{)}^{\frac{3}{4}}}=2-\frac{1}{4\times 2^3}=2-0.031=1.969$
$\therefore (15{)}^{\frac{1}{4}}\simeq 1.969$

$\left(26{)}^{\frac{1}{3}}Letf\right(x)=x^{\frac{1}{3}}\Rightarrow f^{\prime }(x)=\frac{1}{3}{x}^{\frac{-2}{3}}$
Let $x=27and\Delta x=-1,$
Now, $f(x+\Delta x)\simeq f\left(x\right)+\Delta f^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{1}{3}}\simeq x^{\frac{1}{3}}+\frac{\Delta x}{3x^{\frac{2}{3}}}\Rightarrow (27-1{)}^{\frac{1}{3}}\simeq (27{)}^{\frac{1}{3}}+\frac{(-1)}{3(27{)}^{\frac{2}{3}}}=3-\frac{1}{3\left\{\right(27{)}^{\frac{1}{3}}{\}}^2}=3-\frac{1}{3\times 3^2}=3-0.037=2.963\Rightarrow (26{)}^{\frac{1}{3}}\simeq \mathrm{2.963.}$

$\left(255{)}^{\frac{1}{4}}Considerf\right(x)=x^{\frac{1}{4}}\Rightarrow f^{\prime }(x)=\frac{1}{4}{x}^{\frac{-3}{4}}Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow (x+\Delta x{)}^{\frac{1}{4}}+\frac{\Delta x}{4x^{\frac{3}{4}}}\Rightarrow (256-1{)}^{\frac{1}{4}}\simeq (256{)}^{\frac{1}{4}}+\frac{(-1)}{4(256{)}^{\frac{3}{4}}}=4+\frac{-1}{4\left\{\right(256{)}^{\frac{1}{4}}{\}}^3}=4-\frac{1}{4\times 4^3}=4-\frac{1}{256}=4-0.004=3.996\Rightarrow (255{)}^{\frac{1}{4}}\simeq 3.996$

$\left(82{)}^{\frac{1}{4}}Considerf\right(x)=x^{\frac{1}{4}}\Rightarrow f]\left(x\right)=\frac{1}{4}{x}^{-\frac{3}{4}}Letx=81and\Delta x\simeq 1Now,f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{1}{4}}\simeq x^{\frac{1}{4}}+\frac{\Delta x}{4x^{\frac{3}{4}}}\Rightarrow (81+1{)}^{\frac{1}{4}}\simeq (81{)}^{\frac{1}{4}}+\frac{1}{4(81{)}^{\frac{3}{4}}}=3+\frac{1}{4\left\{\right(81{)}^{\frac{1}{4}}{\}}^3}=3+\frac{1}{4\times 3^3}=3+\frac{1}{108}=3+0.009=3.009\Rightarrow (82{)}^{\frac{1}{4}}\simeq \mathrm{3.009.}$

$\left(401{)}^{\frac{1}{2}}Considerf\right(x)=\sqrt{x},\Rightarrow f^{\prime }(x)=\frac{1}{2\sqrt{x}}Letx=400and\Delta x=1Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow \sqrt{x+\Delta x}\simeq \sqrt{x}+\frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \sqrt{400+1}\simeq \sqrt{400}+\frac{1}{2\sqrt{400}}=20+\frac{1}{2\times 20}=20+0.025=20.025\Rightarrow \sqrt{401}\simeq 20.025$

$\left(0.0037{)}^{\frac{1}{2}}Considerf\right(x)=\sqrt{x},\Rightarrow f^{\prime }(x)=\frac{1}{2\sqrt{x}}Letx=0.0036and\Delta x=0.0001Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow \sqrt{0.0036+0.0001}=\sqrt{0.0036}+\frac{0.0001}{2\sqrt{0.0036}}=0.06+\frac{0.0001}{2\times 0.06}=0.06+0.0008=0.0608\Rightarrow \sqrt{0.0037}\simeq 0.0608$

$(81.5{)}^{\frac{1}{4}}$
$Considerf\left(x\right)=x^{\frac{1}{4}}\Rightarrow f^{\prime }\left(x\right)=\frac{1}{4}{x}^{-\frac{3}{4}}=\frac{1}{4x^{\frac{3}{4}}}Letx=81and\Delta x=0.5Now,f(x+\Delta x)\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{1}{4}}\simeq +\frac{1}{4x^{\frac{3}{4}}}\times \Delta x\Rightarrow (81+0.5{)}^{\frac{1}{4}}\simeq (81{)}^{\frac{1}{4}}+\frac{0.5}{4(81{)}^{\frac{3}{4}}}=3+\frac{0.5}{4\times 3^3}=3+\frac{0.5}{108}=3+0.0046=3.0046\Rightarrow (81.5{)}^{\frac{1}{4}}\simeq 3.0046$

$\left(3.968{)}^{\frac{3}{2}}Considerf\right(x)=x^{\frac{3}{2}}\Rightarrow f^{\prime }(x)=\frac{3}{2}{x}^{\frac{1}{2}}=\frac{3}{2}\sqrt{x}Letx=4and\Delta x=-0.032Now,f(x+\Delta x{)}^{\frac{3}{2}}\simeq f\left(x\right)+\Delta x{f}^{\prime }\left(x\right)\Rightarrow (x+\Delta x{)}^{\frac{3}{2}}\simeq x^{\frac{3}{2}}+\Delta x(\frac{3}{2}\sqrt{x})\Rightarrow (4-0.032{)}^{\frac{3}{2}}\simeq (4{)}^{\frac{3}{2}}+\frac{3(-0.032)}{2}\sqrt{4}=8-0.096=7.904\Rightarrow (3.968{)}^{\frac{3}{2}}\simeq 7.904$

$\left(32.15{)}^{\frac{1}{5}}Considerf\right(x)=x^{\frac{1}{5}}\Rightarrow f^{\prime }(x)=\frac{1}{5}{x}^{\frac{-4}{5}}=\frac{1}{5x^{\frac{4}{5}}}Letx=32and\Delta x=0.15Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow (x+\Delta x{)}^{\frac{1}{5}}\simeq x^{\frac{1}{5}}+\frac{1}{5x^{\frac{1}{5}}}\simeq x^{\frac{1}{5}}+\frac{1}{5x^{\frac{4}{5}}}\times \Delta x\Rightarrow (32.15{)}^{\frac{1}{5}}\simeq (32{)}^{\frac{1}{5}}+\frac{0.15}{5(32{)}^{\frac{4}{5}}}=2+\frac{0.15}{5\times 2^4}=2+\frac{0.15}{80}=2+0.0019=2.0019\Rightarrow (32.15{)}^{\frac{1}{5}}\simeq 2.0019$