Question

Use differentials to approximate the cube root of $127$.

• A. $5.05$
• B. $5.026$
• C. $6.02$
• D. $4.09$

Answer 1

Answer: (B)

Since, we have to find the approximate value of the cube root of $127$. So, we consider the function
$y=f(x)=x^{1/3}$
Let $x=125$ and $x+\Delta x=127$.
Then, $\Delta x=127-125=2$
For $x=125$, we have
$y-(125{)}^{1/3}=5$.
[Putting $x=125$ in $y=x^{1/3}$]
Let $dx=\Delta x=2$
Now, $y=x^{1/3}$
$\Rightarrow \frac{dy}{dx}=\frac{1}{3x^{2/3}}$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=125}=\frac{1}{3{\left(125\right)}^{{}^{2/3}}}$
$=\frac{1}{3{\left(5^3\right)}^{{}^{2/3}}}=\frac{1}{75}$
$\therefore dy=\frac{dy}{dx}dx$
$\Rightarrow dy=\frac{1}{75}\left(2\right)=\frac{2}{75}$
$\Rightarrow \Delta y=\frac{2}{75}$
$\left[\because \Delta y\cong dy\right]$
Hence, ${\left(127\right)}^{1/3}=y+\Delta y=5+\frac{2}{75}$
$=5.026$.