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Please solve RD Sharma class 12 chapter Differentials Errors and Approximations exercise 13.1 question 9 sub question (xvi) maths textbook solution

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Answer: 3.074

Hint: Here, we use

        \Delta y=f(x)+\Delta x)-f(x)

Given: 29^{\frac{1}{2}}

Solution:  y=f(x)=(x)^{\frac{1}{3}}

        \begin{aligned} &\Rightarrow \text { let } \\\\ &x=27 \\\\ &x+\Delta x=29 \end{aligned}


        \begin{aligned} &\Delta x=2 \\\\ &\text { For } x=27 \\\\ &y=(27)^{\frac{1}{3}}=3 \end{aligned}

Let d x=\Delta x=2

Now, y=(x)^{\frac{1}{3}}

            \Rightarrow \frac{d y}{d x}=\frac{1}{3 x^{\frac{1}{3}}}

        \begin{aligned} &\left(\frac{d y}{d x}\right)_{x=27}=\frac{1}{27} \\\\ &\Delta y=d y=\frac{d y}{d x} d x=\frac{1}{27} \times 2=0.074 \end{aligned}

        \begin{aligned} &\Delta y=0.074 \\\\ &\text { Hence, } 29^{\frac{1}{3}}=3+0.074=3.074 \end{aligned}


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