Question

How do you estimate the quantity using linear approximation and find the error using a calculator of ${\displaystyle \frac{1}{\sqrt{95}}}-{\displaystyle \frac{1}{\sqrt{98}}}$ ?

Answer 1

Hint: For answering this question we will assume a function $f\left(x\right)={\displaystyle \frac{1}{\sqrt{x}}}$ and find the slope$f^{{}^{\prime }}\left(x\right)$ of the following curve. We will use the equation of the curve and find the approximate value and compare with the calculated value.

Complete step by step solution:
Now considering from the question we have been asked to estimate the quantity using linear approximation and find the error using a calculator of ${\displaystyle \frac{1}{\sqrt{95}}}-{\displaystyle \frac{1}{\sqrt{98}}}$ .
We will assume a function $f\left(x\right)={\displaystyle \frac{1}{\sqrt{x}}}$ with slope ${\displaystyle \frac{d}{dx}}f\left(x\right)={\displaystyle \frac{-1}{2}}{x}^{{\displaystyle \frac{-3}{2}}}$ .
Let us consider $x=100$ then we will have $f\left(100\right)={\displaystyle \frac{1}{10}}$ and slope
$\begin{array}{rl}& {\displaystyle \frac{d}{dx}}f\left(100\right)={\displaystyle \frac{-1}{2}}{\left(100\right)}^{{\displaystyle \frac{-3}{2}}}\\ & \Rightarrow {\displaystyle \frac{d}{dx}}f\left(100\right)={\displaystyle \frac{-1}{2}}{\left(10\right)}^{-3}\end{array}$ .
Now we will try to get the equation of the curve. The equation of the curve will be given as
$\begin{array}{rl}& (y-f\left(x_1\right))=f^{{}^{\prime }}\left(x_1\right)(x-x_1)\\ & \Rightarrow (y-{\displaystyle \frac{1}{10}})={\displaystyle \frac{-1}{2}}{\left(10\right)}^{-3}(x-100)\\ & \Rightarrow y={\displaystyle \frac{-1}{2000}}x+{\displaystyle \frac{3}{20}}\end{array}$ .
Now we need to get the value of
$\begin{array}{rl}& f\left(95\right)-f\left(98\right)={\displaystyle \frac{-1}{2000}}\left(95\right)+{\displaystyle \frac{3}{20}}-({\displaystyle \frac{-1}{2000}}\left(98\right)+{\displaystyle \frac{3}{20}})\\ & \Rightarrow {\displaystyle \frac{-1}{2000}}\left(95\right)+{\displaystyle \frac{3}{20}}+{\displaystyle \frac{1}{2000}}\left(98\right)-{\displaystyle \frac{3}{20}}={\displaystyle \frac{1}{2000}}(98-95)\\ & \Rightarrow {\displaystyle \frac{3}{2000}}=0.00150\end{array}$.
Hence we can say that the estimated value is $0.00150$ and the calculator value is given as $0.00158$ .
The error percentage will be $$ .
Hence we can say that the percentage error is $5$ .
Therefore we can conclude that the estimated value of the given quantity ${\displaystyle \frac{1}{\sqrt{95}}}-{\displaystyle \frac{1}{\sqrt{98}}}$ using linear approximation is $0.00150$ and the value found with calculator is $0.00158$ and the error is $5$.

Note: While answering this question we should be sure with our concept that we apply and the calculations we make. This is a question related to errors and approximations chapter. We can find the value of ${\displaystyle \frac{1}{\sqrt{95}}}$ and ${\displaystyle \frac{1}{\sqrt{98}}}$ as given
$\begin{array}{rl}& {\displaystyle \frac{1}{\sqrt{95}}}={\displaystyle \frac{-1}{2000}}\left(95\right)+{\displaystyle \frac{3}{20}}\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{95}}}={\displaystyle \frac{-95}{2000}}+{\displaystyle \frac{3}{20}}\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{95}}}=-0.0475+0.15\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{95}}}=-0.1025\end{array}$
and similarly
$\begin{array}{rl}& {\displaystyle \frac{1}{\sqrt{98}}}={\displaystyle \frac{-1}{2000}}\left(98\right)+{\displaystyle \frac{3}{20}}\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{98}}}={\displaystyle \frac{-98}{2000}}+{\displaystyle \frac{3}{20}}\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{98}}}=-0.049+0.15\\ & \Rightarrow {\displaystyle \frac{1}{\sqrt{98}}}=-0.101\end{array}$.