Answer
452.7k+ views
Hint: The differentiation of \[x\] is represented by \[dx\] is defined by \[dx = x\] where \[x\] is the minor change in \[x\]. The differential of \[y\] is represented by \[dy\] is defined by \[dy = \dfrac{{dy}}{{dx}}x\]. As \[x\] is very small compared to \[x\], so \[dy\] is the approximation of \[y\]. Hence the increment in \[y\] corresponding to the increment in \[x\], denoted by \[\Delta y\], is given by \[\Delta y = f\left( {x + \Delta x} \right) - f\left( x \right)\].
Complete step-by-step answer:
Let \[y = \sqrt x \] where \[x = 49\& \Delta x = 0.5\]
Since \[y = \sqrt x \]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\sqrt x } \right)}}{{dx}} = \dfrac{1}{{2\sqrt x }}\]
Now,
\[
\Delta y = \dfrac{{dy}}{{dx}}\Delta x \\
\Rightarrow \Delta y = \dfrac{1}{{2\sqrt x }}\left( {0.5} \right) \\
\Rightarrow \Delta y = \dfrac{1}{{2\sqrt {49} }} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{1}{{2 \times 7}} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{1}{{14}} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{{0.5}}{{14}} \\
\therefore \Delta y = 0.036 \\
\]
Also,
\[\Delta y = f\left( {x + \Delta x} \right) - f\left( x \right)\]
By substituting the above values, we have
\[
\Delta y = \sqrt {x + \Delta x} - \sqrt x {\text{ }}\left[ {\because f\left( x \right) = y = \sqrt x } \right]{\text{ }} \\
0.036 = \sqrt {49 + 0.5} - \sqrt {49} \\
{\text{0}}{\text{.036}} = \sqrt {49.5} - 7 \\
\sqrt {49.5} = 0.036 + 7 \\
\therefore \sqrt {49.5} = 7.036{\text{ }} \\
\]
Thus, the approximate value of \[\sqrt {49.5} \] is 7.036
Note: In this problem we have solved the approximation up to 3 places of decimals. And also, we have rounded off the decimal numbers up to three places. We use differentiation to find the approximate values of the certain quantities.
Complete step-by-step answer:
Let \[y = \sqrt x \] where \[x = 49\& \Delta x = 0.5\]
Since \[y = \sqrt x \]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{d\left( {\sqrt x } \right)}}{{dx}} = \dfrac{1}{{2\sqrt x }}\]
Now,
\[
\Delta y = \dfrac{{dy}}{{dx}}\Delta x \\
\Rightarrow \Delta y = \dfrac{1}{{2\sqrt x }}\left( {0.5} \right) \\
\Rightarrow \Delta y = \dfrac{1}{{2\sqrt {49} }} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{1}{{2 \times 7}} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{1}{{14}} \times 0.5 \\
\Rightarrow \Delta y = \dfrac{{0.5}}{{14}} \\
\therefore \Delta y = 0.036 \\
\]
Also,
\[\Delta y = f\left( {x + \Delta x} \right) - f\left( x \right)\]
By substituting the above values, we have
\[
\Delta y = \sqrt {x + \Delta x} - \sqrt x {\text{ }}\left[ {\because f\left( x \right) = y = \sqrt x } \right]{\text{ }} \\
0.036 = \sqrt {49 + 0.5} - \sqrt {49} \\
{\text{0}}{\text{.036}} = \sqrt {49.5} - 7 \\
\sqrt {49.5} = 0.036 + 7 \\
\therefore \sqrt {49.5} = 7.036{\text{ }} \\
\]
Thus, the approximate value of \[\sqrt {49.5} \] is 7.036
Note: In this problem we have solved the approximation up to 3 places of decimals. And also, we have rounded off the decimal numbers up to three places. We use differentiation to find the approximate values of the certain quantities.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)