For the function
the value of
and
are as follows:
![\fbox{\begin\\\ \math dy=\dfrac{1}{2}\ \text{and}\ \Delta y=0.414\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5C%5C%20%5Cmath%20dy%3D%5Cdfrac%7B1%7D%7B2%7D%5C%20%5Ctext%7Band%7D%5C%20%5CDelta%20y%3D0.414%5C%5C%5Cend%7Bminispace%7D%7D)
Further explanation:
In the question it is given that the function is
. The term radical represents the square root symbol
.
The function is expressed as follows:
![y=\sqrt{x}](https://tex.z-dn.net/?f=y%3D%5Csqrt%7Bx%7D)
The above function is also represented as follows:
![f(x)=\sqrt{x}](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7Bx%7D)
It is given that the value of
is
and the value of
is
.
Consider that for small change in the value of
as
there occur a small change in
as
.
The change in the dependent variable
is expressed as follows:
![\fbox{\begin\\\ \Delta y=f(x+\Delta x)-f(x)\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5C%5C%20%5CDelta%20y%3Df%28x%2B%5CDelta%20x%29-f%28x%29%5C%5C%5Cend%7Bminispace%7D%7D)
The change in the dependent variable
with respect to change in independent variable is expressed as follows:
![\fbox{\begin\\\ \dfrac{\Delta y}{\Delta x}=\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5C%5C%20%5Cdfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cdfrac%7Bf%28x%2B%5CDelta%20x%29-f%28x%29%7D%7B%5CDelta%20x%7D%5C%5C%5Cend%7Bminispace%7D%7D)
Substitute
and
in the above equation.
![\dfrac{\Delta y}{\Delta x}=\dfrac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D-%5Csqrt%7Bx%7D%7D%7B%5CDelta%20x%7D)
Rationalize the above expression by multiplying and dividing the term
.
![\begin{aligned}\dfrac{\Delta y}{\Delta x}&=\dfrac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\\&=\dfrac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\times\dfrac{\sqrt{x+\Delta x}+\sqrt{x}}{\sqrt{x+\Delta x}+\sqrt{x}}\\&=\dfrac{x+\Delta x-x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}\\&=\dfrac{1}{\sqrt{x+\Delta x}+\sqrt{x}}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cdfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%26%3D%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D-%5Csqrt%7Bx%7D%7D%7B%5CDelta%20x%7D%5C%5C%26%3D%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D-%5Csqrt%7Bx%7D%7D%7B%5CDelta%20x%7D%5Ctimes%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%7D%5C%5C%26%3D%5Cdfrac%7Bx%2B%5CDelta%20x-x%7D%7B%5CDelta%20x%28%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%29%7D%5C%5C%26%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%7D%5Cend%7Baligned%7D)
Substitute
for
and
for
in the above equation.
![\begin{aligned}\dfrac{\Delta y}{1}&=\dfrac{1}{(\sqrt{1+1}+\sqrt{1})}\\\Delta y&=\dfrac{1}{\sqrt{2}+1}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cdfrac%7B%5CDelta%20y%7D%7B1%7D%26%3D%5Cdfrac%7B1%7D%7B%28%5Csqrt%7B1%2B1%7D%2B%5Csqrt%7B1%7D%29%7D%5C%5C%5CDelta%20y%26%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%2B1%7D%5Cend%7Baligned%7D)
Rationalize the above expression to obtain the value of
.
![\begin{aligned}\Delta y&=\dfrac{1}{\sqrt{2}+1}\\&=\dfrac{1}{\sqrt{2}+1}\times\dfrac{\sqrt{2}-1}{\sqrt{2}-1}\\&=\sqrt{2}-1\\&=1.414-1\\&=0.414\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5CDelta%20y%26%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%2B1%7D%5C%5C%26%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7B2%7D%2B1%7D%5Ctimes%5Cdfrac%7B%5Csqrt%7B2%7D-1%7D%7B%5Csqrt%7B2%7D-1%7D%5C%5C%26%3D%5Csqrt%7B2%7D-1%5C%5C%26%3D1.414-1%5C%5C%26%3D0.414%5Cend%7Baligned%7D)
Therefore, the value of
is
.
For an infinitesimally small change in
i.e., as
then the equation (1) is expressed as follows:
![\fbox{\begin\\\ \dfrac{dy}{dx}=\lim_{x\to0}\dfrac{(f(x+\Delta x)+f(x))}{\Delta x}\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5C%5C%20%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Clim_%7Bx%5Cto0%7D%5Cdfrac%7B%28f%28x%2B%5CDelta%20x%29%2Bf%28x%29%29%7D%7B%5CDelta%20x%7D%5C%5C%5Cend%7Bminispace%7D%7D)
Substitute
and
in the above equation.
![\dfrac{dy}{dx}=\lim_{x\to0}\left(\dfrac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\right)](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Clim_%7Bx%5Cto0%7D%5Cleft%28%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D-%5Csqrt%7Bx%7D%7D%7B%5CDelta%20x%7D%5Cright%29)
Rationalize the above expression as follows:
![\begin{aligned}\dfrac{dy}{dx}&=\lim_{x\to0}\left(\dfrac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x}\times\dfrac{\sqrt{x+\Delta x}+\sqrt{x}}{\sqrt{x+\Delta x}+\sqrt{x}}\right)\\&=\lim_{x\to0}\left(\dfrac{x+\Delta x-x}{\Delta x(\sqrt{x+\Delta x}+\sqrt{x})}\right)\\&=\lim_{x\to0}\left(\dfrac{1}{\sqrt{x+\Delta x}+\Delta x}\right)\\&=\dfrac{1}{\sqrt{\Delta x}+\Delta x}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cdfrac%7Bdy%7D%7Bdx%7D%26%3D%5Clim_%7Bx%5Cto0%7D%5Cleft%28%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D-%5Csqrt%7Bx%7D%7D%7B%5CDelta%20x%7D%5Ctimes%5Cdfrac%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%7D%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%7D%5Cright%29%5C%5C%26%3D%5Clim_%7Bx%5Cto0%7D%5Cleft%28%5Cdfrac%7Bx%2B%5CDelta%20x-x%7D%7B%5CDelta%20x%28%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5Csqrt%7Bx%7D%29%7D%5Cright%29%5C%5C%26%3D%5Clim_%7Bx%5Cto0%7D%5Cleft%28%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%2B%5CDelta%20x%7D%2B%5CDelta%20x%7D%5Cright%29%5C%5C%26%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7B%5CDelta%20x%7D%2B%5CDelta%20x%7D%5Cend%7Baligned%7D)
Substitute
for
and
for
in the above equation.
![\begin{aligned}\dfrac{dy}{1}&=\dfrac{1}{\srqt{1}+1}\\dy&=\dfrac{1}{2}\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%5Cdfrac%7Bdy%7D%7B1%7D%26%3D%5Cdfrac%7B1%7D%7B%5Csrqt%7B1%7D%2B1%7D%5C%5Cdy%26%3D%5Cdfrac%7B1%7D%7B2%7D%5Cend%7Baligned%7D)
Therefore, the value of
is
.
Thus, for the function
the value of
and
are as follows:
![\fbox{\begin\\\ \math dy=\dfrac{1}{2}\ \text{and}\ \Delta y=0.414\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5C%5C%20%5Cmath%20dy%3D%5Cdfrac%7B1%7D%7B2%7D%5C%20%5Ctext%7Band%7D%5C%20%5CDelta%20y%3D0.414%5C%5C%5Cend%7Bminispace%7D%7D)
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Answer details:
Grade: Senior school
Subject: Mathematics
Chapter: Curve sketching
Keywords: Curve, graph, radical, quadratic, expression, roots, y=rootx, delta y, dy, derivative, dx, delta x, round off, decmials, decimal places, rationalize.