Question

If \[y=\sin x\] and x changes from \[\dfrac{\pi }{2}\] to \[\dfrac{22}{14}\], what is the approximate change in \[y\]?

Answer 1

Hint: First, before proceeding for this, we suppose the change in the value of x as $\vartriangle x$ and assumed the starting value of x as \[\dfrac{\pi }{2}\].Then by using the above assumption, we get the expression as \[x+\Delta x=\dfrac{22}{14}\]. Then, by differentiating both sides with respect to x in the given function, we get the value of change in y.

Complete step by step answer:
In this question, we are supposed to find the approximate change in \[y\]when \[y=\sin x\] and x changes from \[\dfrac{\pi }{2}\] to \[\dfrac{22}{14}\].
So, before proceeding for this, we suppose the change in the value of x as $\vartriangle x$ and assume the starting value of x as \[\dfrac{\pi }{2}\].
So, by using the above assumption, we get the expression as:
\[x+\Delta x=\dfrac{22}{14}\]
Then, by rearranging the value of the expression and substituting the value of x as:
\[\begin{align}
  & \dfrac{\pi }{2}+\vartriangle x=\dfrac{22}{14} \\
 & \vartriangle x=\dfrac{22}{14}-\dfrac{\pi }{2} \\
\end{align}\]
Now, we are given with the equation in the question as \[y=\sin x\].
Then, by differentiating both sides with respect to x, we get:
\[\begin{align}
  & \dfrac{dy}{dx}=\dfrac{d}{dx}\sin x \\
 & \Rightarrow \dfrac{dy}{dx}=\cos x \\
\end{align}\]
Now, by calculating the value of derivative of the function at x as \[\dfrac{\pi }{2}\], we get:
$\cos \dfrac{\pi }{2}=0$
So, we get the value of change in y as:
$\vartriangle y=\dfrac{dy}{dx}\times \vartriangle x$
Now, by substituting the values in the above expression, we get:
$\begin{align}
  & \vartriangle y=0\times \left( \dfrac{22}{14}-\dfrac{\pi }{2} \right) \\
 & \Rightarrow \vartriangle y=0 \\
\end{align}$
Hence, we get the value of change in y as 0.

Note:
Now, to solve these types of questions we need to know some of the basics of the differentiation so as to get the answer correctly. So, the basic required formula for differentiation for this question is as:
$\dfrac{d}{dx}\sin x=\cos x$