Question

If $y={{x}^{3}}+5$ and $x$ changes from $3$ to $2.99$ , then the approximate change in $y$ is
1)$2.7$
2)$-0.27$
3) $27$
4) None of these

Answer 1

To solve this question use the concept of approximations of applications of derivatives. In a two variable equation we have one dependent variable and one independent variable. If we change the value of an independent variable then the value of the dependent variable will also change. To find change in dependent variable we use the concept of approximation.

Complete step-by-step solution:
Given equation is
$y={{x}^{3}}+5$……..$(1)$
As we know,
$\dfrac{\Delta y}{\Delta x}=\dfrac{dy}{dx}$
$\Delta y$ Denotes the change in $y$
$\Delta x$ Denotes the change in $x$
$\Rightarrow \Delta y=\dfrac{dy}{dx}.\Delta x$ ……….$(2)$
Differentiating the given equation $(1)$with respect to$x$, we get
$\Rightarrow$ $\dfrac{dy}{dx}=3{{x}^{2}}$
Put this value in equation $(2)$
$\Rightarrow \Delta y=3{{x}^{2}}\Delta x$ ………$(3)$
Since it is given that
$\Delta x=2.99-3$ $($ $x$ is changing from $3$ to $2.99$ $)$
$\therefore \Delta x=-0.01$
Put this value in equation $(3)$

& \Rightarrow \Delta y=3{{x}^{2}}\times -0.01 \\\ & \Rightarrow \Delta y=3.{{(3)}^{2}}\times -0.01 \\\ & \Rightarrow \Delta y=27\times -0.01 \\\ & \Rightarrow \Delta y=-0.27 \\\ \end{aligned}$$ **Hence the correct answer is option $2$ .** **Note:** Derivative of a function tells us the value of change in function by changing the value of an independent variable. So $\dfrac{dy}{dx}$ tells us about the change in value of $y$ with respect to change in $x$ . So, the derivative of a function will tell us the change in value of a function. This derivative can also be used to find the maximum and minimum value of the function.