Instantaneous Rate of Change Formula

VSAT 2022

Instantaneous Rate of Change

The instantaneous rate of change is the change in the concentration of rate that occurs at a particular instant of time. The variation in the derivative values at a specific point also denotes the instantaneous rate of change.  The instantaneous rate of change at a point is equal to the derivative function evaluated at that point. 


Further, the average and instantaneous rate of change at a specific point can map in the graph as the tangent slope line, which shows like a curve slope. The value of the instantaneous rate of change is also equal to the slope of the tangent at a point of a curve.


The instantaneous rate of change formula can also be defined with the differential quotient and limits. The average rate of $y$ shift with respect to $x$ is the quotient of difference. The instantaneous rate of change formula represents with limit exists in,


$f’(a)$ = \[\lim_{\Delta x \rightarrow 0}\] \[\frac{\Delta y}{\Delta x}\] = \[\lim_{x \rightarrow 0}\] \[\frac{t(a+h)-(t(a))}{h}\]


How to Calculate Instantaneous Rate of Change in Graph?

The tangent straight line at a point can be drawn, which touches a curve at the point without crossing over the curve. Or else, the tangent should touch only one point of the curve. The slope of the tangent line can be calculated at a point, which represents the instantaneous rate of change, or derivative of the particular point.


The tangent line can be drawn in the graph to calculate the instantaneous rate of change as shown in the above image. Further, the slope of the tangent line can calculate using the formulas;

\[\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\] 

Or 

f’(\[x_{0}\]) = \[\lim_{(x_{0} \rightarrow x_{1})}\] \[\frac{f(x_{1})-f(x_{0})}{x_{1}-x_{0}}\] = \[\lim_{\Delta x \rightarrow 0}\]  \[\frac{f(x_{0}-\Delta x)-f(x_{0})}{\Delta x}\] 


Problems Related to Instantaneous Rate 

Problem 1: Calculate the Instantaneous rate of change of the function $f(x) = 4x^2 + 24$ at $x = 5$?

Answer: Give that, 

$y = f(x) = 4x^2 + 24$

$f'(x) = 4(2x) + 0$

$f'(x) =8x$


As per the given date, we need to calculate the instantaneous rate of change at the value $x = 5$.

$f'(5) = 8(5)$

$f'(5) = 40$

So, the instantaneous rate of change for the given function at $x = 5$ is $40$. 


Problem 2: Derive the Instantaneous rate of change for the function $f(x) = 7x^3 – 3x^2 + 2x + 5$ at $x = 3$?

Answer: Given that, 

$y = f(x) = 7x^3 – 3x^2 + 2x + 5$

$f'(x) = 7(3x^2) – 3(2x) + 2 + 0$

$f'(x) = 21x^2 – 6x + 2$


As per the given date, we need to calculate the instantaneous rate of change at the value $x = 3$.

$f'(3) = 21(3)2 –  6(3) + 2 =  189 – 18 + 2 = 173$

$f'(3) = 173$

So, the instantaneous rate of change for the given function at $x = 3$ is $173$. 


Conclusion

In this post, we have studied about the instantaneous rate of change of formula and discussed how it will be derived. We also come to know that the variance in derivative values at a given position also represents the instantaneous pace of change. We have also discussed the solved problem so that the application of this formula should be understood properly.

FAQs on Instantaneous Rate of Change Formula

1. What is the Instantaneous Rate of Change According to Physics?

According to physics, the instantaneous rate of change defines the change in the rate at a particular instant of time, and it is also the same as the change in the derivative values at a specific point. As per the graph, the instantaneous rate of change in a specific point shows the tangent slope line. 

2. What is the Major Difference Between Average Rate and Instantaneous Rate?

The changes in average rate can concentrate over a selected period of time. Where else, the instantaneous rate shows the vast changes at a particular time. The average and instantaneous rate of change can be determined by finding the slope of the tangent to the concentration to the time taken. 

3. What is the Instantaneous Rate?

The instantaneous rate of change reaction shows the change in concentration within an infinitely small interval of time. The instantaneous rate is the changes in the rate at some instant of time. This instantaneous rate can be measured by calculating the slope of the tangent of concentration in the graph to the time. 

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