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Introduction of Rate of Change

A slope may be a gradient, inclination, or a pitch. And the formulas are needed to calculate the steepness of a straight line. The higher would be the slope, the steeper it would be. This slope is popular as the rate of change in mathematics and physics. The slope is responsible for connecting multiple points together over a line. The rate of change is easy to calculate if you know the coordinate points.

The Rate of Change Formula

With Rate of Change Formula, you can calculate the slope of a line especially when coordinate points are given. The slope of the equation has another name too i.e. rate of change of equation.

The Rate of Change Formula

The rate of change between the points (x1, y1) and (x2, y2) in mathematics is given as,

\[\large Rate\;of\;Change= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

The value may be either positive or negative that signified the increase or decrease ratio between two data points. If there is some quantity whose value is the same overtime then it is named as the zero rates of change. Briefly, the rate of change simply tells the amount by which one entity affects the other. Well, the rate of change at its simplest denotes the amount at which one entity is affected by another.

Take an example of a Company simply interested in job creation or employment. So, how many jobs he needs to create currently and how it will be expanded in the near future is necessary to know before you start the process. For this purpose, the ROC (Rate of Change) model works the best. It will give a perfect idea of numbers where you should start and how can you expand immediately.

With an ROC model, this is also easy to calculate the growth rate of the population or salary revised rate, etc. With a careful application of ROC mode, you would get to know how applicable it can be to compute the tough problems.

Example Of Rate Of Change

Question: By how much has the value of y changed between the two points (-4, -7) and (-2, -6)?
Solution:
Here,
\[ x_{1}, y_{1} = (-4, -7) \]
\[ x_{2}, y_{2} = (-2, -6) \]
With the formula we can evaluate the rate of change: \[Rate\;of\;Change= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]
\[Rate\;of\;Change= \frac{-6-(-7)}{-2-(-4)}=\frac{1}{2}\]

Type of Rate of Change

1). Positive Rate Of Change

2). Negative Rate Of Change

Negative Rate Of Change

3). Zero Rate Of Change

Zero Rate Of Change

Average Rate of Change Formula

The average rate of change is defined as the average rate at which quantity is changing with respect to time or something else that is changing continuously. In other words, the average rate of change is the process of calculating the total amount of change with respect to another. In mathematics, the average ROC is given as A (x). It signified the average rate of change with

\[\large Average\;Rate\;of\;Change= \frac{f(a)-f(b)}{b-a}\]

f(a) and f(x) is the value of the function f(x) and a and b are the range limit.

Example Of Average Rate Of Change

Question 1: Calculate the average rate of change of a function, f(x) = 3x + 12 as x changes from 5 to 8 ?

Solution:Given,
f(x) = 3x + 12
a = 5
b = 8f(5) = 3(5) + 12
f(5) = 15 + 12
f(5) = 27f(8) = 3(8) + 12
f(8) = 24 + 12
f(8) = 36The average rate of change is,
A(x) = \[\frac{f(b)-f(a)}{b-a}\]A(x) = \[\frac{f(8)-f(5)}{8-5}\]

A(x) = \[\frac{36-27}{3}\]

A(x) = \[\frac{9}{3}\]

A(x) = 3

Constant Rate of Change

If the value of one coordinate increases significantly but the value of the other coordinate is the same then the rate of change is constant here means it always is the same. Basically, the graph would be a straight line either horizontal or vertical line. So, constant ROC can also be named as the variable rate of change. In the case of constant ROC, acceleration is absent and graphing the solution is easier.

Distance (Km)1002004008001000
Time (Hours)2481620
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