Rate is a ratio where the units of the denominator and numerator are different. For example speed is a rate with units like Km per hour or miles per hour. Rate of change is essentially a rate which compares the change in two quantities.

Many mathematical and physical phenomena are interpreted as rates of change and problems related to these are solved using Calculus. Rate of change also occurs in many commercial applications and we can thus say that every real life situation is influenced or determined by Rates of change.

When two quantities change simultaneously, rate of change is the change in one quantity compared to the corresponding change in the other.

Suppose we have an independent variable x, and y the dependent variable, then

Rate of change of y with respect to x = $\frac{Change\ in\  y}{Change\ in\ x}$

Velocity is defined as the rate of change of displacement and acceleration is defined as the rate of change of velocity.
Suppose x changes from $x_1$ to $x_2$ and correspondingly y changes from $y_1$ to $y_2$, then

Rate of change = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

Speed is a rate of change and the formula for speed in terms of distance and time is :

Speed = $\frac{Distance}{Time}$

Based on this the formula for velocity can be written as:

Velocity = $\frac{d}{t}$ where 'd' is the displacement in time 't'.

Solved Example

Question: When Harris started his car at 3:00 PM he noted the Odometer reading as 7458 Km.  When he stopped the Car at a Highway Restaurant at 7:00 PM the Odometer read as 7856 Km. Find the average speed at which he was driving. The distance driven is given by the difference in the odometer readings.
Solution:
 
Distance = 7856 - 7458 = 398 Km

This distance is covered in four hours.

Hence the average speed = $\frac{Distance}{Time}$ = $\frac{398}{4}$ = 99.5 Km/hr.

 


Slope and Rate of Change

The slope of a straight line is defined as the ratio between "Rise" and Run" of the line. As Rise and Run represent the vertical and horizontal change in the position of the point, Slope is the rate of change in y coordinate with respect to the change in the x coordinate of the point.
The formula for finding the slope of a line passing through two points (x1, y1) and (x2, y2) is given by
Slope = $\frac{Rise}{Run}$ = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$
Using function notation, the slope of the line can be given as:

Slope = $\frac{f(b)-f(a)}{b-a}$   where (a, f(a)) and (b, f(b)) are two points on the line.

Average Rate of Change

Suppose f is a function of x. Consider two points P(a, f(a)) and Q(b, f(b)) on the graph of the function f(x).

The slope of the secant line PQ = $\frac{f(b)-f(a)}{b-a}$ is called the difference quotient.

The difference quotient of the two points P and Q is defined to be the average rate of change of function from x = a to b.

When the two points are taken as (x, f(x)) and (x + $\Delta$ x, f(x+$\Delta$ x)), then the average rate of function as x changes from x to x+$\Delta$x is
Average rate of change = $\frac{f(x+\Delta x)-f(x)}{\Delta x}$

When we use the variable y to represent the function, that is when y = f(x) and if the change in y is Δy corresponding to a change of Δx in x, then the average rate of change of y with respect to x is written as $\frac{\Delta y}{\Delta x}$

$\frac{\Delta y}{\Delta x}$ = $\frac{f(x+\Delta x)-f(x)}{\Delta x}$
Consider the image mentioned in subtopic "Average Rate of Change".

The secant line PQ tends to coincide with the tangent line at P as the point Q comes closer to P. Thus the tangent at P can be considered as the limiting line of secant PQ. This can be stated in other words as $\Delta$ x $\rightarrow$ 0, secant PQ approaches the tangent at P. Hence the slope of the tangent at P is the limit of the difference quotient

$\frac{f(x+\Delta x)-f(x)}{\Delta x}$
. This limit is called the instantaneous rate of change of the function f(x).

Instantaneous rate of change = $\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}$ = $\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$ provided the limit exists.

The instantaneous rate of change of a function f(x) at x = a is also defined to be the derivative of the function at the point which is represented as f'(a).
When the change in the variable y, Δy = f(x + $\Delta$ x) - f(x) is a constant for all values of x, then the rate of change of y with respect to x is a constant.This means $\frac{\Delta y}{\Delta x}$ is a constant for all values x.

We come across constant rate of change in the case of linear functions, as the slope of a linear function y = mx + b is a constant m, independent of x.

Solved Example

Question: Find the constant rate of change for the values given in the table
  Time
 t in hrs
Distance
d in miles
    5     300
   10
    600
   15
    900
   20
   1,200

Solution:
 
The change in the variable t can be observed a constant = 5 hrs. Corresponding to every change of 5 hrs, the distance increases constantly by 300 miles. 

Constant rate of change = $\frac{(Change\ in\ distance)}{(Change\ in\ time)}$ = $\frac{300}{5}$ = 60 mph.

This is a situation of traveling at a constant speed of 60 mph.
 

The relative rate of change of a quantity is the ratio of the absolute rate of change to the value of the quantity. When dealing with instantaneous rate of change, the relative rate of change of a function is $\frac{f'(x)}{f(x)}$. You may note here the instantaneous rate of change is expressed as the derivative of the function f'(x). The expression representing the relative rate of change is the derivative of the logarithm of f(x).

That is $\frac{f'(x)}{f(x)}$ = [ln(f(x))]'

Relative rate of change finds application in Economics like the elasticity of demand.
Percentage rate of change is the relative rate of change expressed as a percentage.
Percentage rate of change = 100 x $\frac{f'(x)}{f(x)}$
For functions of two or three variables, the directional derivative for a given direction u represents the rate of change of the function in the direction of u and is represented by$D_u$ f(x) where x denotes the variable vector of two or three dimension.

The Maximum value of the directional derivative is | $\triangledown$f(x) | and it occurs in the direction of the gradient vector $\triangledown$f(x).
The graph of the function f(x) is given below:

     Rate of Change Graph
  1. Find the average rate of change as x varies from 1 to 2
  2. Find the instantaneous rate of change at x = 1.

The average rate of change as x varies from 1 to 2 = $\frac{f(2)-f(1)}{2-1}$ = $\frac{9-1}{1}$ = 8

The instantaneous rate of change is same as the slope of the tangent at x = 1. It can be seen that the tangent at P (1,1) passes through another point (2,4)

Slope of the tangent at P = $\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ = $\frac{4-1}{2-1}$ = 3.

Hence instantaneous change at x = 1, is 3.