Rate of Change of Quantities: Overview, Questions, Preparation

Rate of change of a quantity can be described as how a function varies with respect to its input variable. It is a measure of how the value of a quantity changes with respect to another. Mathematical representation can be given by

dy/dx = (y₂ – y₁) / (x₂ – x₁)

i.e. rate of change of y with respect to x.

Graphical Representation: In the case of graphs, it's slope is used to calculate the rate of change. For this, the conditions are:

1. Positive slope:The quantity 'y' increases as 'x' increases. 
2. Negative slope:The quantity 'y' decreases as 'x' increases. 
3. Zero slope:The quantity 'y' remains constant (no change) as 'x' changes. 

Rate of change if calculated can help us analyze trends, predict future values, make well informed decisions and many more. Business and stock market growth heavily rely on the concept of rate of change. Analysing strategies, calculating marginal cost, maximizing profits, etc. can be easily possible using this technique of derivatives that allows us to take calculated risks and grow our businesses. It helps study the changes and behaviour of the function, which tells us how one quantity can vary with respect to another and take planned decisions according to that.

There are mainly two types of rate of change:

1. Average Rate of Change:

This is the rate which tells the change over a particular time.

Formula: f(x2) – f(x1)/x2 – x1

Secant slope joining two points on the curve detremines the value.

Example: Average speed of a vehicle travelling at 150 km in 3 hours will be 150/3 = 50 km/hour

2. Instantaneous Rate of Change:

This is the rate which tells us the change at a specific point.

Formula: dy/dx​= limΔx→0 ​Δy/Δx

Tangent slope to the curve determines the value.

Example: If s = t^2, then ds/dt = 2t. And If t = 5, then velocity = 5*2 = 10 m/s.