# Conceptual Understanding of Rates of Change

A rate is a comparison between two quantities with different units.

We often judge performances by using rates. For example:

- Average speed of a motor vehicle is $90$ km per hour.
- Average typing rate is $55$ words per minute.
- Average score of a basketball player is $34$ points per game.

Speed is a commonly used rate. It is the rate of change in distance per unit of time. It is a well-known formula:

$$\text{average speed}=\dfrac{\text{distance travelled}}{\text{time taken}}$$

However, if a car has an average speed of $80$ km/h for a trip, it does not mean that the car travels at exactly $80$ km/h for the whole time.

In fact, the speed will probably vary continuously throughout the trip.

So, how can we calculate the car’s speed at any particular time?

A heavy particle falls from a tower, 50 metres high, and crashes into the ground. How can we predict the damage the spanner causes when it hits the ground? We will begin by attempting to develop a mathematical model for the speed of the particle as it falls.

Speed is a rate – a measure of distance travelled per unit time. Our modelling thus begins with a review of rate and the concept of a constant rate.

## Constant Rates of Change

When the rate of change of one quantity with respect to another does not alter, the rate is constant.

For example, if petrol is $80$ cents per litre then every litre of petrol purchased at this rate always costs $80$ cents. This means $10$ litres of petrol would cost $8.00$ dollars and $100$ litres of petrol would cost $80.00$ dollars.

## Variable Rates of Change

The particle travels $50$ metres in $3$ seconds and then stops moving. In other words, it falls $50$ metres and then hits the ground. We also notice that the speed of the particle is not constant but is variable.

If a rate is not constant but is changing then it must be a variable rate.

For example, the amount of electricity used per hour by a household is most likely to be a variable rate as the need for electricity will change throughout the day.

## Average Rates of Change

If a rate is variable it is sometimes useful to know the average rate of change over a specific interval. For example, a tree grew from $5$ m this time last year to $6$ m now.

$$ \begin{align}

\text{average rate of growth} &= \dfrac{\text{change in height}}{\text{change in time}} \\

&= \dfrac{(6-5) \text{ m}}{1 \text{ year}} \\

&= 0.2 \text{ m/year} \\

\end{align} $$

This means that the tree grew $0.2$ metres over the past year but not necessarily constantly as that rate during the year.

## Instantaneous Rates of Change

If a rate is variable, it is often useful to know the rate of change at any given time or point, that is, the instantaneous rate.

For example, a police radar gun is designed to five an instantaneous reading of a vehicle’s speed. This enables the police to make an immediate decision as to whether a car is breaking the speed limit or not.

Instantaneous rates can be found from a curved graph by:

- drawing a tangent to the curve a the point in equation,
- calculating the gradient of the tangent over an appropriate interval, that is, between two points whose coordinates are easily identified.

## Derivative in terms of Rates of Change

$$f'(x) = \lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}$$

The gradient of the tangent at the variable point $(x,f(x))$ is the limiting value of $\dfrac{f(x+h)-f(x)}{h}$ as $h$ approaches $0$.

This formula gives the gradient of the tangent to the curve $y=f(x)$ at the point $(x,f(x))$, for any value of the variable $x$ for which this limit exists. Since there is at most one value of the gradient for each value of $x$, the formula is actually a function.

## Algebraic vs Calculus in terms of Rates of Change

As stated above,

*average rates of change*is often referred as

*algebraic rates of change*. Algebraic acceleration is the average rates of change, while acceleration is the term which is instantaneous rates of change of speed of velocity which obtained by differentiating velocity.