What does the term 'rate' mean in maths

Question

For example birth rate, death rate in population modelling. Correct me if I'm wrong, but does this mean a certain number of births or deaths over a period of time?

Aslo, rate of change in calculus when dealing with derivatives. Does this mean infinitesimal change with respect to another quantity. How much y changes when we change x quantity. Can someone please give me some laymen term examples to confidently understand this.

I've looked elsewhere online for clarification, but nothing seems to be resolving my Confusion.

Answer 1

Rate = Speed

If your birth function varied continuously the rate at which it varied would be the derivative of the function.

For discrete functions the rate is usually synonymous to the difference or the difference of consecutive terms divided by time units.

Answer 2

Rate implies it's a relative measure, typically a ratio, compared to some other quantity. For example death rate could be per unit of time or could be per hundred thousand people or per war. A rate of change is a special case where a function changes with respect to a variable and we compare the relative rates of change between the input and the output.

Answer 3

You're right; birth and death rates are measures of how often, people perhaps, are born and die. What's important, though, is what this measure is with respect to. It's valid to say about 4 people are born per second, but it's also valid to say about 2 people are born per death.

What's interesting is that the statistic that we're told is an average of the actual birth/death rate. Maybe as you're reading this, the death rate is 1.7 people per second, and perhaps now it's 1.9 people per second. So it averages, more or less, to almost 2 people per second.

The derivative gives information about how fast something is changing and in what direction. What's neat is Earth's population is huge, ~7.6 billion people, but both the birth and death rates are very small in comparison, about 4 and 2 people per second respectively. So we can use that to predict that Earth's population is overall increasing (direction) at a rate of ~2 people per second (speed).