Mean speed theorem

Okay, so let's say you're riding your bike from your house to the park. You might go really fast at times and slower at other times, right? But if we want to know how fast you were going on average, we can use something called the mean speed theorem.

The mean speed theorem says that if you travel a certain distance at different speeds, your average speed over the whole distance will always be somewhere in between the fastest and slowest speeds you went.

So, let's pretend your house is 2 miles away from the park. On your way there, you might ride really fast down a hill and reach a speed of 20 miles per hour. But then you might have to slow down when you're going up a hill or when you need to turn a corner. Maybe at those times, you only go 5 miles per hour.

If we add up all the time it takes you to travel those 2 miles, and then divide by how long it took you, we get your average speed. And because of the mean speed theorem, we know that your average speed will always be somewhere between 5 miles per hour (the slowest you went) and 20 miles per hour (the fastest you went).

So even though you might not have gone the same speed the whole way, we can use the mean speed theorem to figure out how fast you were going on average during your whole bike ride!