Mean value theorem
Okay kiddo, imagine you are on a playground that has a very high slide. You want to know how fast you can slide down it. The mean value theorem can help you figure out the average speed you were going while sliding down.
The mean value theorem says that if you have a function (which is like a rule that tells you how to find one thing when you know another thing), and that function is continuous (which means it doesn't have any breaks or jumps in it), and it's differentiable (which means you can draw a line tangent to any point on the curve), then there will be a point on that curve where the tangent line is parallel to the line that connects the endpoints.
In plain English, this means that if you draw a line between the starting point and the ending point of your slide, and you draw a tangent line to the curve of your slide at some point, then that tangent line will be the same slope (or angle) as the line you drew between the starting and ending points.
So if we go back to the playground example, let's say you slide down the slide for 10 seconds and you know you went from the top of the slide (50 feet high) to the bottom (0 feet high). Your function would be your position (how high up you are) as a function of time (how long you've been sliding). Using the mean value theorem, you can find the average speed you were going between the starting and ending points.
Here's how: let's say you started sliding at time t=0 and ended at t=10. The line connecting the starting and ending points would have a slope of -5 (since you went down 50 feet in 10 seconds). The mean value theorem says there must be a point on the curve where the slope of the tangent line is also -5. Let's say that point is at t=5 (halfway through your slide), and at that point, the tangent line has a slope of -5 as well.
Now we know that you were going down the slide at a constant speed of -5 feet per second during that 5-second interval, because the slope of the tangent line (the instant speed at t=5) was the same as the slope of the line connecting the starting and ending points (the average speed during the whole slide). So your average speed for the whole slide was also -5 feet per second.
That's how the mean value theorem can help you find the average rate of change of a function, like how fast you were sliding down a slide. Pretty cool, huh?
The mean value theorem says that if you have a function (which is like a rule that tells you how to find one thing when you know another thing), and that function is continuous (which means it doesn't have any breaks or jumps in it), and it's differentiable (which means you can draw a line tangent to any point on the curve), then there will be a point on that curve where the tangent line is parallel to the line that connects the endpoints.
In plain English, this means that if you draw a line between the starting point and the ending point of your slide, and you draw a tangent line to the curve of your slide at some point, then that tangent line will be the same slope (or angle) as the line you drew between the starting and ending points.
So if we go back to the playground example, let's say you slide down the slide for 10 seconds and you know you went from the top of the slide (50 feet high) to the bottom (0 feet high). Your function would be your position (how high up you are) as a function of time (how long you've been sliding). Using the mean value theorem, you can find the average speed you were going between the starting and ending points.
Here's how: let's say you started sliding at time t=0 and ended at t=10. The line connecting the starting and ending points would have a slope of -5 (since you went down 50 feet in 10 seconds). The mean value theorem says there must be a point on the curve where the slope of the tangent line is also -5. Let's say that point is at t=5 (halfway through your slide), and at that point, the tangent line has a slope of -5 as well.
Now we know that you were going down the slide at a constant speed of -5 feet per second during that 5-second interval, because the slope of the tangent line (the instant speed at t=5) was the same as the slope of the line connecting the starting and ending points (the average speed during the whole slide). So your average speed for the whole slide was also -5 feet per second.
That's how the mean value theorem can help you find the average rate of change of a function, like how fast you were sliding down a slide. Pretty cool, huh?
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