The Basic Idea of Calculus

You may or may not have already done calculus, and for those of you have done it, it might have seemed like a really difficult subject. I struggled with it a lot in high school. Maybe you find it hard to recall what it was about. The funny thing is though; the actual idea behind calculus is really easy and familiar. Its all about approximation. Let me demonstrate. Let's look at the classic problem in calculus, finding the slope of a graph. Say you want to know just how steep a certain point is on a curve. The curve could look like anything. Why don't we start with an easier problem: the slope of a straight line. This is straightforward. In this case, the formula, actually more like the definition of slope is rise divided by run. Let's return to our original problem and use our newfound knowledge to tackle it. Let's draw a straight line from A to a close by point and figure out how steep that line is, to get an approximation of the actual slope at A. That looks kinda rubbish, so lets try again with a closer point. Even worse! How about another one. And you see this one's not too bad. Actually the points really close to A seem to give really really close looking approximations! So what's the actual slope? Calculus basically says the obvious thing, the actual steepness of the curve at A is the number that our approximations are getting closer and closer to. What about the other classic calculus problem? The area under a curve We have the same problem here, this could be any crazy graph and we don't have formulas for those. The only area I really really know of by heart is the area of a rectangle. Well let's use the rectangles then. Why not guess the area of the curve like this? Divide up the graph like this and draw a straight line from the curve to left. Now we can just find out the area of each of these bits. That's not an awful approximation I guess, but there are all these hanging out bits and missing stuff. Well we can make a better approximation. Why not divide the curve up smaller? See, the error gets smaller now. So we keep slicing smaller, and Calculus tells us the area we're limiting to is the one we want. There are lots of other problems that can be solved by Calculus in a similar way. I'll talk about more in some other videos on the topic. But now I want to bring up an issue with this whole thing. Let's go back to the area under the graph stuff. There's another, seemingly better, approximation I could have done. Why not, instead of using rectangles, use these things, called trapezoids. They have very simple area formulas too and as you can see the error is way less. These are better approximations, so wont they give a better, more accurate answer for the area? Well the trapezoid way gets you close the actual area very quickly, but both ways will limit towards the same area in the end, as you divide the graph smaller and smaller. Still, this is a bit worrying. If there are better and worse approximations, are there some approximations so bad that they're wrong? I'm sorry to say that there is. We'll be seeing a really surprising one next time.