## Don’t Drink & Derive

Well, here we go – the last gasp of calculus for the year, and it is *neat* trick.

By now, you are (or should be!) confident and competent with differentiation, but you should be beginning to wonder about the reverse of the process – it is possible to get the gradient function from the function, but how do you go the other way? Can you get the original function from the gradient function?

The answer is yes, but it is weirder than that (this is MAAATHSA! It is always weirder than that!). Lets start from a simple example. We know the derivative of x^2 is 2x, so the reverse (called the “integral”) leads to the fact that 2x integrated is x^2 (not quite, but it’ll do for now.

Check that for a second. If you take any point, x, on the function 2x, you get a height (y value) of 2x. The line f(x) = 2x makes a triangle of base x, and height 2x, which gives it an area of…

x^2!

Spooky! So what we are seeing (so far) is that the area between a function and the x-axis can be determined by evaluating the integral. But why? Well, that’s both complex – and incredibly simple – at the same time. It is as simple as realising that, if you find a lot of narrow rectangles and add them together you can approximate the area under the curve.

But this does not answer why :

Gives the area under the curve. The answer to that is a cool trick called “collapsing series”. If you want to understand this check out this resource, and notice what would happen if you added all the sections together… (this is not part of the course, so don’t stress if you don’t understand!)

Well, as I said before, we’re coming to the end, so I guess this is a good time for a review. Have you got twenty minutes? Try this:

And here are the two files I promised: Bookcheck 3 and Extra Notes.

Last of all, because pain can help you remember (not to do) things, I offer you this musical compilation:

See you in class.

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August 14, 2011 at 8:33 PM

In 20mins??? Really??? And I thought I could talk fast… That’s just crazy…

August 17, 2011 at 8:17 AM

Nice