Why is integral opposite of derivative

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Email Address. Skip to content. Home About Faq. Q: Is there a formula to find the Nth term in the Fibonacci sequence? Posted on April 4, by The Physicist. Email Print Facebook Reddit Twitter. This entry was posted in -- By the Physicist , Equations , Math. Bookmark the permalink. Amphiprion says:. March 4, at pm. The Physicist says:. Shuchi Goyal says:. August 4, at pm. Rex Balsarin says:.

December 27, at pm. If not, thanks for reading. Chris says:. January 19, at pm. David Whitelaw says:. If we take the integral of that new function we get the original function back. So I see how they are opposite operations but I don't see how they are opposite in the sense that the integral is supposed to be the area under the curve and the derivative is supposed to be the instantaneous slope.

It seems like they are separate; deriving one function is the opposite of integrating it but when we look at it graphically it doesn't make sense. Also, if I start with some function, differentiate it and then take the integral I get the original function so how is that the area under the curve?

That non-obviousness is why it's called a "theorem" indeed, more formally stated, it's called "the fundamental theorem of calculus". But maybe I can help out with the intuition a little. Well, if you draw a picture, you'll see that we expect it to be.

That's true for any "nice" smooth, etc. Let me define such a function, the "accumulated area" function:. So we get. Hint: it doesn't! So "differentiate then integrate" doesn't necessarily bring you back to the original function. On the other hand, for nice enough functions e. The derivative is the instantaneous rate of change.

The second fundamental theorem of Calculus says, intuitively, that "the total change is the sum of all the little changes". The geometric interpretations of the derivative and the integral, are not always the best way to think about them. Now try to find A' x with limits.

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