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Fundemental theorem of Calculus intuition
August 24, 2016 at 5:28 pm
I've always been curious why the fundamental theorem of calculus works, I'm talking about the one which talks about the derivative of an integral of a function gives the function itself. Why does this happen? Am I missing something completely obvious?
In other words, why does the slope of the tangent line of every point on an area function result in the original function?
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 5:46 pm
(This post was last modified: August 24, 2016 at 5:47 pm by Alex K.)
It's more intuitive if you look at it the other way around - If you look at how the integral function F (as a function of the upper bound of the integral) arises geometrically as the area under the curve f, I find it very intuitive that the rate of change of the function F is given by the height of the graph of f. The larger f is at a point, the quicker the area, and hence F, increases when integrating a bit further.
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 5:50 pm
(August 24, 2016 at 5:46 pm)Alex K Wrote: It's more intuitive if you look at it the other way around - If you look at how the integral function F (as a function of the upper bound of the integral) arises geometrically as the area under the curve f, I find it very intuitive that the rate of change of the function F is given by the height of the graph of f. The larger f is at a point, the quicker the area, and hence F, increases when integrating a bit further.
Oh, wow. I never looked at it that way. That helps me a LOT. Thanks!
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 5:54 pm
Any time...
The fool hath said in his heart, There is a God. They are corrupt, they have done abominable works, there is none that doeth good.
Psalm 14, KJV revised edition
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 9:51 pm
You smart people can be really fucking annoying sometimes.
Especially when I'm only just smart enough to realise that I'm suffering from big brain envy.
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 9:57 pm
Nice explanation Alex 
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 10:06 pm
I like bananas.
At the age of five, Skagra decided emphatically that God did not exist. This revelation tends to make most people in the universe who have it react in one of two ways - with relief or with despair. Only Skagra responded to it by thinking, 'Wait a second. That means there's a situation vacant.'
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 10:18 pm
Um . .
keep in mind everyone brings something different to the table in cognitive skills.
I'm terrible at math at the level of algebra and above. Period. And it seems to be an inherent wiring issue. Some years ago I had dire need to master some HO scale model railroad planning software. I wasn't thrilled about it, and as I am almost daily reminded, the folks that write software (and design computers) are profoundly different cognitively than me. Conversely, the people that 'designed' home electrical wiring, (T1, T2, ground, neutral, red, black, allowable wire ampacity, fuses, breakers, etc) God bless 'em, they did everything the way I would have done it. I've wired houses, 3 and 4 way light switches, grain bins, motors, garages, and a bunch other stuff) and other than being slow (work by myself) I'm damn good at it. But in sparky school, complex arithmetic is right at the limits of what I can do with numbers.
Geometry, haven't touched since high school, but I remember liking it and doing well with it. Most I've done with it since was with the model train software and laying out grain bin locations for my farm for access and grain drying.
Well anyhow, the gist is, I bet everyone wishing they could unnerstan integrals and such, most likely has skills and skill sets most of the folks that can integrate e to the x in their sleep can only dream of. And vicety versa in regards to art and cooking and mining, and materials science and baseball and music and etc.
No one is good at everything, and spending time worrying about other peoples cleavage, or hairline, or benchpressing ability, or height or skinniness or braininess in otherwise obscure endeavors and fields is
well, we all know.
(and notice I didn't list penis size)
The granting of a pardon is an imputation of guilt, and the acceptance a confession of it.
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 10:20 pm
I like cake too.
At the age of five, Skagra decided emphatically that God did not exist. This revelation tends to make most people in the universe who have it react in one of two ways - with relief or with despair. Only Skagra responded to it by thinking, 'Wait a second. That means there's a situation vacant.'
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RE: Fundemental theorem of Calculus intuition
August 24, 2016 at 10:42 pm
Someone writes something and I can't understand a word of it.
I think, ok, I'll just google it and find a dumbed down version.
Still can't understand. Ok, I'll try and find the basics and start there.
Still can't understand. Ok, I'll find the basics of the basics and start there.
Still can't understand.
That's when I realise, hey, I'm never going to be smart enough.
Emotionally, that hurts. I like to think of myself as a smart guy.
I'm not.
So it makes me feel a little bit better that I'm smart enough to realise that.
Which makes me not really dumb, at least.
Anyway, I love banana cake. :-)
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