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Fundamental Theorem of Calculus

david.alvarezrosa.com

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Re: Fundamental Theorem of Calculus

#41

Post by dalvrosa »

ironSkillet wrote: post #20
mchinen wrote: post #12

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions. I feel similar about the trace of a matrix being equal to the sum of eigenvalues. Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

It is not determined by the derivative, it's the antiderivative, as someone else mentioned. The derivative is the rate of change of a function. The "area under a curve" of the graph of a function measures how much the function is "accumulating", which is intuitively a sum of rates of change (taken to an infinitesimal limit).

Thanks for bringing some intuition!

Re: Fundamental Theorem of Calculus

#42

Post by enriquto »

mchinen wrote: post #12

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions. I feel similar about the trace of a matrix being equal to the sum of eigenvalues. Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative

the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values

     f'(n) = f(n+1) - f(n)
Then the fundamental theorem is just a telescopic sum:

     f(b) - f(a) = \sum_a^b f'(n)

Re: Fundamental Theorem of Calculus

#43

Post by WCSTombs »

bikrampanda wrote: post #2

What is the font used on the site?

Technically "it depends on the browser settings," but the body font Alegreya is served directly by the site, so I think it would be the one used in almost all cases.

The math fonts used in the formulas are just the ones provided by KaTeX, which I think are just TeX's default math fonts.

Re: Fundamental Theorem of Calculus

#44

Post by 1980phipsi »

magicalhippo wrote: post #32
mchinen wrote: post #12

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions. I feel similar about the trace of a matrix being equal to the sum of eigenvalues. Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

The antiderivative at x is defined as the area under the curve from 0 to x, which the Riemann sum gives a nice intuition for how you can get from the derivative. So to get the area under the curve between a and b, you calculate the area under the curve from 0 to b (antiderivative at b) and subtract the area under the curve from 0 to a (antiderivative at a). At least that's my sleep deprived take.

I took calculus in high school and college, and I don't think any of my instructors explained the intuition as well. So sleep-deprived or not, it's a great one!

Re: Fundamental Theorem of Calculus

#45

Post by globalnode »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

Here's my understanding: 1: In the 'olden days' the area A(x) under the graph f(x) used to be approximated as a Riemann sum. 2: Using limits, as the delta x in the Riemann sum->0, we'd call that an integral and set it to be the exact area A(x). 3: If we then look at some small change in A(x), we might notice f(x) = A'(x)... mind blown. 4: since we can now say A is an anti-derivative of f, we have A(x)=F(x)+C (we have to add the C because the derivative of a constant is 0). 5: Using logic and geometry we have C=-F(a) which leads to... 6: The area under the graph f between [a,b] is A = F(b)-F(a). 7: We don't have to cry anymore about pages of Riemann sum calculations.

Re: Fundamental Theorem of Calculus

#46

Post by chadcmulligan »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

If you dive into Analysis (the underlying theory behind calculus) this book - "How to Think About Analysis" by Lara Alcock is the book I wish I had when I studied it. Calculus by Spivak is the book I learnt from but it is probably not the easiest, it is very thorough though.

Re: Fundamental Theorem of Calculus

#47

Post by aquafox »

mchinen wrote: post #12

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions. I feel similar about the trace of a matrix being equal to the sum of eigenvalues. Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

If I tell you I have function f with f(a) = 10 and on it's path from a to b, the graph first increaes by 5 units then by another 10, and then later on drops by 25 units, you can immediately deduce that f(b) = f(a) + (+5 +10 -25) = 0. The fundamental theorem of calculus uses the same concept:

To see why \int_a^b f(x) dx = F(b) - F(a) with F'(x) = f(x),

we replace f with f' (and hence F with f) and get

\int_a^b f'(x) dx = f(b) - f(a).

Re-arranging terms, we get

f(b) = f(a) + \int_a^b f'(x) dx.

The last line just says: The value of function f at point b is is the value at point a plus the sum of all the infinitely many changes the function goes through on its path from a to b.

Re: Fundamental Theorem of Calculus

#48

Post by ulrikrasmussen »

EdwardDiego wrote: post #26

> This post introduces the Riemann integral Sweet! I'm keen to learn about the basic fundamentals of calculus! > For each subinterval ...(bunch of cool maths rendering I can't copy and paste because it's all comes out newline delimited on my clipboard) ... and let m k and M k denote the infimum and supremum of f on that subinterval... Okay, guess it wasn't the kind of introduction I had assumed/hoped. Very cool maths…

3Blue1Brown has an excellent video series that introduces calculus using very intuitive animations and explanations: https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53...

Re: Fundamental Theorem of Calculus

#49

Post by quibono »

viscousviolin wrote: post #6
bikrampanda wrote: post #2

What is the font used on the site?

Today I learned there's a CSS property for styling the first letter of a paragraph, neat. ( https://css-tricks.com/almanac/properties/i/initial-letter/ ) --edit: The font used for those initials is called Goudy Initialen: https://www.dafont.com/goudy-initialen.font

I love this -- I'll have to do something like that for my site. I always liked the big initials on the start of a paragraph. Though it feels a bit more prose-applicable than for non-fiction writing.

Re: Fundamental Theorem of Calculus

#50

Post by dalvrosa »

enriquto wrote: post #42
mchinen wrote: post #12

I've studied the proofs before but there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative at only two points, especially for wobbly non monotonic functions. I feel similar about the trace of a matrix being equal to the sum of eigenvalues. Probably this means I should sit with it more until it is obvious, but I also kind of like this feeling.

> there's still something mystical and unintuitive for me about the area under an entire curve being related to the derivative the discrete version is much clearer to me. Suppose you have a function f(n) defined at integer positions n. Its "derivative" is just the difference of consecutive values f'(n) = f(n+1) - f(n) Then the fundamental theorem is just a telescopic sum: f(b) - f(a) = \sum_a^b f'(n)

Very. Very nice
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