Math and science

[Elbereth she/her]

But there has to be an explanation for it! Why does the antiderivative of a function have the same value as y times x for any given interval? I don't understand this.

[Silverblade5 he/him]

Just once... right?

I really don't see what's so bad about

 

a(ei-fh)-b(di-fg)+c(dh-eg)

[Hood he/him]

So we just did the Fundamental Theorem of Calculus today, which was cool. Also, for clarification, because my teacher isn't particularly good at explaining things: An antiderivative is the same thing as an indefinite integral, right? Why?

 

These are the only things that I hate in maths. I mean, in most of the cases, such distinctions don't matter. It hasn't till now, in my case. But in an exam (esp. viva) such questions can bring down grades.

[Curiosity he/him]

But there has to be an explanation for it! Why does the antiderivative of a function have the same value as y times x for any given interval? I don't understand this.

 

Have you looked up proofs for this? I had my math TA walk me through it, and it's pretty cool. The idea of an indefinite integral (from what I know) was not necessarily used until the Fundamental Theorem of Calculus was derived. Integrals and derivatives are two separate entities, but the FTC joins them and shows that they are inverse processes to a degree. It's just that often times, the symbol for integration is used as a magic wand to say, "This is the symbol for antidifferentiation", when, in fact, it's actually just computing the integral from 0 to [insert variable] for whatever function the integrand is. 

 

Definite integrals are your friends! (most of the time)

[Elbereth she/her]

I mean, I've seen proofs of the Fundamental Theorem, but not why taking the antiderivative of a function gives you the indefinite integral. I'll look around online, though. Maybe I can find soemthing.

[Cold Fusion he/him]

Today was awesome.  I was up until about 11:00 last night studying for today's test on optimization, Linearization, and Related Rates.  I took it, felt awesome, and I'm pretty sure I got 100% on it.  BC AP Calculus is awesome.

[Silverblade5 he/him]

Working out a proof for a seven by seven matrix determinate

[Creccio he/him]

Working out a proof for a seven by seven matrix determinate

 

I wonder why you do this to yourself...

[Silverblade5 he/him]

I wonder why you do this to yourself...

Because it's fun and better than spanish Also, got to have something to do after finishing early... Most of my notes are mental, and I retain the concepts fairly well, so I work on this to keep myself busy. 

[Hood he/him]

Working out a proof for a seven by seven matrix determinate

 

I'd use row reduction. I used to hate it earlier, but now it seems a boon.

[Dunkum he/him]

Can't stand matrices. matrix multiplication was always so complicated that I would get frustrated with them and just give it up. that said, I once read my Abstract Algebra and Number Theory textbooks for fun (and did something like 1/4 of the problems). theoretical classes were always much more interesting to me than the more practical calculations based ones, and differential equations was far and away the hardest math class I took.

On the integral/antiderivative question: I will give it a shot, though I am not 100% sure how close I am to it.

1. When I took calculus, we were taught integration like this: you start by looking at the area under a curve and dividing it up in to rectangles reaching from the axis up to the curve, each with the same width. then you make the rectangles thinner, allowing you to fit more into the area, and more closely match it. the integral itself was the limit of this process. I assume everyone else learned it that way, but retsting just in case.

2. the derivative, meanwhile, shows you the rate of change of a function.

3. so given function f with antiderivative F, why does the integral of f turn out to be related to F, and in particular, why is the indefinite integral of f equal to F (plus or minus a constant)

4. to start with, imagine integrating f using the process I describe in 1. for a given interval, what do the rectangles you divide it into actually represent? since f is d/dx(F), f shows the rate of change of F. so the area of a rectangle with height h=f(a) and width w is h*w which is the rate of change times that width. that is going to be equal to the total change across that width w, and the sum of all the rectangles (the total area under the curve) is going to be the change across the whole interval (F(b )-F(a))

- for a practical example: if F shows velocity as a function of time, then f is going to be acceleration as a function of time. using the notation above, h=f(a) is some measure of acceleration, w, the width is on the x axis, so it represents a length of time, and h*w is acceleration times time, which gives the change in velocity. so the area under the curve f along some interval reperesents the change in the value of F along the same interval (this is where the +c comes in. since this is just the change in the value, what that initial value was won't affect it. the change of the value of x^2 between 0 and 1 is the same as the change in the value of x^2+10973 along the same interval)

5. now for the indefinite integral, as near as I can tell after consulting both wikipedia and my old Calc textbook, the reason it is equal to an antiderivative is basically just down to definition. it looks like they determined that definite integrals were related to antiderivatives, and so just decide that, as a matter of definition, the indefinite integral would be an antiderivative function.

all of the above would be about 8000 times easier to describe with a whiteboard. or if I were better at describing things. that said, I think it is all correct

Edited November 16, 2015 by Dunkum

[Elbereth she/her]

Okay. I think I got at least some of it. It's confusing, but I think it makes sense. You're right that it would be so much easier with a whiteboard.

Thanks!

[Hood he/him]

Is Latex supported in this forum ?

 

It would make formulae easy to understand. 

[Creccio he/him]

Okay. I think I got at least some of it. It's confusing, but I think it makes sense. You're right that it would be so much easier with a whiteboard.

Thanks!

https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/indefinite_integrals/v/antiderivatives-and-indefinite-integrals

 

I think that works too... 3 minutes of your time

[navybrandt he/him]

Is Latex supported in this forum ?

 

It would make formulae easy to understand. 

If you like LaTeX, you should try this site for formulas for web pages/forums. http://quicklatex.com/

[Guest]

[Silverblade5 he/him]

In progress proof for a 7x7 matrix determinant. Identity is [y] Will update here periodically. Once finished, I'll repost the entire completed proof.

[A B C D E F G]

[H I J K L M N]

[O P Q R S T U]

[V W X Y Z a b]y

[c d e f g h i]

[j k l m n o p]

[q r s t u v w]

A(I(Q(Y()-Z()+a()-b())-R(X()-Z()+a()-b())+S(X()-Y()+a()-b())-T(X()-Y()+Z()-b())+U(X()-Y()+Z()-a()))

-J(P(Y()-Z()+a()-b())-R(W()-Z()+a()-b())+S(W()-Y()+A()-b())-T(W()-Y()+Z()-b())+U(W()-Y()+Z()-a()))

+K(P(X()-Z()+a()-b())-Q(W()-Z()+a()-b())+S(W()-X()+a()-b())-T(W()-X()+Z()-b())+U(W()-X()+Z()-a()))

-L(P(X()-Y()+a()-b())-Q(W()-Y()+a()-b())+R(W()-X()+a()-b())-T(W()-X()+Y()-b())+U(W()-X()+Y()-a()))

+M(P(X()-Y()+Z()-b())-Q(W()-Y()+Z()-B())+R(W()-X()+Z()-b())-S(W()-X()+Y()-b())+U(W()-X()+Y()-Z()))

-N(P(X()-Y()+Z()-a())-Q(W()-Y()+Z()-a())+R(W()-X()+Z()-a())-S(W()-X()+Y()-a())+T(W()-X()+Y()-Z()))

-B(H(Q(Y()-Z()+a()-b())-R(X()-Z()+a()-b())+S(X()-Y()+a()-b())-T(X()-Y()+Z()-b())+U(X()-Y()+Z()-a()))

-J(O(Y()-Z()+a()-b())-R(V()-Z()+a()-b())+S(V()-Y()+a()-b())-T(V()-Y()+Z()-b())+U(V()-Y()+Z()-a()))

+K(O(X()-Z()+a()-b())-Q(V()-Z()+a()-b())+S(V()-X()+a()-b())-T(V()-X()+Z()-b())+U(V()-X()+Z()-a()))

-L(O(X()-Y()+a()-b())-Q(V()-Y()+a()-b())+R(V()-X()+a()-b())-T(V()-X()+Y()-b())+U(V()-X()+Y()-a()))

+M(O(X()-Y()+Z()-b())-Q(V()-Y()+Z()-b())+R(V()-X()+Z()-b())-S(V()-X()+Y()-b())+U(V()-X()+Y()-Z()))

-N(O(X()-Y()+Z()-a())-Q(V()-Y()+Z()-a())+R(V()-X()+Z()-a())-S(V()-X()+Y()-a())+T(V()-X()+Y()-Z())))

C(H(P(Y()-Z()+a()-b())-R(W()-Z()+a()-b())+S(W()-Y()+a()-b())-T(W()-Y()+Z()-b())+U(W()-Y()+Z()-a()))

-I(O(Y()-Z()+a()-b())-R(V()-Z()+a()-b())+S(V()-Y()+a()-b())-T(V()-Y()+Z()-b())+U(V()-Y()+Z()-a()))

+K(O(W()-Z()+a()-b())-P(V()-Z()+a()-b())+S(V()-W()+a()-b())-T(V()-W()+Z()-b())+U(V()-W()+Z()-a()))

-L(O(W()-Y()+a()-b())-P(V()-Y()+a()-b())+R(V()-W()+a()-b())-T(V()-W()+Y()-b())+U(V()-W()+Y()-a()))

+M(O(W()-Y()+Z()-b())-P(V()-Y()+Z()-b())+R(V()-W()+Z()-b())-S(V()-W()+Y()-b())+U(V()-W()+Y()-Z()))

-N(O(W()-Y()+Z()-a())-P(V()-Y()+Z()-a())+R(V()-W()+Z()-a())-S(V()-W()+Y()-a())+T(V()-W()+Y()-Z()))

-D(H(P(X()-Z()+a()-b())-Q(W()-Z()+a()-b())+S(W()-X()+a()-b())-T(W()-X()+Z()-b())+U(W()-X()+Z()-a()))

-I(O(X()-Z()+a()-b())-Q(V()-Z()+a()-b())+S(V()-X()+a()-b())-T(V()-X()+Z()-b())+U(V()-X()+Z()-a()))

+J(O(W()-Z()+a()-b())-P(V()-Z()+a()-b())+S(V()-W()+a()-b())-T(V()-W()+Z()-b())+U(V()-W()+Z()-a()))

-L(O(W()-X()+a()-b())-P(V()-X()+a()-b())+Q(V()-W()+a()-b())-T(V()-W()+X()-b())+U(V()-W()+X()-a()))

+M(O(W()-X()+Z()-b())-P(V()-X()+Z()-b())+Q(V()-W()+Z()-b())-S(V()-W()+X()-b())+U(V()-W()+X()-Z()))

-N(O(W()-X()+Z()-a())-P(V()-X()+Z()-a())+Q(V()-W()+Z()-a())-S(V()-W()+X()-a())+T(V()-W()+X()-Z()))

+E(H(P(X()-Y()+a()-b())-Q(W()-Y()+a()-b())+R(W()-X()+a()-b())-T(W()-X()+Y()-)+U(W()-X()+Y()-a()))

-I(O(X()-Y()+a()-b())-Q(V()-Y()+a()-b())+R(V()-X()+a()-b())-T(V()-X()+Y()-b())+U(V()-X()+Y()-a()))

+J(O(W()-Y()+a()-b())-P(V()-Y()+a()-b())+R(V()-W()+a()-b())-T(V()-W()+Y()-b())+U(V()-W()+Y()-a()))

-K(O(W()-X()+a()-b())-P(V()-X()+a()-b())+Q(V()-W()+a()-b())-T(V()-W()+X()-b())+U(V()-W()+X()-a()))

+M(O(W()-X()+Y()-b())-P(V()-X()+Y()-b())+Q(V()-W()+Y()-b())-R(V()-W()+X()-b())+U(V()-W()+X()-Y()))

-N(O(W()-X()+Y()-a())-P(V()-X()+Y()-a())+Q(V()-W()+Y()-a())-R(V()-W()+X()-a())+T(V()-W()+X()-Y())))

-F(H(P(X()-Y()+Z()-b())-Q(W()-Y()+Z()-b())+R(W()-X()+Z()-b())-S(W()-X()+Y()-b())+U(W()-X()+Y()-Z()))

-I(O(X()-Y()+Z()-b())-Q(V()-Y()+Z()-b())+R(V()-X()+Z()-()-S(V()-X()+Y()-b())+U(V()-X()+Y()-Z()))

+J(O(W()-Y()+Z()-b())-P(V()-Y()+Z()-b())+R(V()-W()+Z()-b())-S(V()-W()+Y()-b())+U(V()-W()+Y()-Z()))

-K(O(W()-X()+Z()-b())-P(V()-X()+Z()-b())+Q(V()-W()+Z()-b())-S(V()-W()+X()-b())+U(V()-W()+X()-Z()))

+L(O(W()-X()+Y()-b())-P(V()-X()+Y()-b())+Q(V()-W()+Y()-b())-R(V()-W()+X()+b())+U(V()-W()+X()-Y()))

-N(O(W()-X()+Y()-Z())-P(V()-X()+Y()-Z())+Q(V()-W()+Y()-Z())-R(V()-W()+X()-Z())+S(V()-W()+X()-Y())))

+G(H(P(X()-Y()+Z()-a())-Q(W()-Y()+Z()-a())+R(W()-X()+Z()-a())-S(W()-X()+Y()-a())+T(W()-X()+Y()-Z()))

-I(O(X()-Y()+Z()-a())-Q(V()-Y()+Z()-a())+R(V()-X()+Z()-a())-S(V()-X()+Y()-a())+T(V()-X()+Y()-Z()))

+J(O(W()-Y()+Z()-a())-P(V()-Y()+Z()-a())+R(V()-W()+Z()-a())-S(V()-W()-Y()-a())+T(V()-W()+Y()-Z()))

-K(O(W()-X()+Z()-a())-P(V()-X()+Z()-a())+Q(V()-W()+Z()-a())-S(V()-W()+X()-a())+T(V()-W()+X()-Z()))

+L(O(W()-X()+Y()-a())-P(V()-X()+Y()-a())+Q(V()-W()+Y()-a())-R(V()-W()+X()-a())+T(V()-W()+X()-Y()))

-M(O(W()-X()+Y()-Z())-P(V()-X()+Y()-Z())+Q(V()-W()+Y()-Z())-R(V()-W()+X()-Z())+S(V()-W()+X()-Y))))

That's compressed to a 4x4 state. Will finish later.

Edited November 30, 2015 by Ookla the pony

[Chaos he/him]

I'm a college professor who teaches math. I can answer questions but don't have time to scroll through now. But after this post I'll check.

LaTeX is not supported yet but may in the future.

So we just did the Fundamental Theorem of Calculus today, which was cool. Also, for clarification, because my teacher isn't particularly good at explaining things: An antiderivative is the same thing as an indefinite integral, right? Why?

 

Two names for the same thing. We just write the integral symbol to denote "take the antiderivative." They are the same thing.

Now, the reason why you would want to use the integral symbol is because antiderivatives are related to area, specifically by the Fundamental Theorem of Calculus. So that's why we use that notation for taking an antiderivative.

 

But there has to be an explanation for it! Why does the antiderivative of a function have the same value as y times x for any given interval? I don't understand this.

That's actually not what the antiderivative of most functions are. Antiderivatives are just, hey, you had a derivative, now find which function it came from. They are derivative rules in reverse essentially.

[Oversleep]

I had two semesters of mathematical analysis. The first one was pretty simple, sequences and series, derivatives, integrals, basic stuff like that. I also had a semester of 'algebrae and analitycal geometry', loved it, scored something like... A+, I guess? (we have a much different scale) I had to take additional exam to get a mark above A. On the exam, I proved that there is no such matrix X which obeys X^T= A*X*B. A,X,B are all matrixes of nth grade, n>=2, X^T means X transponed. I was pretty proud since I was the only one to solve this particular task

The second semester of mathematical analysis was much worse, since I focused on other classes and almost failed it. Still, I don't understand 75% of what was in this course -.- Double and triple integrals, partial derivatives and such.

But why am I writing here?

Does anybody know a science article on programming/IT? I have to review it for my English class... And I am running out of time.
 

Edited November 30, 2015 by Oversleep

[Silverblade5 he/him]

Have compressed my 7x7 matrix to a 4x4 state. Will finish solving for determinant later.

[Haelbarde he/him]

On the exam, I proved that there is no such matrix X which obeys X^T= A*X*B. A,X,B are all matrixes of nth grade, n>2, X^T means X transponed. 

 

I'm not sure what's meant by matrices of nth grade, but you couldn't pick A=X^T, B=X^-1 ?

[Oversleep]

Matrix of Nth grade - NxN matrix. The question was if there are such matrixes A and B that for any possible matrix X this equation is fulfilled. Since not every matrix has its inversion, therefore I couldn't pick B=X^(-1).

[Briar King]

I just noticed this thread for the 1st time and when I read it I could have sworn it was titled Meth and science. I was like "whaaaaa?"

I sucked at math in HS and college but paid lots of attention in science.

[Silverblade5 he/him]

I just noticed this thread for the 1st time and when I read it I could have sworn it was titled Meth and science. I was like "whaaaaa?"

I sucked at math in HS and college but paid lots of attention in science.

Well this is the place to come for help.

[Chaos he/him]

There's a lot more matrix theory here than I expected. Did you know that in my graduate linear algebra class, we used matrices maybe... twice? Turns out that you don't need (or want) matrices for proving powerful results.