/sci/ - Science & Math

You'll witness the limit of your abilities once you see the midterm grade

Start off by writing down every formula and get a vague idea of what they're used for and scribble it down
Then do old midterms or tests related to the chapters you need for the midterm

>>11444471
limitpilled

>>11444465
A limit is the formal definition of when something can be approximated as well as desired.

>>11444465
first thing they tell you is that x^2/x does not equal x
second thing they tell you is that x^2/x = x if you do some limit thing
so it cancels out, you'll do fine

>>11444584
What are you talking about x^2/x = x for all real numbers without needing to introduce limits

If I remember they usually hand wave limits along the line of "the limit is the value you get really close to as you get really close to a given point on a function" then they give a confusing epsilon delta definition for derivatives then the limit definition and after making you do tedious calculations with the they finally tell you theres a way faster way to take the derivatives of some functions and off you go

It isn't until you take a class in analysis that you rigorously go over what a limit is

>>11444465
Calc 1 is easy, you should be able to solve this midterm without any trouble.

Now, when you get to Calc II you might need to start showing up to class.

>>11444465
Did you not learn basic calculus and limit in high school?

>>11444465
Bump.
So how'd it go OP?

>>11445683
I'm also curious

>>11444675
The absolute state of calculus I

>>11444694
obviously he didn't you moron.

>>11444675
So 0 isn't a real number?

>>11444465
I have a feeling I will never get limit. Like the number keeps getting bigger and it reaches infinity, but at what point does that happen?

>>11445840
That's unfortunate

US education fucking sucks.
Imagine if math majors could just take math classes, spending time studying would actually be justified. Undergrads in the US either don't get A's, study just enough to get A's or study just enough to get A's and then waste all their time on contest problems. Now imagine math majors actually spent enough time that they could do every problem, and prove every theorem in a couple of calculus books. Now imagine when those students start teaching calculus. Lower division math classes will stop looking like such a shit show.
The problem with the above is that lower division classes are designed for engineers. As they fund the endeavor.

>>11444465
>What is a limit?
yours is calc 1 apparently

Can someone post the kid gloating about how he's an advanced calc 1 student and calls his professor by Dr. out of respect copypasta?

>>11445952
The worst part is when engineers try to do a math minor and they harass the prof with questions like "so what are the physical applications of this?" Had one of those fucks in my first group theory class, they finally dropped after failing 2 midterms. Engineers are "special".

>>11446049
The worst part is in a lower division course where the entire class is engineers or cs the lectures
1) Assume you cannot read
2) Often contain mistakes
3) Autistic engineer asking retarded questions every fucking day. "Is that kinda like this movie where the earths magnetic field disappears"
and the teacher doesn't just tell him to shut the fuck up.

>>11444465
Just pretend you're sick, reschedule the midterm, and bang out Khan academy videos for the next few days

The main point is a little buried in a modern treatment. The point is that it is consistent to imagine little itty-bitty numbers, infinitesimals, adjoined to your conception of the real numbers, and these infinitesimals contain the idea of limit and asymptotics. So for example:

[math] (3 + dx)^2 = 9 + 6dx [/math]

where dx is an infinitesimal, so I dropped the dx^2, because the square of an infinitesimal is twice more infinitesimal than the infinitesimal and can be ignored. By definition, then, 6 is the derivative of squaring at 3. That means that

[math] 3.001^2 = 9.006 [/math]

up to certain negligible corrections. You can use this for party tricks:

[math] (1 + dx)^n = 1 + n dx [/math]

so that

[math] \sqrt{ 1.01} = 1.005 [/math]

You can use this to do arithmetic well, after you internalize the idea. You can also do calculations with trigonometry. Once you know enough, you see that

[math] \sin{dx} = dx [/math]

for infinitesimal dx (in radians) so that

[math] \sin{10 ^{\circ}} = 10 * 2\pi/ 360 [/math]

to a good approximation, because 10 degrees is small. It allows you to approximate quickly.

This infinitesimal idea is due to Cavalieri, it was developed by Leibnitz (Newton always thought in terms of limits), and it was given it's permanent final form inside modern mathematical logic by Abraham Robinson, after a century of suppression. It's a very exciting idea, it really is one of the greatest ideas humanity ever had.

The next idea is that these infintesimals capture the notion of velocity. So that [eqn] x(t +dt) = x(t) + v(t) dt [/eqn] The velocity of the velocity is the acceleration: [eqn] v(t+dt) = v(t) + a(t) dt [/eqn] When [math] dt [/math] is infinitesimal, that's calculus. When [math] dt [/math] is [math] .001 [/math], that's what you do on your computer to simulate physics. You can do it, because [math] a(t) [/math] is known from Newton's law [math] a = F/m [/math] and F is given as a function of the position. That means, knowing [math]x[/math] and [math]v[/math], you can calculate [math]a[/math], and then update [math]x[/math] and [math] v [/math] at the next [math] dt [/math].

This "closes" the system of equations, it allows you to simulate the motion. This was understood already by Newton, but the clear statement everyone remembers is by Lagrange.

The next idea is that infinite power-series converge in series to a class of functions of high importance, so that you have infinite series of successive corrections when [math] dt [/math] is not infinitesimal. [eqn] x(t+dt) = x(t) + v(t) dt + 1/2 a(t) dt^2 +... [/eqn] when [math] dt [/math] is not infinitesimal, there are all these orders. It allows you to indentify certain functions as infinite polynomials, and treat them as polynomials. This idea is due to Newton, it was greatly developed by Euler, and it was made stick by Cauchy and others in the 19th century, in the development of complex analysis and analytic function theory.

The next idea is that areas and derivatives are related. If you look at the area under a curve from [math] 0 [/math] to [math] x [/math]: [math] A(x) [/math], then [math] A(x + dx) = A(x) + f(x) dx [/math] (you can see this by drawing rectangles), and therefore [math] f(x) [/math] is the derivative of [math] A(x) [/math]. This allows you to give a systematic calculus for areas. This theorem is due to Isaac Barrow, Newton's advisor. It was what led Newton and Leibnitz both to run with the idea.

The next idea is that of differential equations: you can express algorithms with steps which are infinitesimals as equations. For example, if you write down:[eqn] df = f(x) dx [/eqn] Where [math] df [/math] means [math] f(x+dx) - f(x) [/math], then you can compute [math] f [/math] given an initial value. This allows you to speak about algorithms--- a differential equation plue a little stepsize defines an algorithm to compute [math] f [/math], and if you iterate it, you do physics. This idea was developed by Newton, Euler, a million people each focusing on a different differential equation, and today there is an industry for understanding these equations.
The next idea is of partial derivatives, that if you have a function of several variables:[eqn] F(x + dx,y +dy) = F(x,y) + F_x dx + F_y dy [/eqn]One set of ideas here are the Legendre transform, swapping out [math] y [/math] for [math] F_y [/math], which is ultimately explained by statistics and Gaussian integrals.

Then there is the idea of vector spaces, and linear tangent spaces, and differential geometry, which leads to General Relativity.

In another generalization, these linear spaces extend to infinite linear spaces, the Taylor polynomial series can be swapped out for better behaved Fourier series and other polynomial series, like those of Tschebycheff, the function classes expand to include random walks, and non-smooth monsters that are convergent in the 19th century, the notion of integration becomes universal in the 20th century due to Lebesgue Cohen and Solovay. And you are in the modern world.

>>11444465
Rip
Hope your schools lets you take a retake.

>>11444465
>>11447608
>>11447626
>>11447630
Each of these topics I mentioned above deserves at least a month or two of serious study, and they all intellectually begin either with Newton doing differential equations and power series, or with Leibnitz doing infinitesimals. This is what gave birth to modern mathematics. The development can be seen as the point of calculus. There are extensions of the idea that were worked out recently. Ito calculus describes the motion of random walks, and it is related to the Feynman path integral, which describes integration over spaces of paths. The main idea here is renormalization, which is the taking of infinitesimal limits inside Feynman path integrals--- these ideas are being worked out today, they were worked out internally to physics in the 1970s, but they need to turn into rigorous mathematics very badly.

>>11447634
I hope you got a good review out of this for yourself, as this dumby dumb doesn't give two fucks clearly or he would have learned it the first time.

>>11444675
the key number is 0 bruv, understand that cae and you understand calc 1 limits

>>11447662
case*

>>11447634
quality shitpost to mention this stuff to a calc 1 that has never been to class

>>11445952
As an engineering student back in uni I was desperately trying to find a good teacher who could actually teach me the concepts and uses with the problems. Unfortunately all the teachers (women as well) were only motivated to say this is concept, this is how it looks like written down, read up on it in the uni issued book (which didn't explain shit).
Also the problems in the uni issued books were substantially underdesigned for studying for the mid terms and finals, you could be sure that none of the type of problems will show up in the exam. Half of the exam problems we never talked or read about in class and books.
It wasn't fun for a freshman who could only rely on the uni issued books. Not an english speaking country, and internet was not very widespread at the time.
Teaching math in uni is fundamentally flawed, they are using it as a filter instead of employing proper teaching methods and use it to make better engineers and also being a filter on its own, proper one this time.

>>11444465
limit
[ˈlimit]
NOUN

a point or level beyond which something does not or may not extend or pass.
"the limits of presidential power" · [more]
a restriction on the size or amount of something permissible or possible.
"an age limit" · [more]
synonyms:
maximum · ceiling · limitation · upper limit · restriction · curb · check · control · curtailment · restraint · damper · brake · rein

>>11444690
>Now, when you get to Calc II you might need to start showing up to class.
I'm in Calc II. It's not nearly the monster I had been led to believe. Actually, it's hardly any different than Calc I. "oooh, sequences and series, so scary."

>>11448077
For me the study material was easy as well. The hard part in passing is where most of the exam problems were types we never did in lecture and practice and the practice book didn't covered them either.
Full on fuck you from the maths department. Also using earlier exams as practice was practically illegal, you could get dropped out if someone found one on you.

>>11447957
this, but this is me but in high school, I didnt really understand the very basics and concept about calculus, teachers just gave dry theory explanation from some textbook, in the end I just memorized the equations and most common questions or problems usually asked at school, and managed to pass final exam with average grade, then next year at uni I had no choice, either I got it or I drop out, literally spend two weeks going through every different calculus explanation I could find on the internet (books, sites, comments on forums etc)

fuck standardized education

>>11444675
lol

>>11447608
6 is the derivative of squaring at 3? What the fuck does that mean? Is this whole string of posts an epic shitpost or is it actually valid? I want to know before I try internalizing it.

>>11444465

is this it ? is this the EXTENT OF YOUR POWER?

ace that fucking exam and show this >>11444471
anon what you are capable of

>>11447608
This is actually making me angry, it's like fucking junk data that almost makes sense but is meaningless, like listening to that Italian song that imitates English

>>11448215
>>11448220
>6 is the derivative of squaring at 3? What the fuck does that mean?
Expand (x+y)^2 to get x^2 + 2xy + y^2 now replace x and y with 3 and dx like in the example: 3^2 + 6dx + dx^2
Now observe:
>because the square of an infinitesimal is twice more infinitesimal than the infinitesimal and can be ignored
Leaving us with: 3^2 + 6dx
We now define "derivative" as the "6dx" part, meaning 3 + "some dx" (like 0.001) will equal 6 times that dx (so 6*0.001 = 0.006).
That means that 3.001^2 = 9.006 (ish). Get it?

>>11448215
>>11448220
>6 is the derivative of squaring at 3? What the fuck does that mean?
Expand (x+y)^2 to get x^2 + 2xy + y^2 now replace x and y with 3 and dx like in the example: 3^2 + 6dx + dx^2
Now observe:
>because the square of an infinitesimal is twice more infinitesimal than the infinitesimal and can be ignored
Leaving us with: 3^2 + 6dx
We now define "derivative" as the "6dx" part, meaning 3 + "some dx" (like 0.001) squared will equal 3^2 plus 6 times that dx (so 6*0.001 = 0.006).
That means that 3.001^2 = 9 + 0.006 = 9.006 (ish). Get it?

>>11448231
No. I was with it until
>3 + "some dx" (like 0.001) will equal 6

Where is that coming from?

>>11448243
I edited that a bit for clarity, see >>11448239

>>11448243
Also, "dx" means "some small change", such as 0.001, just so you know.

>>11448252
Oh shid, I think I understand that now. I understood the concept of a differential, but they were never really used "arithmetically" and I always wanted to build my intuition with freely manipulating them. I still got an A in calc I and II, currently taking DiffEqs. Other places say that it makes no sense to interpret a differential outside of the context of integration or an infinitesimal ratio (derivative?)

I will continue trying to assimilate what is written

>>11448215
>Is this whole string of posts an epic shitpost or is it actually valid?
I can tell you it's valid, but you'll never know if I'm lying until you verify it yourself. That's the hard truth. Always question what you're told. You have to check it yourself to be certain.

>>11448268
That is a bit of a misguided statement. I was sitting there and trying to comprehend it. All I wanted was confirmation that my efforts were not pointless. Someone merely telling me that it's valid would be useless to me. The whole reason for me asking was because I wanted to internalize it into my own knowledge.

>>11447608
>>11447626
>>11448257
I require elaboration on most of what is being stated. There is a great deal being assumed.

I do not understand how this conclusion was reached:\sin{10 ^{\circ}} = 10 * 2\pi/ 360

Next, how can you say that these are equal?:
x(t +dt) = x(t) + v(t) dt

>>11448301
whoops forgot math tag lol

>>11447626
fucking physics majors
if anyone tries to introduce someone to calculus with anything besides limits, they should be slapped in the nuts

>>11444465
Guys im in the same situation and Im freaking out, please help me!!!!!

>>11448320
If you unironically believe Calc I is hard, you should just quit (being alive) while you're behind

>>11444499
look at those digits!

>>11448301
>I do not understand how this conclusion was reached:\sin{10 ^{\circ}} = 10 * 2\pi/ 360
First notice that in [math]\color{#781b86}{\tt{r}}\color{#3e5dcf}{\tt{a}}\color{#7ab77a}{\tt{d}}\color{#b1bd4d}{\tt{i}}\color{#b1bd4d}{\tt{a}}\color{#dea63b}{\tt{n}}\color{#da2121}{\tt{s}}[/math] [math] \sin{dx} = dx [/math] for infinitesimal [math] dx ~ [/math]("numbers very close to the value of x when the value of x is zero")
Once you've understood that part, we can use the formula for any [math] dx [/math] values that reasonably satisfy our definition: "values very close to the value of x when the value of x is zero"
Since our formula is for radians, and because there are [math] 2\pi [/math] radians for every [math] 360 [/math] degrees, we can still use degrees as [math] dx [/math] values as long as the equivalent value in radians satisfy our definition: "values very close to the value of x when the value of x is zero" We expect that [math] 10 * 2\pi/ 360 [/math] will be very close to the value of zero, so we can apply the formula to approximate its value, we just have to make sure our [math] dx [/math] value is expressed in radians: [math] 10 * 2\pi/ 360 [/math] and so we use [math] \sin{10 ^{\circ}} = 10 * 2\pi/ 360 [/math]
>Next, how can you say that these are equal?: x(t +dt) = x(t) + v(t) dt
This is the same idea expressed in the first example >>11448239

Consider this problem from a typical IQ test: [eqn] \text{2 5 10 17 26} ? [/eqn] What's the next number you expect in the sequence (this is not hard, you should do it). The n-th term in the sequence is given by: [eqn] n^2 + 1 [/eqn] as you can see by substituting [math] n=1,2,3,4,5, [/math] so the next term is [math] 37 [/math]. But if you did the problem, you probably noticed first that the differences are: [eqn] \text{5-2 = 3 10-5 = 5 17-10 = 7 26-17 = 9} [/eqn] and then filled in [math] 37 [/math] by adding [math] 11 [/math] to [math] 26 [/math]. This thing you did above, of finding the difference between successive terms, is called "taking the first difference", and given any sequence of numbers [math] A_n [/math], the derived sequence [eqn] \Delta A_n = A_{n+1} - A_{n} [/eqn] From the definition, you can check
[eqn] \Delta 1 = 0 [/eqn]
[eqn] \Delta n = 1 [/eqn]
[eqn] \Delta n^2 = 2n+1 [/eqn]
[eqn] \Delta n^3 = 3n^2+3n+1 [/eqn]
[eqn] \Delta n^4 = 4n^3 + 6n^2 + 4n + 1 [/eqn]
[eqn] \Delta 2^n = 2^n [/eqn]
[eqn] \Delta {1\over n} = - {1\over n(n+1)} [/eqn]
and you can prove the general properties
[eqn] A_n + \Delta A_n = A_{n+1} [/eqn]
[eqn] \Delta (A + B) = \Delta A + \Delta B [/eqn]
[eqn] \Delta cA = c \Delta A [/eqn]
This says that [math] \Delta [/math] is a linear operator. Further, you have a product rule
[eqn] \Delta (AB) = A \Delta B + B \Delta A + \Delta A\Delta B [/eqn]
[eqn] \Delta (AB)_n = A_{n+1} \Delta B_n + B_n \Delta A_n [/eqn]
So now you can see that
[eqn] \Delta (n^2 - n) = (2n+1 - 1) = 2n [/eqn]
[eqn] \Delta (n^2 2^n) = (2n+1) 2^{n+1} + n^2 2^n = (n^2 + 4n + 2) 2^n [/eqn]
And so on. It is good practice toward calculus to find the derived sequence of all common functions. This was done by early modern mathematicians, and this calculus of finite differences directly inspired calculus.

>>11444465
>>11444675
>>11445885
>>11446049
>>11446081
>>11447675
>>11447957
>>11448057
>>11448077
>>11448215
>>11448220
>>11448243
>>11448257
>>11448309
>>11448320
Consider this problem from a typical IQ test: [eqn] \text{2 5 10 17 26} ? [/eqn] What's the next number you expect in the sequence (this is not hard, you should do it). The n-th term in the sequence is given by: [eqn] n^2 + 1 [/eqn] as you can see by substituting [math] n=1,2,3,4,5, [/math] so the next term is [math] 37 [/math]. But if you did the problem, you probably noticed first that the differences are: [eqn] \text{5-2 = 3 10-5 = 5 17-10 = 7 26-17 = 9} [/eqn] and then filled in [math] 37 [/math] by adding [math] 11 [/math] to [math] 26 [/math]. This thing you did above, of finding the difference between successive terms, is called "taking the first difference", and given any sequence of numbers [math] A_n [/math], the derived sequence [eqn] \Delta A_n = A_{n+1} - A_{n} [/eqn] From the definition, you can check[eqn] \Delta 1 = 0 [/eqn]
[eqn] \Delta n = 1[/eqn]
[eqn] \Delta n^2 = 2n+1[/eqn]
[eqn] \Delta n^3 = 3n^2+3n+1 [/eqn]
[eqn] \Delta n^4 = 4n^3 + 6n^2 + 4n + 1 [/eqn]
[eqn] \Delta 2^n = 2^n [/eqn]
[eqn] \Delta {1\over n} = - {1\over n(n+1)} [/eqn]
and you can prove the general properties[eqn] A_n + \Delta A_n = A_{n+1} [/eqn]
[eqn] \Delta (A + B) = \Delta A + \Delta B [/eqn]
[eqn] \Delta cA = c \Delta A [/eqn]
This says that [math] \Delta [/math] is a linear operator. Further, you have a product rule
[eqn] \Delta (AB) = A \Delta B + B \Delta A + \Delta A\Delta B [/eqn]
[eqn] \Delta (AB)_n = A_{n+1} \Delta B_n + B_n \Delta A_n [/eqn]
So now you can see that
[eqn] \Delta (n^2 - n) = (2n+1 - 1) = 2n [/eqn]
[eqn] \Delta (n^2 2^n) = (2n+1) 2^{n+1} + n^2 2^n = (n^2 + 4n + 2) 2^n [/eqn]
And so on. It is good practice toward calculus to find the derived sequence of all common functions. This was done by early modern mathematicians, and this calculus of finite differences directly inspired calculus.

The main identity is the fundamental theorem of derived sequences--- the sum of the derived sequence is found from the original sequence. For example, 3+5+7+9+11 = 37 - 2 (because adding the differences steps up the sequence). Convince yourself that it is true (or prove it by induction). So that
[eqn] \sum_{k=a}^{b} \Delta A_k = A_{b+1} - A_a [/eqn]
This is a remarkable formula, because now you learn a summation formula from each difference formula above:
[eqn] \sum_{n=a}^b (2n+1) = (b+1)^2 - a^2 [/eqn]
[eqn] \sum_{n=a}^b (2n) = 2 \sum_{k=a}^b n = b(b+1) - a(a-1) [/eqn]
[eqn] \sum_{n=a}^b 2^n = 2^{b+1} - 2^a [/eqn]
The second difference is defined as the difference of the difference:
[eqn] \Delta^2 A_n = \Delta \Delta A_n = (A_{n+2} -2 A_{n+1} +A_n) [/eqn]
So that
[eqn] \Delta A_n + \Delta^2 A_n = \Delta A_{n+1} [/eqn]
and this says
[eqn] A_{n+2} = A_n + \Delta A_n + \Delta A_{n+1} = A_n + 2\Delta A_n + \Delta^2 A_n [/eqn]
and so on for third differences etc. You can prove that if two sequences have all of
[eqn] A_0, \Delta A_0, \Delta^2A_0, \Delta^3 A_0 , ... [/eqn]
equal, then the two sequences are equal, since the only way for the n-th differences to agree is if the first n terms are equal.

There is a nice quantity you can define:[eqn] n^{(k)} = n(n-1)(n-2)...(n-k+1) [/eqn]and for completeness, [math] n^{(0)}=1 [/math] The factorial [math] n!=n^{(n)} [/math] by definition. This alternate definition of raising to a power has the property that[eqn] \Delta n^{(k)} = k n^{(k-1)} [/eqn]And in terms of this quantity, there is a formal expression for the n-th term of any sequence
[eqn] A_n = A_0 + \Delta A_0 n + \Delta^2 A_0 {n^{(2)}\over 2!} + \Delta^3 A_0 {n^{(4)}\over 4!} + ... [/eqn]
and this gives an explicit polynomial expression whose first n differences at 0 coincide with those of the sequence A. This allows you to fit a polynomial to any evenly spaced points easily.
The above looks like an infinite sum, but on an integer position, only finitely many terms are nonzero. If it is convergent as an infinite series, you might expect it to interpolate good non-integer values for a reasonably well behaved sequence.
This is called the Gregory series, and it was developed by Gregory in the early half of the 17th century. Gregory used this to give infinite polynomial series expansions for common trigonometric functions, including the arc-tangent. This stuff seems like a bag of formal tricks that is not particularly more insightful than what you can see just by piddling around with intuition. Still, it allows you to quickly prove all the annoying sum identities you learn in high-school.

https://www.mathsisfun.com/calculus/
OP and whoever struggles with basic calculus I suggest to read this site, I have not found a simpler and easier to understand explanations anywhere else, also used this teach it to my niece.

>>11448454
>>11448455
>>11448458
This is why nobody likes mathfags. This won't help a 105 IQ guy who's going to fail Calc 1. He needs to not lim(x ->1)(f(x)) = f(1) or if incalculable, f(.99999999999) AND f(1.000000000000001) and if they don't match, limit does not exist. Boom, limits explained without faggot deltas that have more vacuous and slippery of a meaning than the word "intelligence"

>>11445885
It doesn't. The point of the limit is that it never reaches the value it's approaching. That's why you say "as x approaches" not "when x equals".

Consider now a sequence defined not at the points [math] 1,2,3,..., [/math] but on a very fine grid of points spaced [math] \epsilon [/math] apart, so that the n-th point is at position [math] n\epsilon [/math] All the ideas of the previous section transfer to this situation, since all you have to do is rescale everything to make the unit of length [math] \epsilon [/math] and the points lie on top of the integers.
In this case, the sequence [math] A(n) [/math] turns into a function [math] A(x) [/math] defined on all x's on the lattice. The derived sequence is [eqn] \Delta_\epsilon A = A(x+\epsilon) - A(x) [/eqn] And you can see that as [math] \epsilon [/math] goes to zero, it goes to zero. For a typical function, like multiplication or raising 2 to a power, we can ask, how does it go to zero? [eqn] \Delta x^2 = (x+\epsilon)^2 - x^2 = 2x \epsilon + \epsilon^2 [/eqn] This is just the rescaled version of the first difference of [math] n^2 [/math] (you can work it out directly, and it is good if you do). The lesson is that the thing tends to vanish linearly, meaning that if [math] \epsilon [/math] is small, and it is made twice as small, your derived sequence is generally made about twice as small.
So you can take out this scaling and define the derivative of f [eqn] {df\over dx} = {\Delta f \over \epsilon} = {\Delta f\over \Delta x} [/eqn] Where the idea here is that you define the derivative for each [math] \epsilon [/math], and let [math] \epsilon [/math] become so miniscule that the derivative stops changing. This never happens for any finite nonzero positive value of [math] \epsilon [/math], so it is formally useful to introduce the concept of an infinitesimal [math] \epsilon [/math]

An infinitesimal [math] \epsilon [/math] is an [math] \epsilon [/math] that is so small, that it behaves as if it were zero for the purpose of order of magnitude comparison, but it is not yet zero. It can be formally defined as a procedure: given any quantity you can calculate with [math] \epsilon [/math], the quantity for infinitesimal [math] \epsilon [/math] is defined as the limiting value a[math] \epsilon [/math] gets smaller of the finite quantity.
I will call the limiting infinitesimal [math] dx [/math], as in "the difference between successive allowed values of x". It is important not to read this as "d times x", but as a rounder version of [math] \Delta A [/math], which is not [math] \Delta [/math] times [math] A [/math], but [math] \Delta [/math] of [math] A [/math]. Then the derivative can be calculated from the finite differences: [eqn] { d x^2 \over dx} = 2x + dx = 2x [/eqn] Where I have thrown away the infinitesimal term. Likewise the analog of [math] x^{(k)} [/math] is [eqn] x^(k) = x(x-dx)(x-2dx)...(x-ndx) = x^k [/eqn] So that [eqn] {d\over dx} x^k = k x^{k-1} [/eqn] Further, [eqn] {d\over dx} {1\over x} = - {1\over x^2} [/eqn] Since the small lattice analog of [math] {1\over x(x+1)} [/math] is [math] 1\over x(x+dx) [/math].
The derivative of a function f is also called f'. There is a notion of a second derivative, derived from the second difference--- it is the derivative of the derivative. On a lattice: [eqn] f''(x) = { f(x+\epsilon) - 2f(x) + f(x-\epsilon)\over \epsilon^2} [/eqn]

In the limit as [math] \epsilon [/math] goes to zero. This is the formula for the second difference, but now divided by [math] \epsilon^2 [/math] as required from the typical way the second difference vanishes. The difference vanishes as the first power of [math] \epsilon [/math], and if you were to divide out by [math] \epsilon [/math], you would get something constant, and the difference of this thing vanishes as [math] \epsilon [/math]. So the second difference goes to zero as the second power of [math] \epsilon [/math].
You can define third derivatives, and so on. A physicist generally has to be familiar with these discrete forms of the second derivative, since there are many cases, like atomic lattices in a solid, where there is a real, actual [math] \epsilon [/math], and you are only dealing with an approximate continuum. It is likely that every notion of spatial continuum is related to a limiting quantity which in our universe is large, but finite.
The properties of calculus of finite differences translate to derivatives very simply:
[eqn] {d\over dx}(f+g) = f' + g' [/eqn]
[eqn] {d\over dx} (fg) = f'g + fg' [/eqn]
Anyway, going to infinitesimal calculus gives you a few new things: 1. The formulas simplify somewhat, since you are only interested in asymptotics. 2. The derivative gives a meaning to the notion of "how far do you go in an infinitesimal amount of time", and this defines the notion of velocity at a given time. 3. The derivative obeys the chain rule.
The chain rule is a rule for composite functions, f(g(x)). In the discrete case, you couldn't do anything regarding this, because there is no relation between f(g(n+1)) and f(g(n)) that is simple regarding f and g, since the steps g takes might be large. You can write this as [eqn] f(g(n+1)) = f(g(n) + \Delta g) [/eqn] but now you are stuck, since [math] \Delta g [/math] is not necessarily an integer.

>>11448541

>>11448552
>>11448549

Also: check'd and mathfags BTFO >>11448555

But for infinitesimal lattices, [math] \Delta g [/math] is still infinitesimal, and this problem vanishes. We know that [math] \Delta g [/math] is small, so [eqn] f(g(x+\epsilon)) = f(g(x) + g'(x)\epsilon) = f(g(x)) + f'(g(x))g'(x)\epsilon [/eqn] so you learn the derivative of composite functions. From this, you learn [eqn] {1\over x^n} = - {1\over (x^n)^2} n x^{n-1} = -{n\over x^{n+1}} [/eqn]
This derivative fits the same pattern as positive powers, except plugging in a negative number in the exponent. From the chain rule, you have the following theorem. If f(x) and g(x) are inverse functions, then:
[eqn] f(g(x)) = x [/eqn] Differentiating both sides: [eqn] f'(g(x))g'(x) =1 [/eqn] so the derivative of the inverse function g is determined by the derivative of f at the location of g: [eqn] g'(x) = {1\over f'(g(x))} [/eqn]
Using this formula for [math] f(x) = x^2 [/math], you learn that [eqn] {d\over dx} \sqrt{x} = {1\over 2\sqrt{x}} [/eqn] again, the same pattern [math] k x^{k-1} [/math], except now with half-integer powers! You can now prove this in general by using inverse functions for 1/n and continuity.
The summation theorem becomes more breathtaking: [eqn] \int_a^b f'(x) dx = f(b) - f(a) [/eqn] Where the integral simply means the sum of all values of f on the lattice, multiplied by the lattice spacing. [eqn] \int f(x) dx = \sum_x f(x) \epsilon [/eqn] Where the sum is over x in the interval a,b in steps of [math] \epsilon [/math] beginning at a. This has the interpretation on the graph of f as the area under the curve of f.
Further, for a general function, you expect that if [eqn] f(0) = g(0) [/eqn] [eqn] f'(0) = g'(0) [/eqn] [eqn] f''(0)=g''(0) [/eqn] and so on, you will have f(x)=g(x). This is not true, but it is true for a class of functions of high importance, which are called "analytic functions".

The analytic functions obey the analog of the Gregory series: [eqn] f(x) = f(0) + f'(0)x + f''(0) {x^2\over 2} + f'''(0) { x^3\over 3!} … [/eqn] Which is usually called a Taylor series, but was already known to Newton and contemporaries (who were familiar with Gregory series already).
So you see that the calculus is simply a method of defining a limiting calculation method for finite differences where all the arbitrariness and ugliness of the finite differences go away. It is essential for motion, because it tells you what "velocity" means at any one time. It is essential for physics, because it describes how quantities change continuously, the same way that the finite difference business describes how quantities change discretely.

For a good book, I would recommend Lang's calculus, although it is good to learn everything that appears in every book (there isn't that much).

>>11444465
Too many gay niggers trying to show off their math prowess here, however they dont have the social intelligence required to explain what they know at different levels. lim x->0 of (1/x). 1/0 does not exist, it doesnt equal anything, its undefined. So, instead, we find the limit above by making x APPROACH 0. Set x=1 then solve 1/x, then x=0.5, then 0.1, 0.01, and if the value of 1/x continues to approach some value, that is the limit (in this case, the limit is infinity). Keep in mind you have to also check the limit as it approaches from the other direction (so x= -1, -0.5, -0.1, -0.01). If this limit is the same as the other one, then you have your limit. However, in this case, this limit gives us negative infinity, whereas the other direction was positive infinity, so this limit does not exist.

>>11448552
haha, just give simple example, like the braking distance or acceleration of a car or something

>>11448568
>Too many gay niggers trying to show off their math prowess here, however they dont have the social intelligence required to explain what they know at different levels.
Ironically, all the concepts explained ITT are presented at the 1st-grader reading level and built up from 1st-grader math.

>>11448594
who learns math just by reading? I need graphs and diagrams and visual representations and explanations

>>11448601
words are visual representations of explanations

>>11448594
>partial derivatives, integrals, and sets are 1st grade
Nobody is asking you to write a thesis, OP hasnt attended a single lecture, explain it in terms a retard such as himself can understand

>>11448630
don't you mean text?

>>11448634
>explain it in terms a retard such as himself can understand
this is literally what happened, but you haven't read the posts so shut the fuck up retard

>>11448634
>explain it in terms a retard such as himself can understand
shut the fuck up retard, this is literally what happened in those posts you didn't read

it's been 4 hours, how did you do?

>>11448661
he's not coming back to this thread. in particular he sure as shit isn't reading >>11448454 >>11448455 >>11448458 >>11448541 >>11448549 >>11448552 >>11448558 >>11448563

>>11448454
>>11448455
>>11448458
Fascinating explanation, thanks for that bro (not op btw)

>>11444675
The limit argument you provide by dividing works only for every nonzero real number. Now, if asked to evaluate as x goes to infinity, we find we get asymptotic linear growth

sky IS the limit

>>11448840
Not if you have a rocket

>>11448674
How do you know these things?

>>11448077
The material isn't really anything especially crazy, it's just that the class tends to be structured in such a way as to be more demanding. It's usually the last math class that non-STEM majors are required to take so universities make it into a sort of "filter class".

>>11448750
It really is fascinating. Such a shame the modern intro calc treatment isn't presented this way.

>>11448215
if your the midterm guy try to ignore his post, everything he said is exactly how it is but its written in a way for math literate people and will only confuse you. There is a reason your babbi calc course isn't taught this way but higher up math classes are.

>>11450196
>its written in a way for math literate people and will only confuse you
NO!!!! IT'S THE EXACT OPPOSITE OF THAT!! Oh my GOD! Are you people THIS CLUELESS?? What the FUCK is WRONG WITH YOU!

>>11450247
Are you the person who wrote those posts, by any chance?

>>11444465
annoying orange

>>11450247
If you try teaching babbi calc 1 students like this you will get record number of drop outs. I know because I've tried it.

>>11453279
>If you try teaching babbi calc 1 students like this you will get record number of drop outs. I know because I've tried it.
This is genuinely the most intuitive, straightforward, enlightening introduction to calculus that I've ever seen. How could you possibly present the material any better than this?

>>11453289
Maybe that guy's thinking of the just-for-you retard calculus course that non-scientist non-engineers take. At my school there's one for management and social studies majors. God knows what they do differently.

>>11447608
>>11447626
>>11447630
This is incredibly based.
God bless anon.

ITT: Brainlets get triggered because they can't understand basic math

>>11448309
>fucking physics majors
Anon that isn't a physics major, also you should consider dropping out.

>>11444465

When your character in a final fantasy game takes damage it affects their 'limit' bar. When it maxes out you can perform a 'limit' break which deals a greater amount of damage than a normal attack. Limit breaks are especially useful when fighting tough monster encounters or in a boss fight. I hope that helps and good luck in your exam.

>>11448220

same

you should have gone to class you fucking retard, in the meantime cram like you have never fucking crammed before

>>11444465
18 years

>>11444465
Read the sticky burn boy

>>11448454
>>11448455
>>11448458

Good post.
They should scrap high school calculus and replace it with this.
Then we can teach analysis 3 times.

>>11448531
You view of the world is tiny. Infinitesimal even. When you were in math class, have you ever thought. If you can have cold hard logic vs some box tell you what is true, you choose the box. Your will is weak, and so is your soul.

>>11453279
>If you try teach calculus as built from the ground up using only simple mathematical arguments, that the students are all familar with, you get record numbers of dropouts, versus just handing them "formulas" from on high and seeing if they can use them to pass a test.
WEW LADDY GOD FORBID SOMEONE BE ASKED TO THINK IN A CALCULUS CLASS ITS ONLY LIKE 6 MONTHS

>>11448531
>>11453279
Why are ye fearful, O ye of little faith?