Why is differentiation taught before integration?

Is is taught integration before differentiation in Apostol's calculus text.

>integral of 2x is x^2
try again, retard

I hope you're complaining about the constant

He just copied that from the real retard here
So the joke's on you, brainlet

Mass market calculus books teach differentiation first because it's easier to branch into integration through the fundamental theorem of calculus

because math education is terrible and limits are too hard for most students

Why is modern atomic theory taught before quintessence models? I've been teaching myself physics in chronological order - the order it was discovered. Platonic solids and earth, air, fire, water, quintessence came before modern atomic theory.

>inb4 democritus

You think you're so smart but I bet you can't get this one. Take this integral [math] \int \frac{1}{cabin}[/math] with respect to cabin.

In order to do u-substitution or integration by parts you need to know how to take the derivative of the function.

this.

when i discovered calculus on my own i was investigating the fundamental question "there's got to be a way to analyticly determine the area under a curve, so what is it?". a streamlined version of the reasoning was "well the area under a straight line looks like a sloped line, and the area under a sloped line is parabolic divided by two, so the area under a parabolic curve is cubic and divided by three. with some experimentation i came up with the basic rule that is 1 ; x ; x^2 / 2 ; x^3 / 3 ; etc... and then i worked my way to other rules but then i learned calculus for real and the whole thing was spoiled.

but yeah only after that did i go "it looks like the slope plays a role" but shortly after that i was force fed derivatives and rules and shit so i couldn't find out for myself.

integration comes first.

there isn't a differential though
otherwise ln|cabin|+C

differentiation is much more accesible, and flows more naturally from limits at least IMO it does.

This

>when i discovered calculus on my own

why would u teach someone to walk backwards before forewards

This is a mathematician.

Everybody else is not.
And that's the difference.

x^2 -> 2x is easy to understand
x^2

Genes and Mendelian inheritance were discovered before DNA's structure or role in heredity, and yet we learn about DNA before we learn Mendelian inheritance.

Because you might as well teach about limits first, then differentiation, then only add one new method at a time

>Been teaching myself math in chronological order
Why would you assume that's the best way to do it? There's plenty of shit that was discovered "the hard way" and later became much clearer and easier to understand.

nice

Integration doesn't really needs limits though, darboux integrals and the cauchy criterion aren't limits (Riemann sums come in handy for very few problems and if you want to aproximate build the integral from the ground up in another way). It's just pointless as you will inevitably need to make a pause and start talking about limits, continuity diferenciability and all of that to reach the FTC. Also, even darboux sums are cumbersome and difficult to handle of you are not comfortable with notation, and a lot of good and not so challenging problems can be given in these topics. Spivack's structure is GOAT, but it has shit exposition. If you only care about a geometric view then it doesn't really matter.

>Integration doesn't really needs limits though
Yes it does. Otherwise you're not doing integration, you're just doing finite sums.

>darboux integrals and the cauchy criterion aren't limits
They both require limits. You would need to do an enormous amount of either hand-waving or "techinically-it's-not-a-limit"s to build those without going through limits of sequences and functions first

For the most part, calculus is taught with practical applications in mind. Integration can be taught fairly easily through the use of antiderivatives so its fairly easy to go about it like that in most calculus courses.

For less "practical" applications you might very well see integration first in say a measure theory course.

en.m.wikipedia.org/wiki/Darboux_integral
math.stackexchange.com/questions/1868908/cauchy-criterion-for-riemann-integrability
Cauchy criterion uses something analogous, but no you don't. And again the definition of Rieman integral is taken "as a limit" in the sense that if the integral exists then there is a number you can always make it closer to your Rieman sum for any partition that has a maximum interval value less than that number. This has all the the intuition of delta-epsilon, but it's not the same as you require this for all partitions which is difficult to handle. That's why Darboux integrals are better in my opinion as they let you be rigorous without a lot of mess and it's definition just needs the least upper (lower) bound principle.

>en.m.wikipedia.org/wiki/Darboux_integral
Setting that up and proving anything about it without going through limits would suck. I'm not convinced it's impossible, but it wouldn't be fun.

>the definition of Rieman integral is taken "as a limit" in the sense that if the integral exists then there is a number you can always make it closer to your Rieman sum for any partition that has a maximum interval value less than that number.
>This has all the the intuition of delta-epsilon, but...
"Techinically-it's-not-a-limit".

Have you gone through spivack? It's not really all that hard. I give you that most of the time you use cauchy criterion, which has deltas and epsilons, but everything is easier to prove if you have one class of aproxinations. I was not implying I'm against teaching differentiation first, but saying limits are the reason seems to be off point. The delta epsilon memes work in all contexts and can be explained without much complication eveb if you are not talking strictly about limits.(given that students have the basics of functions and math notation). Hell, just explain how that criterion mean that you can always find a better partition that makes your upper and lower sums better. If you are just saying the integral is a limit to use the limit properties, then your are being half assed with your rigor so there's no point in not going with a basic, intuitive understanding of a limit and just building it like that. All this rambling was because most complaints here came frome the definition of integration, but if you are not being rigorous, the most important thing is FTC. But if you want rigorous proofs then just follow spivack with better exposition.

Also the technically is not a limit is not just math autism, but important to have in mind if you want to give an early exposition of measure (like the lebegue criterion).

Yes, conceptually an integral is nice because it's a really easy way of summing up a series of terms over a continuum, but remember that 14 y/o students are not going to be thinking like a mathematician. Differentiation is easier to pick up, and also has that conceptual link to something's rate of change which plays nicely into the kinematics they're likely also learning.

>anti-derivative
That is also what my Calc Prof called it. At the time, I thought it was just
another of his many eccentricities, but I have seen it in print since then.

>I bet
there is no wagering at Veeky Forums, Grandpa

Anti-derivative is not the same as integration fyi

integration in how many variables?

The theory of integration is deeper, broader, and harder. The fundamental theorem of calculus gives us a very convenient way to integrate some functions via computation of the primitive. In general this is not always possible. However, the cases where it is not possible are typically out of the scope of the classes most people need to take, so it makes sense to use FTC as the primary method of integration since it is easy: after learning to differentiate, once you learn FTC, you've already learned most of the theory of integration you will ever need.

>Cauchy criterion isn't a limit
What? A sequence of real numbers converges (i.e., has a limit) iff it is Cauchy. Defining anything in terms of Cauchy sequences is defining it in terms of limits.