Integral and Differential Calculus

[Bogrune]

If I'm lucky enough to pass Precalculus this semester, I'll be taking Calculus 1 next semester. Just out of curiosity, what is the difference between differential and integral calculus? Also, how does the two of them relate to upper-level mathematics (e.i: Number Theory, Vector Calculus, Real and Complex Analysis)?

 

[paulfr]

Oversimplified a bit but ...

Differentiation is finding Slope
Integration is Multiplication finding Area or Volume

And they are inverse functions.
Do one and then the other and you are back where you started

 

[Mark44]

Oversimplified a bit but ...

Differentiation is finding Slope
Integration is Multiplication finding Area or Volume

It's more like addition.

And they are inverse functions.

"Functions" isn't the right term, since each of these operations takes a function as an input, rather than a number.

Do one and then the other and you are back where you started

 

[micromass]

Hi Bogrune

Integral and differential calculus are two quite different fields that are quite intimately related.

Differential calculus is the study of derivatives. Derivatives can be used to find the "rate of change" of a function. For example, if you plot the functions x² and x³, then you will find the latter to be a lot steeper. Derivatives can be used to make this notion rigourous. In physics, derivatives are often used to find velocity and acceleration. For example, when riding in your car, and looking at the speedometer, you're actually looking at a derivative.
Other uses of derivatives is finding slopes of functions, and actually trying to graph functions. Derivatives are also useful in finding minima/maxima of functions. As such, you can solve cool problems like: "given a rectangle with length l and width w such that the circumference is 10. How to choose l and w such that the area enclosed is maximal". This is a standard problem that can be solved by derivatives, and which would be quite hard to solve without them.
Additionaly, derivatives allow you to forget stupid formula's like $$-\frac{b}{2a}$$ for the top of a parabola.

Integrals are in many ways the inverse of derivatives. Integrals allow you to calculate area and volume of things. Also, you can find lengths of curves, and average values of functions.
For example, consider the function $$y=x^2$$ on [0,1], what is the average value of this function? This is a standard problem that one can solve with integrals.
Solving integrals is quite hard however. Certainly when compared to derivatives. Taking derivatives consists of applying mechanical rules, no thinking is involved. Integrals on the other hand require a lot of creativity, and are much more fun!

Vector calculus and complex analysis use integrals and derivatives a lot! In fact, vector calculus is the study of integrals and derivatives to more dimensions. And complex analysis is the study of derivatives and integrals in complex numbers.
Number theory uses complex analysis. Statements about prime numbers can often be translated as statements about a complex function (called the Riemann-zeta function). And as such, the statements become easier to solve. Since complex analysis uses derivatives and integrals, it becomes obvious that number theory does so as well!

 

[TylerH]

Real analysis is the study of the properties of real numbers, some of which the stuff you do in differential and integral calculus rely on.

EDIT: So you could say real analysis builds the foundations of calculus that aren't necessary just to use it; that's why it's only require for pure math majors.

 

[Bogrune]

This is a standard problem that can be solved by derivatives, and which would be quite hard to solve without them.
Additionaly, derivatives allow you to forget stupid formula's like -b/2a for the top of a parabola.

That sound pretty cool and exciting! :-)
My Precalculus instructor teaches this subject in a very different way than most other Precalculus teachers on campus, and I think it's because he has a PhD in Mathematics and he is qualified to teach any branch of mathematics. He disliskes to teach our class by simply giving us a bunch of formulas and showing us how to use them. Rather, he shows us how formulas were come up with to have us know their "sources". For example, when he taught us quadratics, instead of simply giving us the quadratic formula to work with, he gave us the formula for "completing the square" and we were to apply it to the expression ax²+bx+c=0 to find out where it came from. He did the same thing for showing us the sine/cosine addition identity, the laws of sine and cosine, and a formula for vectors (I don't know the name) that goes a little like: (AD + BC)/ (C² + D²)

And because he said that the class is called Pre-Calculus, he gave us a sneak-peek at derivatives.

 

[TylerH]

He does go about it differently. When I took it, we never got to derivatives or integrals.

I would guess, based on how it was presented in the back of my Precal book, that he's only going into the rule based way to find derivatives eg d/dx of x^n]=nx^n-1. If you're an over achiever, you may want to ask him, after class, about the limit based way to find derivatives.

 

[Bogrune]

Real analysis is the study of the properties of real numbers, some of which the stuff you do in differential and integral calculus rely on.

EDIT: So you could say real analysis builds the foundations of calculus that aren't necessary just to use it; that's why it's only require for pure math majors.

I mentioned Real Analysis because I'm the kind of person who really loves mathematics. Furthermore, no matter how hard it is, I'm thiking of challenging myself and to try to earn a PhD in mathematics.

Also, I'm not really an over-achiever in his class. I did mention a few methods in class that he was probably was never get to (such as Cramer's Rule when he was lecturing how to solve systems of equations using matrices), but it didn't help very much. To be honest, both I and many other students in his class are barely passing his class because of the way he likes to "torment" his students with his crazy way of teaching.

 

[paulfr]

It's more like addition.

Yes.
Multiplication is Addition
Multiplication is repeated addition
3 x 4 = 3 + 3 + 3 + 3

"Functions" isn't the right term, since each of these operations takes a function as an input, rather than a number.

True.
Integration and Differentiation are operations, not functions.

 

[TylerH]

Yes.
Multiplication is Addition
Multiplication is repeated addition
3 x 4 = 3 + 3 + 3 + 3

That's a special case; what's 3.5*4.5 in terms of addition?

Regardless, integration is introduced as the infinite sum of infinitesimal slices of f. You can't say it's the product of infinity and an infinitesimal, as it is when f is constant, in all cases.

 

[micromass]

Oh, please, not the repeated addition thing again
Here's the main topic about it: https://www.physicsforums.com/showthread.php?t=474515

 

[paulfr]

That's a special case; what's 3.5*4.5 in terms of addition?

Regardless, integration is introduced as the infinite sum of infinitesimal slices of f. You can't say it's the product of infinity and an infinitesimal, as it is when f is constant, in all cases.

3.5 x 4.5 = 7/2 x 9/2 = 7 x 9 / 2 x 2 =
9+9+9+9+9+9+9 / 2+2 =

You can take it from there

I agree, describing [not defining] Integration as Multiplication is not
rigorously correct and not appropriate for a Math major.
But for beginning HS students faced with the mystery of The Calculus,
it helps simplify the basic idea and makes them comfortable
learning all that surrounds the topic.

 

[TylerH]

I agree, describing [not defining] Integration as Multiplication is not
rigorously correct and not appropriate for a Math major.
But for beginning HS students faced with the mystery of The Calculus,
it helps simplify the basic idea and makes them comfortable
learning all that surrounds the topic.

If addition is more EDIT:[STRIKE]correct[/STRIKE] rigorous, what makes addition more complicated, and thus too complicated for a beginning HS student?

 

[paulfr]

If addition is more EDIT:[STRIKE]correct[/STRIKE] rigorous, what makes addition more complicated, and thus too complicated for a beginning HS student?

I said Multiplication is not a rigorous DEFINITION of Integration.
It is a useful description of it.
This does not preclude showing that Integration can also be viewed as addition.
In fact I always go on to "the accumulation process" view when introducing
the subject. That is how we routinely talk about it in engineering/physics circles.

The real definition of Integration requires the understanding of what a Limit is,
how to find or calculate it, and applying this to the repeated addition of the multiplication
of the function and delta x, evaluated at an infinite number of points over the interval.

And this only covers Definite Integration

We have not yet introduced the idea of the Antiderivative.This is silly quibbling of theorists who teach advanced minds
at a University. In the trenches of High School Math you will
lose half or more of your students attention if your approach
is ONLY rigorous.

As an electronics engineer, I always found it useful to think of the
Laplace Transform, the Integral of the multiplication of the time domain function
by exp(-st) as a division [the minus exponent putting the exp in the denominator].
Thus it was dividing the function into its complex frequency domain components.
This simplified view of this bedrock tool of the profession made working with it
a breeze, not a convoluted mathematical operation.

And consider this; No one knew what addition was until Peano came along with
his Axioms. Try to prove 1 + 1 = 2 without them.

 

[Bogrune]

Yes.
Multiplication is Addition
Multiplication is repeated addition
3 x 4 = 3 + 3 + 3 + 3

I was told by my instructor on the first day that multiplication should be thought of as the area of a shape.

3.5 x 4.5 = 7/2 x 9/2 = 7 x 9 / 2 x 2 =
9+9+9+9+9+9+9 / 2+2 =

You can take it from there

Clever. I haven't thought to see it that way. Although some might try to make their arguments against that (just like most try to argue against a few statements as if it were a debate...)

 

[TylerH]

I teach nothing; I'm a student. A high school one at that. I don't even know how to perform a Laplace Transform (I'm currently working on learning nth order diff eqs.), but I have studied the Peano Axioms a little. :)

My argument is that if one explanation can be both rigorous and useful, there must be justification, ie removal of complexity, to warrant a less rigorous simplification. The difference in complexity from addition to your "simplification" of multiplication, IMHO, doesn't warrant the simplification.

That was the point of my question; I'm asking what simplification is there to justify the "simplification" from addition to multiplication? If, as you say, multiplication is defined as repeated addition, isn't it inherently more complex that addition?

 

[TylerH]

Clever. I haven't thought to see it that way. Although some might try to make their arguments against that (just like most try to argue against a few statements as if it were a debate...)

Wait... it's not a debate? I must have missed that announcement. :tongue:

Seriously though, speaking as someone just past your position(I took Precal last semester and Calc I this semester), it helps immensely to argue with these guys. Their geniuses, and they'll bust your chops for any misstep. You learn a lot just in the process of researching your argument. I speak from experience when I say it teaches you to debate math on a whole different level than your peers.

 

[Bogrune]

You have a point there. Guess it wouldn't hurt to try. I have been having trouble in my Precalculus class because of the way my instructor teaches it, but he did say he's teaching us the way he does to get us ready for Calculus by taking things apart to its simplest elements. Thanks for the tip though!