A lot of people have written in and asked for a down-to-earth, introductory tutorial on calculus.
This is no small task because of the vastness of the subject, so I’ve decided to write a series of posts to explain the motivation, ideas, and techniques used in calculus!
I’m calling this post Part 0 because here, I just want to tell you guys about the big idea! So the question we want to address:
Whats the big idea?!
And unlike most other things in mathematics, the answer here is both succinct and somewhat straightforward, conceptually:
The big idea is: Limits!
Calculus is an enormous discipline whose reach spans nearly every quantitative discipline beyond basic algebra, but at the heart of it all is the notion of a limit.
Okay…so…what is a limit?
Intuitively, a limit is a tool which allows you to analyze the behavior of a thing (a sequence of numbers, a function, a collection of spaces…) as it gets near a certain thing (an index, a domain value, a fixed other space…).
For instance: What is the behavior of the function near the point where Gandalf’s staff is at? That’s an example of what limits measure!
Algebra on the other hand islanguage through which we describe patterns.
This, in a nutshell, is what separates calculus from other areas of math:
In much the same way that the shift from numbers to variables marks the jump from arithmetic to algebra, the introduction of the limit marks the jump from algebra to calculus.
Said differently, variables allow quantities to vary from one situation to another; limits allow quantities to vary from one situation to another and to change within a single context.
Conceptually, it’s a little strange, but with some practice, it becomes pretty natural.
So, like…is there an example…?
There are tons!
If you have a sequence of numbers like 1, ½ , ¼ , 1/16 , 1/64.. and you want to know what happens to this sequence as we go farther and farther out?
In particular, what do the numbers in the sequence get close to (whatever that means) as you write down more and more numbers which follow the same pattern?
What you’re really asking is what the limit of the sequence is, and since the numbers are getting smaller while never becoming negative, it seems likely that their limit is 0! Voila!
As another example:
Let’s say we have a function which is defined to be the parabola y= x² everywhere except at x=2, where it has a hole.
Graphically, this looks like the parabola with a hole at ( 2,4 )
In this example, it very clearly doesn’t make sense to talk about what the function does at x=2 since it’s undefined there.
However, it does make sense to ask what the function does near x=2 and even though a picture isn’t a proof, we can probably all be convinced that as the x gets closer to 2, the function approaches 4.
So, the obvious next question is:
How do we find a limit?
And there are two answers:
The short answer: Very carefully. Seriously: Don’t muck something up because of carelessness!
The long answer: …
The long answer is far too long to address here, so it’ll be the topic of the next post in the introduction to calculus series! So, you know…stay tuned!
Have a great day everyone!
By the way: I’m Chris and I’m a guest poster here on FYP! I’m a doctoral student in mathematics (I study a branch called topology) and I have a main blog (source) and a side math-type blog (source); the latter needs some freshening up, though. ^_^
A lot of people have written in and asked for a down-to-earth, introductory tutorial on calculus.
This is no small task because of the vastness of the subject, so I’ve decided to write a series of posts to explain the motivation, ideas, and techniques used in calculus!
I’m calling this post Part 0 because here, I just want to tell you guys about the big idea! So the question we want to address:
Whats the big idea?!
And unlike most other things in mathematics, the answer here is both succinct and somewhat straightforward, conceptually:
The big idea is: Limits!
Calculus is an enormous discipline whose reach spans nearly every quantitative discipline beyond basic algebra, but at the heart of it all is the notion of a limit.
Okay…so…what is a limit?
Intuitively, a limit is a tool which allows you to analyze the behavior of a thing (a sequence of numbers, a function, a collection of spaces…) as it gets near a certain thing (an index, a domain value, a fixed other space…).
For instance: What is the behavior of the function near the point where Gandalf’s staff is at? That’s an example of what limits measure!
Algebra on the other hand islanguage through which we describe patterns.
This, in a nutshell, is what separates calculus from other areas of math:
In much the same way that the shift from numbers to variables marks the jump from arithmetic to algebra, the introduction of the limit marks the jump from algebra to calculus.
Said differently, variables allow quantities to vary from one situation to another; limits allow quantities to vary from one situation to another and to change within a single context.
Conceptually, it’s a little strange, but with some practice, it becomes pretty natural.
So, like…is there an example…?
There are tons!
If you have a sequence of numbers like 1, ½ , ¼ , 1/16 , 1/64.. and you want to know what happens to this sequence as we go farther and farther out?
In particular, what do the numbers in the sequence get close to (whatever that means) as you write down more and more numbers which follow the same pattern?
What you’re really asking is what the limit of the sequence is, and since the numbers are getting smaller while never becoming negative, it seems likely that their limit is 0! Voila!
As another example:
Let’s say we have a function which is defined to be the parabola y= x² everywhere except at x=2, where it has a hole.
Graphically, this looks like the parabola with a hole at ( 2,4 )
In this example, it very clearly doesn’t make sense to talk about what the function does at x=2 since it’s undefined there.
However, it does make sense to ask what the function does near x=2 and even though a picture isn’t a proof, we can probably all be convinced that as the x gets closer to 2, the function approaches 4.
So, the obvious next question is:
How do we find a limit?
And there are two answers:
The short answer: Very carefully. Seriously: Don’t muck something up because of carelessness!
The long answer: …
The long answer is far too long to address here, so it’ll be the topic of the next post in the introduction to calculus series! So, you know…stay tuned!
Have a great day everyone!
By the way: I’m Chris and I’m a guest poster here on FYP! I’m a doctoral student in mathematics (I study a branch called topology) and I have a main blog (source) and a side math-type blog (source); the latter needs some freshening up, though. ^_^
By virtue of everyday usage, the fact that (-1) x (-1) = 1 has been engraved onto our heads. But, only recently did I actually sit down to explore why, in general negative times negative yields a positive number !
Intuition.
Let’s play a game called “continue the pattern”. You would be surprised, how intuitive the results are:
2 x 3 = 6
2 x 2 = 4
2 x 1 = 2
2 x 0 = 0
2 x (-1) = ?? (Answer : -2 )
2 x (-2 ) = ?? (Answer : -4 )
2 x ( -3) = ?? (Answer : -6 )
The number on the right-hand side keeps decreasing by 2 !
Therefore positive x negative = negative.
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = ?? (Answer : 3)
-2 x -3 = ?? (Answer : 6)
The number on the right-hand side keeps increasing by 3.
Therefore negative x negative = positive.
Pretty Awesome, right? But, let’s up the ante and compliment our intuition.
The Number Line Approach.
Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y.
Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps.
So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
Ergo, -1 x -1 means to stand on 0, face in the negative direction, and then take 1 backward step. This lands us smack right on +1 !
The Complex Numbers Approach.
The “i” in a complex number is an Instruction! An instruction to turn 90 degrees in the counterclockwise direction. Then i * i would be an instruction to turn 180 degrees. ( i x i = -1 ). where i = √-1
Similarly ( -1 ) x i x i = (- 1 ) x ( -1 )= 1. A complete revolution renders you back to +1.
We can snug in conveniently with the knowledge of complex numbers. But, complex numbers were established only in the 16th century and the fact that negative time negative yields a positive number was well established before that.
Concluding remarks.
Hope you enjoyed the post and Pardon me if you found this to be rudimentary for your taste. This post was inspired by Joseph H. Silverman’s Book – A friendly Introduction to Number Theory. If you are passionate about numbers or math, in general it is a must read.
There are several other arithmetic methods that prove the same, if you are interested feel free to explore.
First Order LDE and that mysterious Integrating Factor.
The integrating factor can be deeply confusing if you don’t understand what it does. This was an attempt to explain the first order linear differential equation to a friend.
(Sorry about the image quality though. I had to make these in a hurry )
Ecstasy Shots 