What a headache created Newton and his German Fellow Leibniz back in the 1660's. All they needed was to follow the regular math from Ptolemy, but We guess, it was due for the sake of the advancement of humankind.
Newton called it the Method of fluxions and fluents, and Der Herr Leibniz just did his own discovery around 1693. He said and the other one said, that there was not even the slightest idea of each others work. The quarrel went for centuries, until our grandparents decided to give them both the credit.
Ever seen a kid react to an ice cream? How his/her mouth get watery and can't wait to have it?
Ever heard that saying, from any action there is a reaction? Sure you did!
In the case of Functions; for every X there is a Y....or F(x)=y
From the SET A of infinitesimal x's called DOMAIN of F
There is a SET B of correlated y's called the RANGE of F
Why this wise minds thought of this? They wanted a mathematical tool for their research in a new Scientific way.
If a car speeds up..then gas will be consumed quickly
If we practice our math skills more often..then the grades will raise up...you follow us?
If we write more hubs..then AdSense will pay accordingly!
The graphic is pretty clear. If we give the value of 2 to X..then Y will be zero
mnemonics: Desperate housewives SET A will be called the DOMAIN of F (not passive!)
Survivor's SET B where all the Y's are held hostage will be called the deranged RANGE of F
Limits are very important to Calculus and was devised in order to study events in nature that cannot be predicted or are limited to a near expected range. For instance, if we fill a bucket with water and we hold it for one hour..there will be a limit for our strength to keep against our muscles. Let's go to our mathematical example.
If you feed 100 people and double the ration, but we take just one meal away
but divide those 199 between the same amount of people... no matter how big the crowd will be:
You will always reach this answer: 1.9999
Then the limit of that function for x= infinite, will be 2. Do we make sense?
This can be silly but we try to make examples with reality, away from abstraction:
Anna 'a' is being enticed by Derek DELTA who is wooing her away from her boyfriend'
Larry 'L', who in turn falls for Ericka EPSILON....now you know from real life, depending how hard Derek works on his last conquest he can win Anna away (-delta,+delta)
Same thing for Larry, depending on Ericka's body language and secret 'turn on' (-epsilon,+epsilon) She can have Larry any minute, if she wants to.
Now according to functions, Jealousy, rebounds and pay back will shape up the graph shown:
Now we can get a little rigorous and apply our example to what mathematicians define in their own words:
- choosing e > 0, indicating that we want the distance between f(x) and L to be less than e
- we can find d > 0
- so that if the distance from x to a is less than d but not equal to 0
- then the distance from f(x) to L will be less than e.
We have paved the way to give the easiest explanation about derivatives and its use. So far we want you to know that we have limited a ratio y/x to their infinitesimal approach. It can be noticed that this ratio is a slope and also a relationship between two physical quantities. Most Scientists around Newton's time studied and gathered information for further study; let's be more practical and compare speed VS time. All we can do is go to the basic equation:
V=Velocity D=distance T=time of course we use it every day..."I was going over 85miles/hour"
Newton and Leibniz wanted to work with instantaneous speeds for practical reasons. Say, you are going around a curve on a Highway, you know you are not going to keep an steady speed, either you slow down so the centrifugal force won't cause your car to flip or you regulate the gas accordingly. Now, that curve can match or represent our classical Cartesian coordinates F(t)=D; for every second there will be a distance traveled.
Let's continue on a second Hub..since we are running out of space...see you!
LOOKING FOR PART 2..?
- CALCULUS DEMISTIFIED PART 2
Getting into real business after giving a smooth intro to Calculus!