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Basic Mathematics - Calculus

Calculus is a style of mathematic invented by Sir Isaac Newton. The word "calculus" which means rock. Interesting choice of names as Calculus is assumed to me one of the harder sub-topics that make up the subject of mathematics.

My Calculus teacher used to tell us that calculus really isn't that hard, all you have to imagine is that calculus is focused more narrowly on a particular equation than a simple a + b = c.

An example he used was a car traveling to the local store. You may have traveled at a top speed of 30 km/h, but you did not travel at that velocity the entire distance. Calculus is used to take into account stops and starts, twists and turns in the road, time it take entering and exiting the vehicle - things like that that will result in a much more realistic equation.

My mathematics is a bit rusty, so if there are errors please let me know.

I like to think of Calculus as three individual topics:

Limits
Derivatives
Integrals

One of the common terms found in this subject is the function:

f(x)

A function is a relation that uniquely describes a set of numbers and associated it with a set of variables. For example:

Limits:

A limit describes a function up to a point. Recall that graphic plots in Algebra are lines that continue on. A limit is a tool that limits (as the name suggests) the duration of a plotted set of values.

where x is the value of the function and a = the value of the limit.

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Derivatives:

A derivative is a rate of change of some example - the the car traveling to the corner store. Its is basic form, a derivative is:

Where A is the house, B is the corner store and T is the total time from A to B.

Now if you want to know how fast you were traveling from a specific part of the trip, the power of the derivative comes through. First you need to splice up your line:

And pick two point - X and Y. Using the term:

To get:

Insert a limit to determine instantaneous velocity:

Now for the biggie: for the derivative of the distance with respect to time, insert limits and functions to assess all the points on the line:

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Integrals:

An integral is a term used to define the sum total of all parts. However, the integral is used to find unknown small bits from a know larger size. To do this, we need to use a differential equation:

dy and dx are "some number," that is we don't know the value yet. If we choose to solve for y, the equation takes form:

Where C is a known value.

This may seem a bit confusing, but to use an example:

You want to know the area of the blue triangle. By using an integral, you can. First we assign the function:

For the integral function from 0 to x=b:

 

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