Saturday, 28 March 2020

Mathematics I


Calculus is a branch of mathematics that explores variables and how they change by looking at them with small values ​​called the infinitesimal quantity.

What is the benefit of infinitesimals?
In order to understand what the infinitesimal quantities mean, let us take the mathematical formula expressing the area of ​​the circle; that is, the following relationship: A = ²r², which Professor Steve Strugats of Cornell University indicated that despite its simplicity it is impossible to derive it without having the finite values ​​in Filigree.

Initially we found that the ratio between the circumference of a circle and its diameter is equal to a constant value of approximately 3.14, which is the ratio we call pi and write in the form (π), and using this information we also write the formula for the circumference of the circle in the form: C = 2πr; (r is the radius).

Halves integration and differentiation

The study of integration and differentiation consists of two aspects.

The first is called "differential calculus" and it focuses on the individual study of infinitesimal amounts, and what happens in infinitesimal parts.

The second aspect of differentiation and integration is called "integral calculus", as it depends on adding an infinite number of infinitesimal amounts together (as in the previous example).

They are two opposite processes and are generally referred to as the basic theory of integration and differentiation. To find out how this theory works, let's take the following example from our daily lives:

We have a ball that we threw upward, in a vertical direction, from an initial height of three feet (0.9144 meters) at an initial speed of 19.6 feet per second.

If we graph the location of the ball’s change over time, we get a familiar shape called parabola.

differentiation
The ball changes its velocity at every point along the curve, and there is no time when the ball maintains a steady velocity, but we can calculate the average velocity in any time period.

For example, to find the average velocity from 0.1 seconds to 0.4 seconds, we find the location of the sphere between these two times and draw a line between them.

We note this line increases with the increase of its width.

This ratio is often called the slope, and is defined as the quotient of the height divided by the width.

On the time graph, the slope represents speed, and the line rises from 4.8 feet to 8.3 feet, or about 3.5 feet.

Time changes from 0.1 seconds to 0.4 seconds, meaning that the duration is 0.3 seconds. The slope of this straight line is the average velocity of the ball during this period, and the sum of the division of the height by the change in time is equal to 3.5 feet divided by 0.3 seconds = 11.7 feet per second.

At the moment 0.1 seconds, we see that the curve in the graph line is slightly sharp compared to the average that we calculated, and this means that the ball was moving at a speed much faster than 11.7 feet / second, while in the moment 0.4 seconds, the curve of the graph line is slightly above the level, and this indicates The ball was moving at a speed less than 11.7 feet / second.

And because the speed was decreasing, this means that we must have a specific moment when the ball was moving at 11.7 feet / second completely, so how do we determine the precise time for this moment?

Let's go back and note that the time range between 0.1 seconds and 0.4 seconds is not the only time that the ball has a velocity of 11.7 feet per second.

So if we maintain the slope we can move it anywhere on the curve and we get the same average velocity equal to 11.7 feet / second in the time range between the two points that intersect the curve.

If we move the straightness further towards the height of the parabola, the time range decreases.

When time reaches zero, the intersection points are in the same place and the straight line becomes touching the cut (barely touching it), and the time range is described as finite to zero.

integration
It is the opposite process of differentiation. The velocity integration of a given object relative to time is where it is.

The derivation is calculated as we found by finding the curves; while the integral is calculated by finding the values ​​of the areas.

Speed ​​corresponds to the time on the graph, and the area represents the distance, finding the areas on the graph is relatively simple when dealing with triangles and lozenges, but when we deal with a winding graph instead of straight lines, it becomes necessary to divide the space into an infinite number of small triangles (this It is similar to adding an infinite number of infinitesimals to calculate the area of ​​a circle.


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