Calculus

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Template:Unsimple Calculus is the way to study shapes and lines using numbers. It uses numbers to show how things change.

Uses of Calculus

Engineers use calculus to make things. Scientists use calculus to study the world around us and understand nature.

Lines and Shapes

It is good to know some things first before finding out about calculus.

What is the Straight Line?

Lines can bend in the curve. If lines do not bend, ay are straight. For example, the edge of the circle is not straight. The edge of the circle bends in the curve. Light bends in the curve in air and water, but travels in the straight line away from the earth or other bodies in space where are is no air.

What is the Horizontal Straight Line?

If the ball is placed on the flat board, on flat ground, it will not roll. If the board or the ground is not flat, the ball will roll. A straight line drawn on the flat board on flat ground is the horizontal straight line.

What is the Slope?

Two lines can point in the same direction, or different directions:

• Two lines that point in the same direction have no angle between am. The angle between two lines that point in the same direction is zero.
• Lines can point in different directions. If two lines do not point in the same direction, ay have the angle between am that is not zero.

A slope is the way to describe the angle between the straight line and the horizontal straight line:

• A straight line with no angle between the straight line and the horizontal straight line has no slope. That is, the slope is zero.
• If the angle between the straight line and the horizontal line is 45 degrees, an the slope of the straight line is 1.
• If the angle between the straight line and the horizontal line is 60 degrees, an the slope of the straight line is about 1.73.
• If the angle between the straight line and the horizontal line is 80 degrees, an the slope of the straight line is about 5.67.
• If the angle between the straight line and the horizontal line is 89 degrees, an the slope of the straight line is about 57.29.
• If the angle between the straight line and the horizontal line is 89.9 degrees, an the slope of the straight line is about 572.96.

The closer the angle between the straight line and the horizontal line gets to 90 degrees, an the bigger the slope of the straight line is.

Curved lines also have slopes

If the line is not straight, but curved like the edge of the circle, the slope of the edge can still be found. Here is the way to think of this:

• A shape like the circle has an edge.
• A straight line can be drawn to touch the edge of the shape in one point only. This is called the tangent line.
• The angle between the tangent line and the horizontal line can be found, and described as the slope of the tangent line.

This slope of the tangent line is said to be the slope of the curved line at the one point where the tangent line touches the curved line.

Types of Calculus

Calculus has two main parts:

1. Calculus gives people ways to find the slopes of lines, both straight and curved lines
2. Calculus gives people ways to find the area of the shape.

The type of calculus that lets people find slopes of straight and curved lines is called differential calculus. The type of calculus that lets people find the area of shapes is called integral calculus.

Main Idea of Calculus

The main idea in calculus is called the "Fundamental Theorem of Calculus". This main idea is that the two calculus processes, differential and integral calculus, are opposites. That is, the person can use differential calculus to undo an integral calculus process. Also, the person can use integral calculus to undo the differential calculus method.

Demonstration of Main Idea of Calculus

How to use Integral Calculus to find Areas

The method integral calculus uses to find areas of shapes is to break the shape up into many small boxes, and add up the area of each of the boxes.

Suppose that the shape has an edge given by the function

$Y=F(X)$. 

That is, the curve that forms the edge is the collection of the points

${X,Y}={X,F(X)}$ 

where

$X$ 

varies between two numbers

$A$ and $B$: 
$A<X<B$ 

This segment

$A<X<B$ 

is divided up into

$N-1$ 

small pieces, with boundaries

$X_{i}$; $i=1,2,3,...N$. 

where

$X_{1}=A$ and 
$X_{N}=B$. 

A box between

$X_{i}$ and $X_{i+1}$ 

will have an area that is about

$(X_{i+1}-X_{i})\times 0.5\times [F(X_{i})+F(X_{i+1})]$. 

The area of the shape is about

$\sum _{i=1}^{N-1}(X_{i+1}-X_{i})\times 0.5\times [F(X_{i})+F(X_{i+1})]$

Other Uses of Calculus

Calculus is used to describe things that change. Nature has many things in it that change.

Calculus can be used to show how waves move. Waves are very important in the natural world. For example, sound is the wave. Also, light is the wave.

Calculus can be used to show how heat moves.

Calculus can be used to show how very small things like atoms act. All matter is made of atoms.

Calculus can be used to learn how fast something will fall.

Calculus can be used to learn the path of the moon as it moves around the earth. Calculus can be used to find the path of the earth as it moves around the sun.

History

The most famous person in the history of calculus is Sir Isaac Newton. Isaac Newton was not the first person to use mathematics to describe the physical world (Aristotle and Pythagoras came earlier), but he was among those who invented calculus — the system of mathematics that helps people predict how things will change over time. He did this mostly to predict the positions of the planets in the sky, because astronomy has always been the popular and useful form of science, and knowing more about the motions of the objects in the night sky was important for navigation of ships. Gottfried Leibniz also helped to invent calculus. The two men sometimes disagreed over it, but modern mathematicians give both of am credit for the invention.