Calculus
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Calculus is a way to study shapes and lines using numbers. It uses numbers to show how things change. Engineers use calculus to make things. Scientists use calculus to study the world and understand nature.
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[change] History
The most famous person in the history of calculus is Sir Isaac Newton. He was not the first person to use mathematics to describe the physical world (Aryabhata (differential calculus), Aristotle, and Pythagoras came earlier), but he was among those who invented calculus — a 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 a 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 them credit for the invention.
[change] Lines and shapes
[change] What a straight line is
Lines can bend in a curve. If lines do not bend, they are straight. For example, the edge of a circle is not straight. The edge of a circle bends in a curve. Light bends in a curve in air and water, but travels in a straight line away from the earth or other bodies in space where there is no air.
[change] What a horizontal straight line is
A vertical straight line looks like this |, a horizontal straight line looks like this —.
[change] What a slope is
Two lines can point in the same direction, or different directions:
- Two lines that point in the same direction have no angle between them. The angle between two lines that point in the same direction is zero. They are called "parallel lines".
- Lines can point in different directions. If two lines do not point in the same direction, they have an angle between them that is not zero.
A slope is a way to describe the angle between a straight line and a horizontal straight line:
- A straight line with no angle between the straight line and a 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, then the slope of the straight line is 1.
- If the angle between the straight line and the horizontal line is 60 degrees, then the slope of the straight line is about 1.73.
- If the angle between the straight line and the horizontal line is 80 degrees, then the slope of the straight line is about 5.67.
- If the angle between the straight line and the horizontal line is 89 degrees, then 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, then the slope of the straight line is about 572.96.
- For any straight line, the slope is the tangent of the angle between the line and the horizontal line.
The closer the angle between the straight line and the horizontal line gets to 90 degrees, then the bigger the slope of the straight line is, just as the tangent function gets closer to infinity as its
[change] Curved lines also have slopes
If a line is not straight, but curved like the edge of a circle, a slope of the edge can still be found. Here is a way to think of this:
- A shape like a circle has an edge.
- A straight line can be drawn to touch the edge of a shape in one point only. This is called a tangent line.
- The angle between the tangent line and a 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.
[change] Types of calculus
Calculus has two main parts:
- Differential calculus gives people ways to find the slopes of lines, both straight and curved lines
- Integral calculus gives people ways to find the area of a 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.
[change] 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, a person can use differential calculus to undo an integral calculus process. Also, a person can use integral calculus to undo a differential calculus method, just like if you add a number to another number you can 'undo' it by taking away that number.
[change] Demonstration of main idea of calculus
[change] 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 a shape has an edge given by a 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 Xi; i = 1,2,3,...N. where X1 = A and XN = B. A box between Xi and Xi + 1 will have an area that is about ![(X_{i+1}-X_i) \times 0.5 \times [F( X_i )+ F(X_{i+1})]](../../../../math/8/6/e/86e9626e24e61a92636c3d1355388711.png)
. 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})]](../../../../math/0/d/4/0d44e8ab0f6b2f30cc3317b326b688af.png)
[change] Other uses of calculus
Calculus is used to describe things that change such as Nature.
Calculus can be used to show how waves move. Waves are very important in the natural world. For example, sound and light are waves.
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.
