In the real world, it can be used to find the speed of a moving object, or to understand how electricity and magnetism work. It is very important for understanding physics—and many other areas of science.
Differential calculus is also useful for graphing. It can be used to find the slope of a curve, and the highest and lowest points of a curve (these are called the maximum and minimum, respectively).
Variables can change their value. This is different from numbers because numbers are always the same. For example, the number 1 is always equal to 1, and the number 200 is always equal to 200. One often write variables as letters such as the letter x: "x" can be equal to 1 at one point and 200 at another.
Some examples of variables are distance and time, because they can change. The speed of an object is how far it travels in a particular time. So if a town is 80 kilometres (50 miles) away and a person in a car gets there in one hour, they have traveled at an average speed of 80 kilometres (50 miles) per hour. But this is only an average: maybe they travelled faster at some times (say on a highway), and slower at other times (say at a traffic light or on a small street where people live). Certainly it is more difficult for a driver to figure out a car's speed using only its odometer (distance meter) and clock—without a speedometer.
Until calculus was invented, the only way to work this out was to cut the time into smaller and smaller pieces, so the average speed over the smaller time would get closer and closer to the actual speed at a point in time. This was a very long and hard process, and had to be done each time people wanted to work something out.
A very similar problem is to find the slope (how steep it is) at any point on a curve. The slope of a "straight" line is easy to work out — it is simply how much it goes up or down ("y" or vertical) divided by how much it goes across ("x" or horizontal). On a "curve", however, the slope is a variable (has different values at different points) because the line bends. But if the curve was to be cut into very, very small pieces, the curve at the point would look almost like a very short straight line. So to work out its slope, a straight line can be drawn through the point with the same slope as the curve at that point. If this is done exactly right, the straight line will have the same slope as the curve, and is called a tangent. But there is no way to know (without complex mathematics) whether the tangent is exactly right, and our eyes are not accurate enough to be certain whether it is exact or simply very close.
What Newton and Leibniz found was a way to work out the slope (or the speed in the distance example) exactly, using simple and logical rules. They divided the curve into an infinite number of very small pieces. They then chose points on either side of the range they were interested in and worked out tangents at each. As the points moved closer together towards the point they were interested in, the slope "approached" a particular value as the tangents approached the real slope of the curve. The particular value it approached was the actual slope.
Given a function formula_1. "f" is short for function, so this equation means "y is a function of x". This tells us that how high y is on the vertical axis depends on what x (the horizontal axis) is at that time. For example, with the equation formula_2, we know that if formula_3 is 1, then formula_4 will be 1; if formula_3 is 3, then formula_4 will be 9; if "formula_3" is 20, then "formula_4" will be 400. The slope of the tangent line produced using this method here is formula_9, or 2 multiplied by "formula_3". So we know without having to draw any tangent line at any point on the curve formula_11 that the derivative, often written as formula_12 (marked with the prime symbol), will be formula_9 at any point. This process of working out a slope using limits is called differentiation, or finding the derivative.
The way to write the derivative in mathematics is
formula_14
Leibniz came to the same result, but called h "formula_15", which means "with respect to x". He called the resulting change in formula_16 "formula_17", which means "a tiny amount of y". Leibniz's notation is used by more books, because it is easy to understand when the equations become more complicated. In Leibniz notation:
formula_18.
Mathematicians have grown this basic theory to make simple algebra rules—which can be used to find the derivative of almost any function.
Integral calculus is the process of calculating the area underneath a graph of a function. An example is calculating the distance a car travels: if one knows the speed of the car at different points in time and draw a graph of this speed, then the distance the car travels will be the area under the graph.
The way to do this is to divide the graph into many very small pieces, and then draw very thin rectangles under each piece. As the rectangles become thinner and thinner, the rectangles cover the area underneath the graph better and better. The area of a rectangle is easy to calculate, so we can calculate the total area of all the rectangles. For thinner rectangles, this total area value "approaches" the area underneath the graph. The final value of the area is called the "integral" of the function.
In mathematics, the integral of the function "f(x)" from "a"  to "b", is written as
formula_19.
The main idea in calculus is called the fundamental theorem of calculus. This main idea says that the two calculus processes, differentiation and integration, are inverses of each other. That is, a person can use differentiation to undo an integration process. Also, a person can use integration to undo a differentiation. This is just like using division to "undo" multiplication, or addition to "undo" subtraction.
In a single sentence, the fundamental theorem runs something like this: "The derivative of the integral of a function "f" is the function itself".
Calculus is used to describe things that change, like things in nature. It can be used for showing and learning all of these:
Coin
A coin is normally a round piece of metal that is used as currency, or money. Coins have been made for about 2600 years; the first place to make coins was Lydia (modern Turkey).[citation needed] These coins were made of Precious metals and allowed people to trade with a standard amount of metal.
Most people use coins as currency. They usually have lower value than banknotes. Most are made in government mints.
Many coins have unique or complicated decorations; one side often has the picture of a famous or important person's head on it.
The different decorations on each side of a coin might be used to decide things randomly. This is called "tossing a coin". A person can throw the coin into the air and catch it. You then look at which side is facing up. If the head is facing up it is called "heads", if the other side is facing up it is called "tails". Before tossing the coin someone has to decide what each side means. Tossing a coin can be a type of gambling, which is illegal (against the law) in some countries.
Some people see coins as a sign of greed, such as some Communists and Puritans, who sometimes condemn over-hoarding of coins, and ascetics, who often keep little in the ways of money (coins), leading a "poor"-lifestyle.
Because coins have been made for a very long time, some people collect old coins. They can be much cheaper than other old things, especially if they are made of cheap metals like copper. Older coins normally cost more than newer ones, but rarity matters more-some coins from the 1920s cost vast sums, while some Roman coins cost very little.
Conceptual metaphor
A conceptual metaphor or cognitive metaphor is a metaphor which refers to one domain (group of ideas) in terms of another. For example, treating quantity in terms of direction: