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| Isaac Newton
developed the use of calculus in his laws of
motion and gravitation. |
Calculus
Calculus is a branch of mathematics which helps us
understand changes between values that are related by a
function. For example, if you had one formula telling
how much money you got every day, calculus would help
you understand related formulas like how much money you
have in total, and whether you are getting more money or
less than you used to. All these formulas are functions
of time, and so that is one way to think of
calculus—studying functions of time. There are two
different types of calculus. Differential calculus
divides things into small (different) pieces and tells
us how they change from one moment to the next, while
integral calculus joins (integrates) the small pieces
together and tells us how much of something is made,
overall, by a series of changes. Calculus is used in
many different areas of study such as physics,
astronomy, biology, engineering, economics, medicine and
sociology.
History
In the 1670s and 1680s, Sir Isaac Newton in England and
Gottfried Leibniz in Germany figured out calculus at the
same time, working separately from each other. Newton
wanted to have a new way to predict where to see planets
in the sky, because astronomy had 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. Leibniz wanted to measure the
space (area) under a curve (a line which is not
straight). Many years later, the two men argued over who
discovered it first. Scientists from England supported
Newton, but scientists from the rest of Europe supported
Leibniz. Most mathematicians today agree that both men
share the credit equally. Some parts of modern calculus
come from Newton, such as its uses in physics. Other
parts come from Leibniz, such as the symbols used to
write it.
They were not the first people to use mathematics to
describe the physical world—Aristotle and Pythagoras
came earlier, and so did Galileo, who said that
mathematics was the language of science. But they were
the first to design a system that describes how things
change over time and can predict how they will change in
the future.
The name "calculus" was the Latin word for a small stone
the ancient Romans used in counting and gambling. The
English word "calculate" comes from the same Latin word. |
|
Differential calculus
Differential calculus is the process of finding out the
rate of change of a variable compared to another
variable. It can be used to find the speed of a moving
object or the slope of a curve, figure out the maximum
or minimum points of a curve, or find answers to
problems in the electricity and magnetism areas of
physics, among many other uses.
Many amounts can be variables, which can change their
value unlike numbers such as 5 or 200. Some examples of
variables are distance and time. 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—they may have been traveling faster at
some times (on a highway) and slower at others (at a
traffic light or on a small street where people live).
Imagine a driver trying 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 (y or vertical) divided by how much it goes across (x
or horizontal). On a curve, though, 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 it 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 very
complicated 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.
They said that this particular value it approached was
the actual slope.
Mathematicians have grown this basic theory to make
simple algebra rules which can be used to find the
derivative of almost any function. |
|
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. This gives an
approximation to the area. If the boxes are made
narrower and narrower, then there are more and more of
them, and the area of all the boxes becomes very close
to the area of the shape. One of the main ideas of
calculus is that we can imagine having an infinite
number of these boxes, each infinitely narrow, and then
we would have the exact area of the shape. |
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Other uses of calculus
Calculus is used to describe things that change, like
things in nature. It can be used for showing and
learning all of these: |
- How waves move. Waves are very
important in the natural world. For example, sound and
light can be thought of as waves.
- Where heat moves, like in a house.
This is useful for architecture (building houses), so
that the house can be as cheap to heat as possible.
- How very small things like atoms
act.
- How fast something will fall, also
known as gravity.
- How machines work, also known as
mechanics.
- The path of the moon as it moves
around the earth. Also, the path of the earth as it
moves around the sun, and any planet or moon moving
around anything in space.
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Visit the Kiddle page for more information. |
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 Kiddle: Calculus
Wikipedia: Calculus |
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