Calculus! For some, it is unimaginably hard math,
the stuff that only geniuses would dare try.
True, doing calculus can be rough. However, the ideas behind it are not that difficult to see and relate to our everyday world – which is why it is useful.
So be prepared to step into a brave new world of math to expand your horizons.
First, a word from our sponsor
Calculus has two parts, as will be explained below. Here both will be discussed in terms of the basic concepts one at a time. Following each of these parts, a more mathematical explanation is given. Even here, the goal is not to be overly "mathy," but if it is too much that you signed on for, feel free to move to the next section.
Getting there is half the fun
We've heard of the story of the Tortoise and the Hare; the animals race each other and the hare, being overconfident of his chances of victory, takes a nap and the tortoise ends up winning.
Suppose that later the hare challenges the tortoise to a rematch where he, having learned his lesson, easily wins.
Further suppose, that one of the people watching this race is a science nerd. Being a science nerd, he cannot just and sit and enjoy the race but must take measurements of both animals as they go on their race and make a graph or chart of their position over the time of the race. It might look something like this.
Notice that the line for the faster moving hare is steeper; that is, it goes up faster than the slow tortoise.
Now suppose that also viewing this race is a math nerd. Being a math nerd, he also is not interested in the race. Instead, he is interested in the measurements and using them to do the math. He measurements how far both animals traveled at different times and uses these to find how fast the animals are moving at various times in the race.
This idea – taking the animals' positions and finding the speed, the rate of travel – is the first part of calculus called differentiation. Some examples of applied differentiation:
Knowing how far a falling object has fallen at different times can tell us the object's speed as it falls.
Using measurements of the water level of a bathtub filling up can tell us how fast the water is rising and how fast the water is going into the tub.
You have a large amount of Halloween candy. By keeping track of the amount of candy eaten over time, a parent can figure out how fast the candy is being polished off.
A more technical explanation
In its purest method, calculus works with math equations. You give me a math equation that tells how far a car has traveled over time, and I can tell how fast it was going at; the speed on the speedometer.
As seen in the graph earlier, the steeper the graph, the greater the rate of change. But suppose you have the graph of some equation and want to find the rate at some point.
The graph of some equation with the point of interest circled.
No problem, simply draw out a line from our point, going in the same direction, and use that – the same way we used the tortoise and hare graph.
This is a proper approach, but it does have its limits. It would also be great to know how to solve the problem without having to draw out a graph.
The credit to the approach to the problem goes to Sir Isaac Newton.
Take two other points on either side of our point of interest and draw a line between them.
Our graph with the two points and a line.
OK, this really doesn't look impressive or helpful.
But here is the first clever bit. Choose two different points that are closer than the ones before.
And two points that are closer still.
Look and see that as we move the points closer in, the line changes and is more and more in the right direction. What's more, this approach allows us to solve the question on paper without any graph at all.
Working out this on paper can be tedious. But the best part is people doing this have found rules or shortcuts that make the math so much easier and faster.
And this is what makes differentiation so useful as a tool.
Look where we've been
In a car is an odometer, giving how many miles it has been driven.
The idea of differentiation is that you can get its speed by knowing how many miles it drove at different times.
But is the reverse also true? If you know the car's speed at different times, can you figure out how many miles have been driven?
There is, it is called integration. To integrate means to blend or combine together. It is a mathematical method that works out the combined distance traveled at every speed, at every moment. Of course, the same math tricks can be applied to a host of different problems.
Knowing how fast a car is using gasoline, and how much gas is in the tank, it is possible to determine how long before it will run out of gas.
The speed of the water flowing down a river gets measured over time. From this, one can determine how much water gushed down the river over the same period.
Again with the technical stuff
Suppose that you have an equation you want to integrate from some starting point to an ending point. The graph of this might be something like this.
If this graph represents a car's speed, then the integration – the distance traveled – must be all the distances taken at every step added together. This works out to be the area between the graph and the axis.
How do we do such a thing?
The main idea is to slice up our area into a bunch of rectangles.
The combined area of all these rectangles will be our area. Of course, those with good eyes may notice that our rectangles don't match perfectly with the curve. True, but as the number of rectangles used increases, this becomes less and less of an issue.
Historical note: Before Newton, Johannes Kepler used a similar approach to calculate volumes of wine barrels.
The Babysitter by Normal Rockwell
As it stands, yes.
But again, there are tricks. Since integration is the opposite of differentiation, sometimes one can easily use the tricks used in differentiation to get the integration rules.
OK, higher math might sound like martian, but its development opened up the pursuit of science to allow for more astonishing discoveries than ever were possible before.
And, it has the power to make a person appear smarter than they really are.
On the web
Calculus for Beginners and Artists
This page presents an online course to learn about calculus, hence the name and other math topics.
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