How would you describe calculus in simple terms?
There came a time in mathematics when people encountered situations where they had to deal with really, really, really small things.
Not just small like 0.01; but small as in infinitesimally small. Think of "the smallest positive number that is still greater than zero" and you'll realize what sort of problems mathematicians began encountering.
Soon, this problem became more than just theoretical or abstract. It became very, very real.
For example, velocity. We know that average velocity is the change in position per change in time (i.e., 5 miles per hour). But what about velocity at a point in time? What does it mean to be going 5 mph at this moment?
One solution someone came up with was to say "it's the change in position divided by the change in time, where the change in time is an infinitesimally small amount of time". But how would you handle/calculate that?
Another problem came about trying to find the area under a curve. The current accepted solution was to divide the curve into rectangles, and add together the area of the rectangles. However, in order to find the exact area under the curve, you'd need to divide it into rectangles that were infinitesimally tiny, and, therefore, add up an infinite amount of tiny rectangles -- to something that was finite (area).
Calculus came about as the system of math dedicated to studying these infinitesimally small changes. In fact, I do believe some people describe calculus as "the study of continuous changes".
One of the greatest achievements of human civilization is Newton's laws of motions. The first law says that unless a force is acting then the velocity (not the position!) of objects stay constant, while the second law says that forces act by causing an acceleration (though heavy objects require more force to accellerate).
However to make sense of those laws and to apply them to real life you need to understand how to move between the following three notions:
- Position
- Velocity (that is the rate of change in position)
- Acceleration (that is the rate of change of the velocity)
Moving down that list is called "taking the derivative" while moving up that list is called "taking the integral." Calculus is the study of derivatives and integerals.
In particular, if you want to figure out how objects move under some force you need to be able to integrate twice. This requires understanding a lot of calculus!
In a first semester class you usually learn about derivatives and integrals of functions of one variable, that is what you need to understand physics in 1-dimension! To understand the actual physics of the world you need to understand derivatives and integrals in 3-dimensions which requires several courses.
Calculus is basically a way of calculating rates of changes (similar to slopes, but called derivatives in calculus), and areas, volumes, and surface areas (for starters).
It's easy to calculate these kinds of things with algebra and geometry if the shapes you're interested in are simple. For example, if you have a straight line you can calculate the slope easily. But if you want to know the slope at an arbitrary point (any random point) on the graph of some function like x-squared or some other polynomial, then you would need to use calculus. In this case, calculus gives you a way of "zooming in" on the point you're interested in to find the slope EXACTLY at that point. This is called a derivative.
If you have a cube or a sphere, you can calculate the volume and surface area easily. If you have an odd shape, you need to use calculus. You use calculus to make an infinite number of really small slices of the object you're interested in, determine the sizes of the slices, and then add all those sizes up. This process is called integration. It turns out that integration is the reverse of derivation (finding a derivative).
In summary, calculus is a tool that lets you do calculations with complicated curves, shapes, etc., that you would normally not be able to do with just algebra and geometry.