In simple words,
- differentiation tells you how something changes, whereas
- integration tells you how something accumulates.
But perhaps you are looking for a few ways that these ideas could arise in the world as you see it.
- If you are climbing a mountain, think of the height you obtain as you walk uphill, then the derivative of that function is the local slope. In fact, this concept of local "slope" is important for optimization schemes used in a huge number of mathematical tools, by scientists, engineers, economists, etc. Thus in an optimization scheme, one needs to be able to walk uphill (or downhill) until you find the top of the peak or very bottom of a valley. The gradient (sort of a derivative in multiple dimensions) helps you choose your direction of search.
- Driving a car, you can think of your position as a function of time. The derivative of that position as you drive in a straight line is your velocity, as measured by the speedometer gauge on your dashboard. The second derivative of position, or the derivative of your speed is your acceleration as you step on the gas petal.
- In fact, derivatives, in the form of differential equations, are used in infinitely many places to build models of the physical world. As a scientist, you want a model of how oxygen is absorbed through a cell wall in your body? A differential equation probably will be used. You fire a cannon ball into the air, and you wish to see where it will land? You could use a differential equation to predict that.
- A structural engineer must choose the proper beam size to build a bridge. How do they know how much a beam will deflect under load? Well normally, the engineer will just refer to a book of tables, tabulating the deflection for any given beam, but those tables were built using differential equation models to predict the behavior of beams.
See also :
WHAT IS THE MEANING OF INTEGRATION?
