Derivative applications



One of the reasons calculus was invented was to be able to optimize functions. When you have some function modeling a real world situation, you often want to find its maximum or minimum. In this tutorial, you will see how information about the derivative of a function can give powerful ways to mathematically describe the "shape" of a function.

When you're looking for the maximum or minimum of a function, a good way to start is by finding points where the derivative equals zero. However, you won't always get the maximum possible value of the function; you might just end up with a point which is maximum *relative* to those points around it. In this tutorial, you will learn about the extreme value theorem, and what it tells us about relative maxima and minima.

One of the neat things about calculus is that it gives us a mathematical way to describe the shape of curves. In this tutorial, you will learn about concavity and inflection points, which describe quantitatively how a curve curves. This will be useful for finding maxima and minima.

Choosing which topic in calculus is the most useful is like asking a parent to choose their favorite child. What you're supposed to say is that there isn't one. That said, between you and me, optimization is quite possible the most important topic from calculus that you should remember.

So, you've probably been told that calculus is useful, but this fact can be hard to believe and remember when you're deep in the weeds of a particularly nasty derivative. In this tutorial, we begin to just scratch the surface as we apply derivatives in fields as disperse as biology and economics.





Here you have the chance to practice thinking about tangent lines when curves are defined in polar coordinates.