# Derivative applications

Contents

The reason we study calculus, and the reason it was invented, is for its many uses in real-world problems. In particular, derivatives let us optimize functions and study their rates of change.
Here we cover optimization, rates of change, L'Hopital's rule, the mean value theorem, and more!

A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, we'll not only find equations of tangent lines, but normal ones as well.

Derivatives can be used to calculate instantaneous rates of change. The rate of change of position with respect to time is velocity and the rate of change of velocity with respect to time is acceleration. Using these ideas, we'll be able to analyze one-dimensional particle movement given position as a function of time.

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.

Have you ever wondered how fast the area of a ripple of a pond is increasing based on how fast the ripple is? What about how fast a volcano's volume is increasing? This tutorial on related rates will satiate your curiosity and then some!
Solving related rates problems using calculus

If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).

Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.

Let's see how we can local linearization can be used to approximate values of functions near values that we know.

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