# Derivatives: Interesting (unexpected?) situations where they arise?

I am re-learning Calculus. Can anyone provide any interesting (unexpected?) situations where Calculus derivatives arise in various situations or real-life careers? I am looking for something more substantial than typical textbook examples such as a car driving down the road, or water flowing in a cone, etc.

So far, I understand the relationship between:

• position function: f(x)
• velocity function: f'(x)
• acceleration function: f''(x)

Velocity is the instantaneous rate of change of the position. Acceleration is instantaneous rate of change of the velocity. Rate of change of the rate of change, if you will. (speeding up, slowing down)

Thanks!

A handout from my vector calculus class.

Consider the social network of seven individuals with the unimaginative names $A,B,C,D,E,F$ and $G$. An edge connects each pair of friends. This network or graph consists of two smaller, distinct graphs or components.

Question: How to write an algorithm to suggest that person $B$ befriend $D$?

The computer program should analyze the two components $\{A,B,C,D\}$ and $\{E,F,G\}$, identify that person $B$ is in the first component and then step through that list to find people to whom person $B$ is not currently linked. To do this, the computer will be fed the graph Laplacian, a matrix defined via the formula: \begin{equation*} L = (a_{ij}) = \begin{cases} \text{degree of vertex $i$ along the diagonal} \\ \text{$-1$ when an edge connects vertices $i$ and $j$}. \end{cases} \end{equation*} For the network of seven friends, the Laplacian matrix looks like: \begin{equation} L = \begin{bmatrix} 3 & -1 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & -1 & 3 & -1 & 0 & 0 & 0 \\ -1 & 0 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 1 \end{bmatrix} \end{equation} where rows are in alphabetical order.

Question: How to determine the components of the graph using this matrix?

Note that the vector $\begin{bmatrix} 1 & 1 & 1 & 1 & 0 & 0 & 0 \end{bmatrix}^T$ is in the nullspace of $L$ and this vector corresponds to the first component. Can you find a second vector in the nullspace? In general, these vectors associated with the components form a basis for the nullspace (and this isn't difficult to prove). So if you find the basis for $N(L)$, you've found the components of the original graph.

In real life, graphs aren't as simple as the one pictured above. In fact, the graph may consist of one giant component with tightly clustered "approximate components" embedded within. (See any of the images in this search.) And if the graph does have a lot of components, there are more computationally efficient methods of finding them. So why introduce the graph Laplacian? It turns out that the graph Laplacian is a basic object in the field of spectral clustering, which has numerous "real life" applications. In fact, I actually used the technique at a previous job while analyzing a large dataset.

I should point out that in spectral graph theory, you analyze all eigenvalues of the graph Laplacian not just $\lambda=0$, as we have done.

Question: What does this have to do with derivatives?

If you take multi-variable calculus, you may learn about the Laplace operator $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$, which I have written in two dimensions. You may not believe it, but there's actually a connection between the Laplace operator and the graph Laplacian which can be explained via the discrete Laplacian!

• This is a very cool example! Thanks! – JackOfAll Aug 14 '13 at 11:37

Reactor design (rates of reaction, heat transfer, etc), Process control system (essentially automatcially control to meet a setpoint), Process optimization methods

These are ways ChemEs and other process engineers use them in manufactering.

Any measure that can be graphed over time: Housing markets -> how fast houses are selling, instantaneous daily sales, change in sales since yesterday etc. others:

Stock markets, Student test scores, girlfriends per year, presidential popularity or any political graph, Video game kill death ratios by minutes

I find that it is easier to teach someone using "nouns" that they can relate to

In my work as an HVAC technician, we use "PID loops," where PID stands for Proportion, Integral, Derivative. These terms are exactly descriptive of the process that the loop performs on each iteration, but rather than doing actual integration or derivation, a typical object will perform approximations of these.

Another good use of a PID loop is in your cruise control in your car, where the loop controller automatically increases and decreases fuel flow rate to your engine in response to the changing speed that it senses based on your setting of how fast you want to go.

My favorite example of an unexpected situation where the derivative comes up is definitely the notion of a "one-hole context" for an algebraic data type.

Here's an example. Let's define the data type of "letter trees" recursively, as follows. A letter tree consists of either

• a letter paired with two more letter trees (the branches), or
• just a letter (a leaf node).

We can write this definition in the form of an equation, using multiplication to indicate the pairing operation (equivalent to a cartesian product), and addition to represent the "either/or" (equivalent to a disjoint union). The definition, in that form, looks like this:

$$T = L T^2 + L.$$

Now, here's a question. What would the data type of "missing-letter trees"—trees with exactly one letter missing—look like?

It's easy enough to write out this definition explicitly, by reasoning through it. In the case of a branching missing-letter tree, the missing letter could be further down either branch, or it could be at the junction. In the case of a leaf node, the missing letter must be at that node.

A missing-letter tree consists of either

• a letter paired with one missing-letter tree and one letter tree (the missing letter is in the first branch);
• a letter paired with one letter tree and one missing-letter tree (the missing letter is in the second branch);
• two letter trees only (the missing letter is at this junction, and both branches are complete letter trees); or
• no information (this is a leaf node and the missing letter is here).

If we write out this definition as an equation, using $T'$ to denote the type of a missing-letter tree, the equation we get is:

$$T' = LT'T + LTT' + T^2 + 1,$$

or, equivalently,

$$T' = 2LTT' + T^2 + 1.$$

But notice that this equation is exactly the derivative of the equation for $T$ with respect to $L$!

This "trick" works for any algebraic data type. I don't know why it does, but I think it's quite interesting.