An article in ScienceNews reports that a mathematician at Hope College observed a curious phenomenon: his dog seems to know calculus.

Tim Pennings (is/was) the chair of the math department at Hope College. According to the tribute website his colleagues made him for his birthday, he an I have a few bits in common:

40. Wrap your bike lock chain around your bike. Take care to NOT actually lock the bike.
...
28. Whistle a show tune.
...
24. Break into song. [Okay, so slightly different songs...]
...
9. Over-analyze a holiday performance of "Sleigh Ride."
8. Argue passionately about anything.

... thankfully only a few bits in common. Anyway, back on subject. He observed that his dog, Elvis, seems to be following Fermat's Principle quite well. Fermat's Principle is a principle in optics that states that the path traveled by a light-beam would be the local minimizer of distance. It can be used to derive Snell's Law of refraction, and to explain why light reflections follow the principle that angle of incidence is equal to the angle of reflection.

When playing fetch at the beach, Pennings noticed that, when he stands on the beach and throws a ball into the shallow waters, Elvis does not run directly to the ball, nor does he run to the point on the beach closest to the ball and swim out from there; Elvis would run along the beach for a while, and then cut across the water at an angle. In a way, he follows Snell's Law! A dog can run more quickly on land than swim in water, so the beach has a lower index of dog refraction than does the water. Elvis's behaviour of traveling at a shallow angle (relative to the shoreline) on land and a steeper angle in water reflects an innate ability to optimize his path toward the target.

Furthermore, if Pennings stands in the water and throws the ball to some other point in water, he notes that rather than swimming directly to the ball (a local minimizer of travel time), Elvis would swim to shore at an angle, run a bit along the shoreline, and hop back in the water to swim to the ball (a global minimizer of time), in the same way that we, humans, sometimes choose to, instead of driving the small, direct country road between two points that has speed limit of 25mph, drive first out to a state highway that allows a speed of 50mph and take an exit near the destination to get back on the small country road. We choose to minimize travel time, rather than distance traveled.

There are some disagreements as to how Elvis makes the decisions: some people claim a simple evolutionary model in which Elvis makes the decision point-wise by traveling, at every point, in the direction that would offer the largest speed of approach toward the target. Though the model describes in fact the solution to the Euler-Lagrange equation corresponding to Fermat's Principle, it would not guarantee a global time minimizer, in seeming contrast to Elvis's behaviour when the ball is thrown from a point in water to another point in water (in contrast, light would not avoid the straight-line path between two points in a cube of ice). It is quite possible, given the evidence, that the dog plots out a simple recipe for traveling (whether to take a land-route or a water-route) while allowing for small perturbative corrections based on the local speed maximization.