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Skip to 0 minutes and 14 seconds So we’ve had a good look at linear, quadratic, and inverse relations, and now we’d like to extend our view to more general powers. And the natural place to start is to think about cubic relations. It’s a topic that goes back to Newton. It’s interestingly connected with some fascinating geometry. It actually has important applications to modern cryptography. And it also connects very strongly to biology. It will also give us a chance to have a look at the shape of polynomial relations more generally. So let’s have a look at some powers of x. The simplest power of x is y equals x to the 0, which is just a constant function 1.

Skip to 0 minutes and 56 seconds y equals x to the 1, or y equals x, is our familiar basic line. y equals x squared is our quadratic relation, in pink. And now we’re going to introduce y equals x cubed. So these are all part of this family of power functions, y equals x to the n. In fact, the inverse function that we’ve been talking about is also of this kind, because that’s the case when n is equal to minus 1. So what does this y equals x cubed curve look like? Well, when x is bigger than 1, it grows a little bit faster than the quadratic, and when x is much bigger, it grows much faster.

Skip to 1 minute and 35 seconds However, when x is between 1 and 0, it hugs the x-axis a little bit more closely. So x cubed will be smaller than x squared. And so when we get to negative values– this is an odd function, so the values become negative, and it’s symmetrical around the origin there. So this is an interesting function that really took the stage only around the 17th century. It wasn’t really studied very much in classical antiquity. But with a Cartesian coordinate system, it’s very natural to ask, well, now that we understand quadratic functions, what about cubics? In fact, the great physicist and mathematician Isaac Newton was very interested in cubics and contributed quite a lot to the study of these things.

Skip to 2 minutes and 21 seconds The most important application of the cubic function is to volume– volume of three-dimensional objects. For example, if we look at a cube of side length l, then its volume is l cubed. On the other hand, its surface area– consisting of those six square sides– will be 6 l squared. The important difference between these two equations is not really the 6. It’s the difference between this exponent 2 and this exponent 3. In one case, we are talking about a quadratic function, like y equals x squared, in the other case, a cubic function, like y equals x cubed. And the crucial thing is that for large values of l, the volume grows much faster.

Skip to 3 minutes and 12 seconds But for small values of l, the l squared term dominates, as you can see from the graph here. So this has a very important application for biology in terms of the biology of a cell. The surface area and the volume of a cell are crucial in terms of the physical, and chemical, and biological properties of it. The cell is actually a factory for creating materials. And so there’s stuff coming into a cell and stuff going out of a cell through the cell membrane. So the surface area of a cell really determines how effective that transmission is.

Skip to 3 minutes and 53 seconds So in order for the surface area to be reasonably big compared to the actual working volume of the cell, the cell actually needs to be very small. That’s why cells are so small, generally, that they’re almost invisible to the human eye. In fact, we have about 40 trillion cells in our human body. So it has a lot to do with the mathematical aspects of these two formulas. It’s interesting to compare the formulas of the cube with the formulas for the sphere. So for a sphere of radius r, the surface area is 4 pi r squared. Again, there’s a square term there, the quadratic dependence on r, while the volume is 4/3 pi r cubed– cubic dependence on the r.

Skip to 4 minutes and 41 seconds And these famous formulas go back to Archimedes. And it’s typical not just of these but of different kinds of shapes. If you have any kind of solid shape and you expand it, you expect its volume to grow cubically while its surface area will only grow quadratically.

Cubic curves I

In this video, we discuss cubic curves and their history, going back to Isaac Newton.

The basic cubic function

The basic cubic function is of the form \(\normalsize y = ax^3\) for some constant \(\normalsize a\). Perhaps the simplest example is the relationship between length and volume for various shapes. Here is a list of some common shapes, along with their volumes and surface areas.

shape volume surface area
Cube of edge length \(\normalsize x\) \(\normalsize x^3\) \(\normalsize 6 x^2\)
Sphere of radius \(\normalsize x\) \(\normalsize \frac{4}{3} \pi x^3\) \(\normalsize 4 \pi x^2\)
Tetrahedron of edge length \(\normalsize x\) \(\normalsize \frac{x^3}{6\sqrt{2}}\) \(\normalsize \sqrt{3} x^2\)

One important application is to biology; the difference between the quadratic aspect of surface area and the cubic aspect of volume has important consequences for the workings of a cell. This explains why cells are as small as they are, and has implications also for metabolic rate, heart rate and even lifespans of animals.

While formulas for volumes and surface areas may appear complicated, the basic pattern is always that volume is a cubic function of linear size, while surface area is a quadratic function.

A colourful and puzzling cube

Speaking of cubes, here is probably the world’s most famous cube, invented in 1974 by the Hungarian sculptor Ernő Rubik:

A 3x3 Rubik's cube Rubik’s cube, By Booyabazooka, GFDL or CC-BY-SA-3.0, via Wikimedia Commons

Let’s look at the volume and surface area of an \(\normalsize n\times n\times n\) Rubik cube.

n volume in \(\normalsize 1\times 1\times 1\) cubes surface area in \(\normalsize 1\times 1\) squares
1 1 6
2 8 24
3 27 54
4 64 96
5 125 150
6 216 216
7 343 294

We see that for small values of \(\normalsize n\) the surface area dominates, but as \(\normalsize n\) gets large, the volume dominates.

More general cubic polynomials

More general cubic polynomials in \(\normalsize x\) and \(\normalsize y\) give degree three curves, sometimes just called cubic curves, or cubics. They are considerably more complicated than the degree two curves, or conic sections, of Apollonius and Descartes. However they also have beautiful properties.

To get an idea for the complexity possible, here for example is the degree three curve \(\normalsize 3x^3+5xy^2−4x^2−10y^2−45x+90=0\):

Graph of 3x^3+5xy^2−4x^2−10y^2−45x+90=0, made of 2 curves - one that loops to the left and one that forms a small closed circle

We see that it is quite different from a conic section, with even a little part of the curve broken off from the main body, forming a little loop all by itself!

Of course when we restrict ourselves to cubic functions, for example of the form \(\normalsize y=x^3-4x^2+x-3\), where there is only one occurrence of \(\normalsize y\) on the left hand side of the equation, then we get a much more predictable situation. We say this is a polynomial function, and it has a rather pleasant graph that rolls up and down:

Graph of y=x^3-4x^2+x-3, which rises, falls and then rises again

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Maths for Humans: Linear, Quadratic & Inverse Relations

UNSW Sydney