Cubic growth and beyond

Unlimited growth: Linear and quadratic growth

Cubic growth and beyond

For cubic growth of a quantity $y(t)$ at time $t$ :

the second derivative of quantity $y(t)$ changes with the same value per unit of time ;
$y(t)$ can be described by a cubic function $y(t)=a\cdot t^3+b\cdot t^2 + c\cdot t +d$ , where $6a$ is the increase or decrease of the second derivative $y''(t)$ per unit of time, $2b$ the value of the second derivative at time $t=0$ ( $y''(0)=2b$ ), $c$ the value of the first derivative at time $t=0$ ( $y'(0)=c$ ) and $d$ the initial value at time $t=0$ ;
the corresponding graph of $(y,t)$ is a third-degree polynomial function;
the third derivative of the quantity $(y,t)$ is a constant.

The first two properties of cubic growth mean that the second derivative $y''(t)$ is a linear function equal to $6a\cdot t + 2b$ . This expression is the formula of a straight line, so $y''$ is s straight line. The first derivative $y'(t)$ is a quadratic function, and equal to $3a\cdot t^2 + 2b\cdot t+c$ . This expression is the same as the formula for the parabola that represents the graph of $y'$ .

So cubic growth of a quantity $y(t)$ is described by: $y'''(t)=6a$ , for some constant $a$ . This is a differential equation, because it has a derivative. In this case, the third derivative plays a role; in mathematical jargon, this is a differential equation of order three.

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Of course, we can continue like this for a while: linear growth, quadratic growth, cubic growth, quartic growth, ... A further generalisation of the dynamic model of linear growth is the differential equation $y'(t)=f(t),$ where $f$ is a function. If a function $F$ is known such that the derivative of $F$ equals $f$ (so $F$ is the antiderivative of $f$ ), then the general solution of the differential equation is given by $y(t)=F(t)+c$ for some constant $c$ . So the solution is obtained by finding antiderivative of $f$ , i.e., a function $F$ with the property that $F'(t)=f(t)$ for all $t$ . Finding antiderivatives is also called integrating functions. Integrating functions is often required for solving other types of differential equations. The graph of a solution of a differential equation is therefore often referred to as an integral curve.