A cubic function is a polynomial function of degree 3.
THE CONCEPT OF DERIVATIVE OF A FUNCTION
The derivative of a function at a point can be defined as the instantaneous rate of change or as the slope of the tangent line to the graph of the function at this point. We can say that this slope of the tangent of a function at a point is the slope of the function.
The slope of a function will, in general, depend on x. Then, starting from a function we can get a new function, the derivative function of the original function.
The process of finding the derivative of a function is called differentiation.
The value of the derivative function for any value x is the slope of the original function at x.
To find the derivative at a point we can draw the tangent line to the graph of a cubic function at that point:
But how can we draw a tangent line?. We can use a magnifying glass!. If we look very near the point in the graph of the function we can see how the function resembles the tangent line. This tangent line is the best linear approximation of the function at that point:
Then we can draw a parallel line to this tangent line through the value x-1 and we get a right triangle:
The derivative of a cubic function is a quadratic function.
A critical point is a point where the tangent is parallel to the x-axis, it is to say, that the slope of the tangent line at that point is zero.
In the following example we can see a cubic function with two critical points. One is a local maximum and the other is a local minimum. In these points, the derivative function (a parabola) cut the x-axis:
These critical points are points where the function stops increasing or decreasing (some times they are called "stationary points"). At these points, the tangent is horizontal.
To find the stationary points we solve the quadratic equation:
In this case, solutions of this equation are:
As we already know (quadratic functions), sometimes a quadratic equation has no real solutions. (the parabola does not cut the x-axis). Then the cubic function has no critical points:
But a parabola has always a vertex. The vertex of the parabola is related with a point of the cubic function. We call this point an inflection point.
An inflection point of a cubic function is the unique point on the graph where the concavity changes
The curve changes from being concave upwards to concave downwards, or vice versa
The tangent line of a cubic function at an inflection point crosses the graph:
To find the inflection point we can calculate the vertex of the parabola:
This is an example of an inflection point of a cubic function without critical points:
The inflection point in this case is also an stationary point (the vertex of the derivative touches the x-axis):
Inflection points may be stationary points, but are not local maxima or local minima
One simple and interesting idea is that when we translate up and down the graph of a function (we add or subtract a number from the original function) the derivative does not change. The reason is very intuitive and we can play with the following version of the mathlet to see this property. When you move the violet dot you are translating up and down the graph and the derivative is the same:
It is important to notice that the derivative of a polynomial of degree 1 is a constant function (a polynomial of degree 0). The derivative of a polinomial of degree 2 is a polynomial of degree 1. And the derivative of a polynomial of degree 3 is a polynomial of degree 2.
When we derive such a polynomial function the result is a polynomial that has a degree 1 less than the original function.
When we study the integral of a polynomial of degree 2 we can see that in this case the new function is a polynomial of degree 2. One degree more than the original function. And the integral of a polynomial function is a polynomial function of degree 1 more than the original function.
These results are related to the Fundamental Theorem of Calculus.
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.