Derivatives of Polynomial Functions
Explore graphically and interactively the derivatives as defined in calculus of third order polynomial functions.
A third order polynomial function of the form
f(x) = x 3 + ax 2 + bx + c
and its first derivative are explored simultaneously and interactively in order to gain deep analytical and graphical meanings of the concept of the derivative. This interactive tutorial assumes that you have some knowledge about functions and their derivatives and the tangent line to the graph of a function.
1 - Click on the button "plot". Three graphs are shown: the graph of the polynomial function f(x) = x 3 + ax 2 + bx + c , in blue, where the parameters a, b and c can be changed in the text boxes above. In black color the tangent line to the graph of f and in red the graph of the first derivative f ' which is drawn as the position of the tangent line is changed using the red button bottom along the green line.
2 - Use the red button, at the bottom, to change the position of the tangent line (black) so that it is at local maximum or local minimum of f(x). Note that the slope of the tangent at such points is close to zero (zero in theory). What is the value of f '(x) at these maximum and minimum points?
3 - Change parameters a, b and c included in the definition of the polynomial above and repeat the same activity of positioning the tangent line at a local minimum or maximum. Take note of the slope and the value of f'(x) at these points and experiment with as many values of the parameters a, b and c as possible.
4 - Change parameter c to different values; does parameter c have any effect(s) on the derivative or the tangent line? Explain.
5 - The first derivative of f(x) = x 3 + ax 2 + bx + c is given by
f '(x) = 3 x 2 + 2 a x + b
which is a quadratic function. The points at which f '(x) is equal to zero are found by solving the quadratic equation
3 x 2 + 2 a x + b = 0
The discriminant D of the above quadratic equation is given by
D = (2 a) 2 - 4 (3)(b) = 4 a 2 - 12 b
Change the values of parameters a and b such that D is positive (Example: a = 2 and b = 0). Locate the two points where f '(x) is equal to zero (the x intercepts of the graph of the derivative in red). Use the red button to position the tangent line at these two points and note that it is (almost) horizontal. Does f have a local maximum or minimum at these points? Use calculus theorems (if you have studied them already) to explain what you have observed.
6 - Enter values for a and b such that D = 0 (Example: a = 0 and b = 0). The quadratic equation 3 x 2 + 2 a x + b = 0 has one solution and the graph of f '(x) touches the x axis at one point, in fact it is a point of tangency. Does f has a local maximum or minimum at this specific point? Use calculus theorem to explain what you have observed.
7 - Set a and b to values such that the discriminant D is negative (Example: a = 1 and b = 2). The quadratic equation 3 x 2 + 2 a x + b = 0 has no solution. Does f have a local maximum or minimum? Is there a position for which the tangent line is horizontal? Explain.
8 - Set parameters a, b and c so that the discriminant D is positive and therefore f has a local maximum and a local minimum. What is the sign of the derivative f '(x) when the graph of f is increasing? What is the sign of the derivative f '(x) when the graph of f is increasing? Where does the change from increasing to decreasing happen? Where does the change from increasing to decreasing happen?
9 - Set parameters a, b and c so that the discriminant D = 0. Does the sign of f'(x) change? Is there a local minimum or maximum? Explain.
10 - Change parameters a, b and c and explore f and its derivative f ' till you understand fully the graphical and analytical properties. e-mail me if you have suggestions on how we can explore other properties of graphs of functions and their derivatives using the above applet.
More References and Links First and Second Derivatives Theorems .
Derivatives of Quadratic Functions . The derivative of quadratic functions are explored graphically and interactively.
Derivatives of Sine (sin x) Functions . The derivative of sine functions are explored interactively.
Derivative of tan(x) . The derivative of tan (x) is explored interactively to understand the behavior of the tangent line close to a vertical asymptote.
Vertical Tangent . The derivative of f(x) = x 1 / 3 is explored interactively to understand the concept of vertical tangent.