A third order polynomial function of the form
f(x) = x 3 + ax 2 + bx + c
and its first derivative are explored simultaneously in order to gain deep understanding of the concept of the derivative and also its graphical meaning.
1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are shown: in blue color the graph of the polynomial function f. In red color the tangent line to the graph of f and in black color the graph of the first derivative f ' which is drawn as the position of the tangent line is changed using the bottom slider ("Change Tangent Position").
2 - Use the slider at the bottom of the left panel to change the position of the tangent line (red) so that it is at local maximum or local minimum. 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 maxima and minima?
3 - Use the three sliders on the top left to 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.
4 - 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
Use the top sliders to choose values of the 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 in black color). Position the tangent line at these two points and note that it is (almost) horizontal. Does f have a local maxima or minima at these points? Use calculus theorem (if you have studied them already) to explain what you have observed.
5 - Use the top sliders so 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.
6 - 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.
7 - Does parameter c have any effect(s) on the behavior of the derivative or the tangent line? Why or Why not?
8 - Set parameters a, b and c so that the discriminant D is positive so that f has a local maximum and a local minimum. What is the sign of f'(x) when the graph of f is increasing? What is the sign of 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?
10 - Change parameters a, b and c and explore f and its derivative f' till you understand fully their properties. e-mail me if you think there are other properties we can explore using the above applet.
More on derivatives:
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.