Vertical Tangent

The vertical tangent is explored graphically.

Function f given by

f(x) = x 1 / 3

and its first derivative are explored simultaneously in order to gain deep the concept of vertical tangent in calculus.

Interactive Tutorial

1 - Three graphs are displayed: in blue color the graph of function f. The tangent line (in red) to the graph of f and in green color the graph of the first derivative f ' which is drawn as the position of the tangent line is changed using the red button slider along the green line.

2 - Use the red button to move the tangent line close to the point whose x coordinate is equal to 0. What happens to the slope of the tangent line? The tangent line is (or almost) vertical. Calculate the first derivative of f(x) = x 1 / 3. Is f '(0) is defined?
Use the last result to explain what happens to the slope of the tangent line at x = 0 and also to find out if the first derivative has any vertical asymptote at x = 0.

More References and Links

Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.
Derivatives of Quadratic Functions. The derivative of quadratic functions are explored graphically and interactively.
Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.
First and Second Derivatives Theorems.
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.