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

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1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are displayed: in blue color the graph of function f. The tangent line (in red) 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 bottom slider 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 and use it 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 on derivatives:

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.

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 behaviour of the tangent line close to a vertical asymptote.