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.

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