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.
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.