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.