Investigate each of the four continuous functions in the app. Notice that each of them have a point of interest on the x-axis with a secant line drawn from this point to a dynamic arbitrary point on the graph of the function. Move the arbitrary point to see what happens to the secant line and its slope as the arbitrary point approaches the point of interest. [br][br]Is the function differentiable at the point of interest? If not, why not?
In each of the four cases the derivative does not exist at the indicated point of interest, even though the function is continuous there.[br][br]There are two possible problems in the third and fourth examples the secant lines are approaching a single legitimate tangent line as the arbitrary point approaches the point of interest. However, that line is a vertical line. Since the slope is undefined for a vertical tangent line, the limit that defines the derivative at that point is undefined.[br][br]The problem with the first and second examples is that if we take a one-sided limit of the difference quotient as h approaches zero, then both the left and right limits exist. However, these are two different tangent lines with two different slopes. Again the full limit defining the derivative does not exist. This situation occurs when there is a sharp corner in the graph.[br][br]Although a function must be continuous at a point to be differentiable there, it could be continuous there and not be differentiable there. The two potential problems are that the graph has a single well-defined tangent line at the point, but it is a vertical line, and that the graph may have a sharp corner at the point.