Related Rates: Filling a Container

As a container of any size or shape is filled with a liquid, the depth of the liquid increases as the volume of liquid increases and both quantities are changing with the passage of time.  This GeoGebra Worksheet invites the student to consider how the rate of change of the depth can vary as the shape of the container changes.  The student can design his or her own container by defining its radius as a function of the variable along its vertical axis.  Then an animation can be run to simulate the process of filling the container, complete with time series graphs for its depth (h) and the rate at which its depth is changing with respect to time (dh/dt).

L'Hopital's Rule Visualization

One case of L'Hopital's Rule states that as if two differentiable function f and g both approach 0 at a number c, then the limit of their ratio is the same as the limit of the ratio of their deriviatives.  The goal of this GeoGebra Worksheet is to give the viewer a good framework for comparing the slopes of the tangent lines to the ratio of the values of the two functions as the points approach the mutual x-intercept.