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.

Difference Quotient versus The Slope Formula

The difference quotient for a function basically gives the slope of the line connecting two points on its graph.  One point is determined by specifying its x-coordinate (x).  The other point is determined by indicating the directed horizontal distance (h) or (delta x) from the first point.  In this GeoGebra Worksheet, students position a secant line so that it passes through these two specific points.  Then they observe that by evaluating the difference quotient using the values x and h, they obtain the same result for the slope of the secant line as could be obtained by using the slope formula to calculate "rise over run."

Explorations with Newton's Law of Cooling

The rate at which an object cools is proportional to the difference between its temperature and that of its environment. 
In this GeoGebra Worksheet you can adjust the initial temperature of the object (u0) and the constant temperature of the environment (T).
Also, you can indicate the temperature of the object (b) after the object has been cooling for (a) minutes.
Finally, observe that the ratio of the current rate of change of the temperature of the object to that of its original rate of change is equal to the ratio of the current difference between the temperature of the object and its environment and the original difference between the temperature of the object and its environment.

Visualizing solution sets to nonlinear (or linear) inequalities

This GeoGebra Worksheet enables students to visualize the solution set to an inequality by dynamically shading the x-axis while viewing the 2-dimensional graph.

Use hyperbolas to locate the point of origin of a loud noise.

This short GeoGebra worksheet is designed to demonstrate how a family of three hyperbolas can be used to pinpoint the location of the occurrence of a loud sound such as an explosion or a lightning strike.  The three points A, B, and C represent locations of devices capable of registering a sound and recording the precise moment in which it was recorded.  Using 1100 ft/s as the speed of sound and expressing the scale in miles, the differences in the times at which the various stations recorded the sound can be combined with the distance between the stations to form unique hyperbolas upon one of whose branches the sound must have originated.  Considering two or three such hyperbolas, it is usually not difficult to pinpoint the sound's point of origin.

Find the area of a triangle using a determinant.

The area of a triangle can be calculated using a determinant of a 3x3 matrix whose entries include the coordinates of the three vertices of the triangle as indicated in the picture below.
GeoGebra Worksheet