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

When the value of a function changes by a (perhaps) small amount delta x, there is a corresponding change delta y in the value of the function.  When the derivative of the function is known, the value of delta y can be approximated by value of dy, a value obtained by the tangent line approximation to the function at the point (a, f(a)).  This GeoGebra Worksheet enables the user to compare the values of delta y and dy for different points on the graph and different values of delta x.  By defining the settings of the graphic window in terms of delta x and delta y, the key aspects of the graph stay largely in place when the sliders are moved, so accurate comparisons between dy and delta y can more easily be made.

The Sampling Distribution

This GeoGebra Worksheet enables the student or instructor to create multiple examples of sampling distributions of different sizes.  Students can instantly compare the shape of the sampling distribution with that of the population.  Also, for a given sample size, multiple 100-sample collections are available, so that students can compare their standard deviations with that which would be predicted by the formula.

Poisson Probability Simulation

Inspired by the article "What does randomness look like?" by Empirical Zeal, I designed
this GeoGebra worksheet to generate random distributions similar to the Poisson probability distribution.  Select the size of the grid and the number of dots to be selected at random on the grid, and the spreadsheet counts the number of squares which receive 0 dots, 1 dots, 2 dots, etc.  Compare the results with what would have been predicted by the Poisson distribution.  Then try it again with a different sized grid or a different number of dots.  Careful!  Large grids with a lot of dots may take a long time to process.

The Domain of a Logarithmic Function

This GeoGebra worksheet illustrates the connection between the argument of a logarithmic function, its domain, and its vertical asymptote.  Move the sliders to change the base, the argument, and the constant outside the argument.  Watch the changes that take place in the domain and the vertical asymptote.

Calculating and Visualizing Binomial Distributions

Students and teachers alike can use this GeoGebra worksheet to compare binomial distributions by instantly changing the value of n or p.  Then specific calculations can be made using particular values of x.

A Manual Approach to Finding the Line of Best Fit

When trying to find the equation of the straight line which best "fits" a set of points in the plane, one approach is to find the line which minimizes the sum of the squares of the vertical distances between the y-coordinates of the data points and the y-coordinates of the corresponding points on the line.  This so-called "least squares" approach produces the "linear regression equation" whose graph is called the "line of best fit."  Precise formulas exist for computing the slope and y-intercept of this line, and graphing calculators can handle it in a snap.  But for students being presented with the concept for the very first time who may be having a hard time seeing exactly what is going on, this GeoGebra worksheet is a manual, yet very visual approach.  As they move sliders or input specific guesses as to what they think the slope and y-intercept of the linear regression equation might be, the graph of the line they produce gives them instant feedback.  When they get close enough to the actual line, a congratulatory message appears with the correct answer.  If  they need a hint, either the slope or the y-intercept can be displayed, greatly simplifying the task.

The Archer and the Discriminant

This GeoGebra worksheet demonstrates a visual application of the discriminant.  When friction is neglected, the path of an arrow can be modeled using a quadratic function.  GeoGebra randomly generates an initial velocity (the vertical component), an initial height, and a target height.  Then it calculates the discriminant of the resulting quadratic equation and asks whether or not the arrow will reach the desired height.  The arrow is then fired so that we can see if our prediction is correct.

A Classic Mixture Problem

Today's GeoGebra Worksheet can be used to generate numerous different versions of a classic mixture problem.  The unit prices of two different blends of coffee, the total weight of the mixture, and the unit price of the mixture can all be adjusted with sliders.  GeoGebra then calculates the amount of each blend which should be used to obtain the correct amount of revenue.  An interesting connection can be made between the ratio of the blends in the final answer and the position of the unit price of the mixture relative to the unit prices of each of the blends.

Use the derivative to determine whether f is increasing, decreasing, or both.

Here is a short sweet GeoGebra worksheet to help students see the connection between the graph of y = f'(x) and the location of intervals where f is increasing or decreasing.  The worksheet randomly generates 3rd-degree polynomial functions and displays the parabolic graphs of their derivatives.  Students then decide whether the unseen original function f is increasing everywhere, decreasing everywhere, or increasing on some intervals and decreasing on others.  It usually does not take long to exhaust the three possible cases which are presented here, but as an added bonus, one of course can emphasize that the real zeros of the derivative (if there are any) correspond to values where the relative extrema of the function f may occur.

Color the solutions to systems of linear inequalities.

It can take a little while to sketch an accurate graph of the solution region of a system of linear inequalities with pencil and paper.  This GeoGebra Worksheet gives the user rapid feedback on whether or not an ordered pair satisfies one or both linear inequalities in a system.  As the point's color changes, the location of the solution region is revealed.  When the tracing feature is used, dragging the point around the screen actually shades the solution region in blue and shades the regions where only one or no inequalities are satisfied in either green or red. Finally, the boundary lines themselves can be toggled off so that for new systems, the solution region can be found in somewhat of a game of hide and seek.  Give it a try!

Visualize the solutions to absolute value inequalities and equations.

Given their apparent simplicity to those who teach such things, absolute value equations and inequalities (the linear variety) have needlessly confounded a disproportionately high percentage of my College Algebra students throughout the years.  In keeping with my ongoing quest to find the key to unlock the mystery which enshrouds this curiously impenetrable topic, I offer this GeoGebra worksheet.
As you use sliders to adjust the values of a, b, and c in either the inequalities |ax - b| <= c and
|ax - b| >= c or the equation |ax - b| = c, GeoGebra dynamically displays the graphs of y = |ax - b|, y = c, and their points of intersection.  And it shades the solution set on the x-axis while simultaneously displaying the solution set using interval notation or listing notation as appropriate.  Designed for use in the classroom, it also provides the enterprising student with a limitless collection of do-it-yourself examples, complete with solutions.  Both the shading along the x-axis and the solution set notation can be turned on and off, so practice problems can be created and attempted before the instantaneous feedback is given.

Find the hidden conic section.

We want our students to make connections between certain relationships that exist among the coefficients of the equation of a conic section and the type of conic section that results. This GeoGebra worksheet can be use to generate numerous examples of different conic sections quickly.  When the graph is displayed, students can see how the graph changes dynamically in response to the changing coefficients.  When the graph is turned off, students can be challenged to predict the type of conic section that will result.  Then, they can test their prediction and search for the conic section by moving the point A to reveal the graph in yellow.

Find the minimum cost of a can.

In the classic problem of minimizing the surface area of a cylinder or minimizing the cost of constructing a can subject to a constraint such as fixed volume, students sometimes have a tough time visualizing cylinders of varying heights and radii and imagining that they all can have exactly the same volume.  This GeoGebra worksheet enables the student to explore the dynamic relationship between the cost and the dimensions of a can.  Also, by adjusting the unit cost of the top and bottom of the can independently of the unit cost of the lateral side, students can see the dynamic effect on the cost function, its minimum value, and the resulting dimensions of the can of minimum cost.