I’ve done this activity once before, and I wrote about it here, but I’m continuing to tweak it. In the late 1800s, Eadward Muybridge published several time-lapse photo essays of animals and people in motion. Because he used a background with a grid on it, it is easy to see how much distance the subject covers in between each frame (which are snapped at 0.031 sec intervals). This sets up a great lesson on calculating average velocity, and approximating instantaneous velocity!
While I still asked my students to plot the cat’s position vs. time by hand, we also used the desmos online grapher to plot the data as verification of their work. It was great to see the light bulbs go off in my students’ heads as they worked out difference quotients for smaller and smaller time intervals.
Here’s the plot generated using desmos:
We had an excellent discussion of how the cat’s motion breaks down into two distinct parts, and how the slope of each corresponds to the velocities of the cat walking and running.
Once again, I must thank Dr. Nell Rayburn, Professor and Chair of Mathematics at Austin Peay State University for sharing this activity with other calculus teachers in Tennessee. Here are her original documents: Cat Photos, Muybridge Cat Worksheet, Muybridge Cat Key.
One of my favorite applications of derivatives is solving related rates problems. In this type of problem, two related parameters are changing at different rates over time. For example, throwing a stone in a pool of water creates ripples that expand outward. The rate of change of the area enclosed by the first ripple is related to the rate of change of the radius of the ripple. In this screencast, you can see how calculus helps us solve this type of problem.
This screencast is all about implicit differentiation. We use this technique on functions and relations that are difficult to solve for one variable in terms of another (usually y in terms of x). For example, an equation like is nigh-impossible to solve for y in terms of x. However, using implicit differentiation, the impossible can be accomplished with ease!
We’re getting into real calculus with this topic: Basic Differentiation. Scared of calculus? You shouldn’t be. If you understand that a slope is a rate of change (i.e. miles per hour, etc.), then you’re well on the way to understanding calculus. Algebra deals with average rates of change, whereas calculus enables us to figure out instantaneous rates of change, called derivatives. What’s really amazing is how calculus can find the slope of a curve at a single point! And applying some very simple techniques gives us these derivatives with very little effort. Impossible, you say? Watch below: