![]() ![]() These are all questions that don’t require any derivative rules and focus on graphing and thinking about the meaning and context of the model. Are there any equilibria? Are they stable? Slope fields. Developing the motivation for dP/dt=rP as a model for growth takes time. A single realistic and accessible model can be used for lots of different topics. While they’re usually taught in a fourth semester (aside from possibly a chapter in calculus II) they are full of problems. I think differential equations offer a lot of possibility. Different teachers will be attracted to different models and the resource should supply a lot of choices. For the teacher resources there needs to be not only summaries and solutions, but extensive bibliographies. ![]() There’s a lot of overhead in introducing real, even simple, models from the sciences. Along with those would be some bank of context-less questions that are optional if students want to practice the mechanics. The sets wouldn’t be identical but the each set ought to capture/cover the same ideas. For any given unit there might be five homework sets and a student could choose a biology themed set and work those problems. Ideally the materials would allow a number of different tracks that students could identify with. Premed, bio, computer scientists, physics, engineering and a couple others. ![]() I teach the first couple semesters of calculus and it’s always a diverse set of students. This video is the exception I love it.Īnyway, holla below in the comments if you got anything. It’s like using integration to do simple addition. But those problems are just like the others: contrived. Related rates applications can be used to answer the focusing problem as well as the elevation problem.Ī number of AP Calculus classes have their students make videos with related rates problems. In addition, the camera-to-rocket distance is changing constantly, which means the focusing mechanism will also have to change at just the right rate to keep the picture sharp. The rocket rises vertically and the elevation of the camera needs to change at just the right rate to keep it in sight. Rockets: A camera is mounted at a point so many feet from a rocket launching pad. So far, though, the closest I can get is here: I am thinking that maybe figuring out how a radar gun calculates the speed of a car, especially if it is being used from a moving car, might have something good there. No contrivances, but something where I can point to and say: “this calculation needed to get done and because it was, we now have _.” I am searching high and low for one true real world problem. But when we’re teaching something and hold it up as “calculus in the real world,” I refuse to believe that this is the best we can come up with. I’m not someone who needs a real world application to justify everything I teach. Weird exemplars - I wonder where they got started and why they still hold so much water in every textbook? Because seriously?!, a ladder sliding down a wall - when is anyone truly going to need to know the rate of change of the angle over time? Same with the melting snowball. Like a snowball melting, a ladder falling, a balloon being blown up, a stone creating a circular ripple in a lake, or two people/boats/planes/animals moving away from each other at a right angle. ![]() Supposedly, related rates are so important because there are so many “real world” applications of it. And the book and the Internets aren’t helping me. I’m about to teach Related Rates in my Calculus class. ![]()
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