I think this switcheroo, where you start thinking about the derivative as a function, is one of the trickiest parts of Calc 1. First you look at the tangent line, then you abstract away the line to just its slope, and then you treat just that number as a function of the point the tangent line went through. It's several levels of abstraction. The more natural thing to do would be to think of a function from points to lines, but then it's tricky because lines are described by two parameters and both are changing! In particular, I think not really wrestling with how tricky and confusing this step is leads to a lot of pain down the road in multivariable calculus.
The graphs on the bottom of page 6 ("Therefore, a plausible graph for f'(x) could be something like so") and at the bottom of page 7 ("This happens in the graph of the absolute value function, f(x)=|x|") won't load properly for me -- they are cut off at the bottom. Is anyone else having this problem?
Thank you for this series, heebie!
3: I was just popping in here to report the same thing -- some of the pictures at the bottom of pages get cutoff. (I suspect because they're not being allowed to relocate to the next page instead.)
Oh no! I changed the setting on those two pictures - did that fix it?
The other thing I could be doing is saving it as a PDF and linking to that, although it makes edits slightly more tedious.
2: say more about how it affects Cal 3? I taught Cal 3 this past semester and was really surprised at how much they struggled with the geometry that had nothing to do with calculus.
I always had trouble with rhomboids.
8:. They are supposed to be loyal, but they always wind up turning against us.
8: Try some Preparation R. If that still doesn't help, maybe you can get into the trials for Preparation R Shiny.
Well it's abusing the fact that a line through a point in the plane is a number, but in higher dimensions you have a tangent plane (or tangent space) and you can't just think of it as a number anymore.
Another confusing point is that you can think of the tangent line itself as a function! But that function is unrelated to the derivative thought of as a function.
And then if you want to get really confused, start thinking about what the variable is in the function describing the tangent line! Is it the same x as before?
Eagerly awaiting Unfogged's crossover series "Conversational Calculus for Ekranoplan Physics and Navigation."
https://thepointsguy.com/news/hawaiian-airlines-sea-gliders/
and the manufacturer:
Though it looks like they are designed to travel a little more than half as cast as our favourite Caspian Sea Monster. I imagine calculus can explain that, too.
I've been finding it easier to start teaching about integration. You start with a constant acceleration, then you get a straight line velocity. The longer you go with your foot mashed on the pedal, the faster you go. Then you explain about the area under the curve which is less abstract than the slope. What is the area under a straight line with a given slope? It's the area of a triangle, and kids already know the formula for the area of a triangle or they can derive it graphically. (I do this on graph paper.) Plot it out, and you get a parabola, surprise, surprise.
Slope is much harder to explain than area, especially when the line curves. My go-to is usually to fire up a drawing program and show them how to play with splines.
It's time for a mathematician joke: "Daddy, what is area?" "Well, first we have to define sigma algebras. They're kinda like topologies, but closed under countable union and intersection instead of arbitrarily union and finite intersection. Anyway, a measure is a ..."
What is the area under a straight line with a given slope? It's the area of a triangle, and kids already know the formula for the area of a triangle or they can derive it graphically. (I do this on graph paper.) Plot it out, and you get a parabola, surprise, surprise.
I have students do this exact exercise as a lead up to the FTOC. It blows their mind (partly though because they've seen derivatives in the traditional order, so it feels like magic that they've recovered x^2 from 2x.)
I might have to show my son this, but never mention how I came across this. He's just learning limits. I learned that kids don't always get it when explain 'e' and say "Euler, Euler, Euler" in a monotone.
I mentioned something about "stupid pet tricks" at work and got blank stares, so I tried "David Letterman?" Nothing.
David Letterman starting his show is closer in time to Steve Allen hosting the Tonight Show than to now.
People should show some respect and not have a stroke unless they are more than two years older than me. I might have to vote Lamb.