It’s the same as with number e. What is special about it? Why do we use it for powers and not the simpler 2^x or 10^x, or as basis for the logarithms? Because is the exponential whose derivative is the same function.

]]>My vision, shared by most math teachers, I would hope, is that proof would be a central part of every math course. Mathematics is different than other disciplines and the thing that makes it different is that mathematics is a *sense-making* discipline. It’s only in math class that you don’t have to answer to authorities–you can verify results yourself. And, in a truly ideal world, a student would encounter *no* unjustified statement in math class.

If math class isn’t logical and doesn’t make sense, what exactly *is* math class offering? This is our entire business, I say.

I should’ve said in my last comment I use basically your first diagram when teaching Calc I students, so I agree with you that it’s a good way to convince students. The fact that an arc approaches the length of its chord as the sub-tending angle gets small seems pretty intuitive for most people. But the more engaged students usually than ask how we know it’s true, which then requires something equivalent to .

It occurs to me a pretty similar proof is to draw a vector tangent to circumference at with magnitude and call it the derivative of sine at 0. Than it’s easy to see that this magnitude must be the same all around the circle, with just the direction changing. And so the derivative must go as . And it’s easy to convince yourself that s depends on your units for the angle, and so you can find units for which the derivative of sin(x) = cos(x). Though it remains to be shown that the angular units so desired are radians.

]]>I do think it’s important to learn as its own theorem. It says something deep about the sine function and codifies what students already know by looking at the sine function near the origin — the slope of the tangent line @ is 1. I used this fact over and over again in my geometric proofs above.

To establish that , you can just draw a picture of the sine of a very small angle, which will perhaps satisfy 95% of people and bother 5% of people. For those that are bothered, there is the more rigorous proof via the squeeze theorem, but as I mentioned above, that also depends on some geometry, arguably some lower-level and therefore more obvious geometry, but it’s still somewhat involved. Perhaps now, 99% of students will be convinced.

So if you insist on proving “rigorously” via the squeeze theorem, you’re targeting only those students who find the naive geometric proof unconvincing and the complicated geometric proof convincing.

What I think is unforgivable though, is the abandonment of geometric reasoning in the actual proof of the derivative of sine. As you can see from the above diagram, the proof basically writes itself once you set up the picture. For this one, at least, no knowledge of is even required. You just use the definition of sine and the definition of the derivative and get to work.

By contrast, the algebraic proof of the derivative of sine is boring, tedious, challenging, and unilluminating. And the proof depends crucially on , which was proven geometrically, so you haven’t really gotten rid of geometry, you’ve just locked it away in the basement.

]]>Which in turn is pretty close to .

]]>These types of identities pop their heads up in introductory calculus and precalculus, and it seems as educators (that usually had a lot of math classes) we have been trained in logic and reasoning, so requesting that students prove or verify a statement (especially in this algebraic manipulation setting) seems like a straight forward task that needs little to no explanation. But, in my personal education logic and reasoning was something I never gave much thought to (especially in a calculus type class) and it wasn’t until later in my education that I learned (and fell in love with) logic, as we use the word…

So after that rant, which I apologize for… The question I raise to many of my cohort is: “Should we teach a ‘logic’ course before we expect students to ‘prove’ things or even comprehend conditional statements as many of the theorems in calculus involve?”

Or (as it feels like it is handled now) do we treat logic like it is something born to each person, that has little to nothing said in the classroom?

[I’m just a low level graduate student at a state school, and I’m sorry if I missed something in your post or others comments that answered this, or if this comment is offensive in any way to professionals as yourselves]

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