myrmepropagandist
Exactly. A “rate of change” is a slope. 20miles/hour is a line with slope 20/1 because, for each 1 hour you go forward in the x-direction, your total total distance goes up by 20 miles in the y-direction.
A tangent is a straight line … again with a slope. So “the slope of a curve at a point” (curves being precisely those lines with non-consonant slopes) we use the tangent to define it in a way so that our straight-line idea of slope works.😄
Robert Link
@futurebird @Infrapink @jacquiharper A tangent is a straight line?
https://mathbooks.unl.edu/PreCalculus/tangent-and-cofunctions.html
Robert Link
@dendari @futurebird @Infrapink @jacquiharper Arguably that's a different use of the word. Of course it's been 30+ years since second semester calculus, which is as far as I got.
"tangent" is defined:
Given an angle θ (in either degrees or radians) and the (x,y) coordinates of the corresponding point on the unit circle, we define tangent as
tan(θ)=y/x
Resulting in this curve:
Dendari
@phaedral @futurebird @Infrapink @jacquiharper
A tangent is a straight line that touches a curve on exactly one point. The slope of that tangent is then considered the slope of the curve at that exact point.
Robert Link
@dendari @futurebird @Infrapink @jacquiharper Dendari: I see you /teach/ math, so maybe you can help dispel my error.
Robert Link
@futurebird @Infrapink @jacquiharper Maybe I misunderstood your post. If so, apologies. I thought you were offering 20mph as an example of rate of change.
myrmepropagandist
@phaedral @Infrapink @jacquiharper
"20mph isn't a rate of change; it's a velocity, unchanging."
I don't know what you are getting at with this.
Robert Link
myrmepropagandist
@phaedral @Infrapink @jacquiharper
It's a rate of change.
The distance changes 20 miles for each 1 hour.
It could be instantaneous, and just the rate at a particular moment on a nonlinear function or it could be the constant rate of change of a linear function.
Robert Link
@dendari @futurebird @Infrapink @jacquiharper I'm totally content with "different tangent," that's sort of what I meant earlier. This has been a /delightful/ exchange!!! Thank you all!!
Dendari
@phaedral @futurebird @Infrapink @jacquiharper this is the tangent from trigonometry not the same tangent.
Futurebird is likely a much better math teacher than I am. I haven't done anything beyond 8th grade algebra in years.
Robert Link
@futurebird @dendari @Infrapink @jacquiharper Futurebird: I've got a /visual/ that I cannot articulate, for how the line at a tangent to the semi-circle could be used to re-create the trig tangent curve...but if that tangential line is tangential to a curve other than a semi-circle...uh... B^)
myrmepropagandist
@dendari @phaedral @Infrapink @jacquiharper
The crazy thing is... in a way it *is* the same tangent, or at least for a circle.
Most people know that you can plot a circle using:
x=cos theta
y=sin theta
And let theta go from 0 to 2pi (or 0 to 360 if you like)
Because the x and y co-ordinates of a circle are parametrized by the sin and cos.
Tangent has a geometric meaning here too. And this is why it increases without bound as theta gets close to pi/4 (90 degrees)
Alexandre Oliva (moving to @lxo@snac.lx.oliva.nom.br)
Robert Link
@llewelly @futurebird @Infrapink @jacquiharper I think when I read "rate of change" I assume the topic is "rate of change of the curve." However, that assumption was proved wrong by OP. Yes, 20mph is "a" rate of change. It is not, however a rate-of-change-of-a-curve, which is one of the mathematical questions I think calculus was invented to answer...the absurdity of "change over zero time."
llewelly
@phaedral @futurebird @Infrapink @jacquiharper
1/2
When traveling 20 miles per hour, your position is changing at the rate of 20 miles per hour. That's a rate of change in your position. A rate of change in position is the first derivative of position, also known as velocity. Velocity, in turn, also has its own first derivative, called acceleration, which is also the second derivative of position.
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