Calculus For The People
geogebra.org
Calculus For The People
1–10 of 78 posts
Re: Calculus For The People
#2"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)"
Huh?
"and therefore, the slope of the line very closely matches the growth rate of the function as well."
Growth rate? What's that?
"Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. "
Not sure what this means...
"For instance, in this situation we can study the limit of the slope of g when h tends to 0. As we can see, the limit of the slope of g as h tends to 0 is 4."
Wait... where is this 4 coming from?
"From this, we can conclude that the growth rate of the function f at x=2 is 4."
What the hell is growth???
"Sometimes limits are obvious like this one"
And now I give up.
Re: Calculus For The People
#3"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can…
To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)).
To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So, (x+h, f(x+h)) gets closer and closer to (x, f(x)), the slope of the line passing through those two points tends closer to the tangent line passing through (x, f(x)).
In other words, we are identifying the limit of the slope as h approaches 0.
Based on the points of confusion you mentioned, I recommend a refresher on algebra. I think that will clear up your confusion
Re: Calculus For The People
#4"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can…
To find the slope of a line tangent to a point (x, f(x)) on a line, you can "draw" a secant line through two points (x, f(x)) and (x+h, f(x+h)). Then, identify the slope of the line passing through these two points. This gives an approximation of the slope of the tangent line passing through (x, f(x)). To get a more and more accurate approximation, you can look at what value the slope tends to as h approaches 0. So,…
I think he is pointing out that this “for the people” tutorial uses math jargon without introducing it.
Re: Calculus For The People
#5My favourite resource for the introduction to Calculus is "Calculus Made Easy" by Silvanus P. Thompson[0].
Re: Calculus For The People
#6"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can…
Quoting the prerequisites:
“On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
Re: Calculus For The People
#7A common failing of these "for the people" guides is that they fail the most basic UX test: User observation.
When you build a UI, at some point you have to test it on actual people, observing them as they try to use it. Without fail, you'll discover a whole bunch of assumptions you'd made without even realizing it. It's only natural, since you've been working on this project for months and have intimate understanding that you've gained during your time of designs and rewrites and refactorings. But your user doesn't have that history, and you can't remember where common knowledge ends and your assumptions begin anymore. So you do observation tests to expose as many of these as you can.
If you want to make a "for the people" instructional site, it's imperative that you offer an easy feedback mechanism so that people can instantly tell you when something confuses them. Simply relying on success stories exposes you to survivorship bias. Understanding is a two-way street. Design your medium with that in mind, and do LOTS of iterations with real people.
Re: Calculus For The People
#8"You might notice that when h is very close to 0, the slope of the line very closely matches the graph of f(x)" Huh? "and therefore, the slope of the line very closely matches the growth rate of the function as well." Growth rate? What's that? "Notice that as h gets close to zero, the secant line almost perfectly matches the growth of f at point A. " Not sure what this means... "For instance, in this situation we can…
Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
Re: Calculus For The People
#9Earlier quoted context omitted.
Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
It's using a naive, informal notion of those. If you were to define it formally, well, you'd have the derivative. Which is what he does quite soon after. This is how definitions frequently work in mathematics -- they're meant to take some naive informal notion and formalize it, by coming up with a formal definition that matches how it should work.
So, it's assuming you already have some informal notion of growth rate in your head, like being able to talk about the velocity of an object even when that velocity is not constant. (Imagine the x coordinate is time, and the y coordinate is position (we'll work in one spatial dimension here); then the "growth rate" is velocity.) Then it discusses how to define this formally.
Re: Calculus For The People
#10Earlier quoted context omitted.
Quoting the prerequisites: “On the matter of prerequisites, this book assumes you are competent, if not a Jedi, at basic algebra and arithmetic. Specifically, an understanding of lines, their equations, slope, y-intercepts, x-intercepts, and so on is more or less assumed. I think this is reasonable.”
And I do understand those. What I don't understand is growth rate and the slope of a function that is a curve, not a line, therefore not a slope.
If it's a curve, that means that the growth rate of the function is itself changing. And using the approach of finding slopes of lines incrementally closer to the tangent line through a point, you can naturally identify the value that the slopes approach. This is the most basic way to demonstrate taking a limit