New answers tagged calculus
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Intuitively, the fundamental theorem of calculus states that "the total change is the sum of all the little changes". $f'(x) dx$ is a tiny change in the value of $f$. We sum up all these little changes to get the total change $f(b) - f(a)$.
I elaborated on this explanation here: https://math.stackexchange.com/a/1537836/40119.
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If you want more clearly "undoing" operations, define "stacking" as the following:
Take some $\Delta x$. Chop the curve into rectangles of width $\Delta x$ and height $f(x)$. (There's some leeway as to take $f(x)$ at the left side, right side, middle, minimum, maximum, etc. The rest of my description will be right side.) Now take each rectangle and put the ...
5
My answer also comes from physics.
Say p(t) is the position of an object in the time t. For concreteness, suppose you are traking a truck and the truck is going forward on a road from A to B. (i.e., all derivatives are positive)
It is very natural to graph p(t), and derive it, arriving at v(t), the speed of the truck for each time. After all, the ...
1
Well, for me, the nicest intuition comes from physics. If $F=F(x)$ is some force applied to an object $O$ casuing $O$ to move, where $x$ is its displacement, when the work $W(x)$ produced by that force from $x_0=0$ to $x$ metres is given by:
$$W(x):=\int_0^xF(s)ds.$$
Now, what if we ask what is the rate of change of that work with respect to displacement? ...
4
I gave a similar post to this one touching on this on Math.StackExchange.
Basically, the way I would go about it is to say that there is a very easy way by which one can think of at least Riemann integration (the usual definition given in a "most courses" calculus course) as an inverse of differentiation by construction: that is, the relationship between ...
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(this is from my calculus notes, see page 233 of: http://www.supermath.info/OldschoolCalculusII.pdf)
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Derivative of integral is the original function:
Let $F$ be the integral function of $f$, so that $F(x)$ is the area under the graph from zero to $x$.
For small $h>0$ the difference $F(x+h)-F(x)$ is the area of a narrow vertical strip.
The width is $h$ and height approximately $f(x)$.
As $h\to0$, this means $F'(x)=f(x)$.
If you like thinking in terms of ...
4
When I first began teaching Calculus, I realized that I really didn't understand the Fundamental Theorem. So I looked for something that would help me have a deep understanding, that would also help me help students to see it.
I found a lovely project, which I have modified over the years. Here are links to the pdf and to a .doc version (in which the ...
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You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the derivative -- still undo each other.
Added: The first sentence is a part of what is called “Discrete Calculus” https://en.wikipedia.org/wiki/Discrete_calculus
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My approach:
Lectures
I use a tablet laptop at home, which I can write on with a bluetooth pen. I use to OneNote app to actually write thngs down, though there are many other options.
To do an online lecture, I use Google Meet.
Students (about 50 of them) gather online in the meeting room (a link I send them 15 minutes before the lecture starts).
I ...
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My home teaching setup. Still to be tested.
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My university (I am in Spain) has Microsoft Teams integrated with student accounts, so I use this. My classes are two hours. I open a chat (there are 60-70 students). (Zoom for free only allows 40 minutes and I am not sure it can handle the 60-70 students; a colleague is using the free video mixing software OBS and broadcasting class via Youtube, but ...
1
Well, one idea would be that of shifting to evaluating students through projects. For instance, there could be created a pool of project topics which will be randomly assigned to students. This may be way off the usual written examination, it makes it, however, necessary for the students to process the knowledge they have obtained conceptually and not only "...
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I'm a full-time faculty member at a community college in NJ. I've been teaching for 12 years but have never taught an online course. When our college left for spring break I was left scrambling for how to make use of the tools that I had at that moment: my course notes and textbooks, my laptop, and my iPhone.
With these tools I write, by hand, a short ...
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The case for WeBWorK
tl;dr - use WebWork as an easy way of giving students as much practice solving problems as they can handle. Learn by doing lots of problems with a tight feedback loop. It engages the game-playing, obsessive nature in us.
Main use case - Homework engine
The two biggest features of a VLE/LMS are the presentation of materials and ...
6
I am surprised that a school would effectively say "go figure it out". You ask for "brainstorming"... here are my thoughts..
You haven't quite defines your goal, although I did hear, loud and clear, you'd like 'free' or close to it. Still, there are a number of outcomes.
Live video - I believe there are many options, but in general, a multiuser ...
1
This is best done via an email list. You write up lecture notes using LaTeX, produce a PDF file which you email to all your students. You can ask your students to submit homework to you via email. This way of teaching is far more efficient than teaching in the traditional way or the live online methods, as you can prepare the lecture notes whenever it suits ...
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I gave a presentation to my department about this today. Like you, as I see in a comment, I am also at a CUNY math department. I haven't done all-online classes before, but I've used Blackboard heavily for ~20 years and have had a hybrid (partly online) class for the last two years.
I have access to Blackboard Collaborate. My only cameras at home are
...
6
In some sense this question is waaaay too broad, but it is attracting a useful collection of hints, and it's super topical for thousands of college math instructors (likely to be followed by primary/secondary ones), so here are a few things which I don't see mentioned yet, collated from the far too much time spent on this subject today.
Web/Doc cam. There ...
1
Though I taught electronic maintenance and simple math years ago, I am now a casual student. I actually clicked on the link to this discussion just out of curiosity. I may have some small insight into the problems you are now facing.
I recently found an online course that was free to take if I did not want a certificate. I thought it would help me bridge ...
2
We're on Blackboard as well, here's some workflow's we've been experimenting with to do everything through that interface:
Lectures: Blackboard Collaborate Ultra works pretty well here. Just remember to turn your mike on, and to go into the settings once you've joined the session, click "Notification Settings" and turn on Browser Popup Notifications for ...
22
Instructor at University of Washington here - we were one of the early closures, so I feel like we're starting to get the hang of it. Here's what I'm using:
Zoom: Zoom is similar to Skype, with better support for many-participant calling and additional features. It has a built-in whiteboard you can write or type on, mechanics for allowing students to "raise ...
3
To enhance my last comment, this is what I was talking about with Blackboard Collaborate Ultra. I did a trial session with some of the other faculty yesterday and it seemed to work alright. We come "back" from spring break next week, so I haven't had a chance to try it with students yet. It might be worthwhile to either do a training session with students ...
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Already put two comments but ideas keep coming to me so I'll just package them here. Keep in mind I'm not an educator, I'm just trying to think of practical solutions to the problem as a whole.
Another answer recommends YouTube to upload source material but I feel this might be inadequate interaction. Other alternatives:
Discord
Recently in response to ...
6
Well, since I have also been obliged to teach from home due to CoVID-19 these days, I will describe here a possible solution to your problem.
As a fast and cheap solution - I have been granted no access to any platform, unfortunately - I use the following:
Skype, as a platform to communicate with my students and conduct the major part of the lesson.
...
9
I like You Tube for posting videos. Once you get started it's pretty simple. There are various levels of privacy possible which you can read about. If your school has a convenient way to post videos and you have broadband (we're talking about 1-1.5 Gb files here, do NOT use HD resolution or worse yet the 4k resolution...). Given all that, basically the thing ...
0
I agree with your assertion this issue is not dealt with in many introductory calculus texts for a variety of reasons. There are two rather different concepts of convergence at play:
Convergence of the Taylor series centered at $x_o$ let's say $\displaystyle \sum_{n=0}^{\infty} a_n(x-x_o)^n$. The Interval of Convergence (IOC) is the set of all $x$ for which ...
1
Functions whose Taylor series converge to the original function are called analytic. How do you know if a function is analytic? For teaching basic calculus it probably suffices to know that $\exp,\cos,\sin$ are analytic and that analytic functions are closed under sums, products, division, composition, inversion, derivation and anti-derivation. In particular ...
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If the power series $\sum_{j=0}^\infty a_j z^j$ converges to some function $f(z)$, then the Maclaurin series of $f(z)$ is $\sum_{j=0}^\infty a_j z^j$.
But the converse need not hold.
It could happen that the Maclaurin series of a function $f(z)$ is $\sum_{j=0}^\infty a_j z^j$, but $\sum_{j=0}^\infty a_j z^j$ converges to some function other than $f(z)$. ...
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