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Wolfram U

[WSG24] Daily Study Group: Introduction to Calculus

A Wolfram U Daily Study Group on "Introduction to Calculus" begins on Monday, August 12, 2024.

Join a cohort of fellow mathematics enthusiasts to learn about the fundamentals of calculus from the recent Introduction to Calculus ebook by John Clark and myself. Our topics will include functions and limits, differential and integral calculus, and practical applications of calculus.

The study group will be led by expert Wolfram U instructor Luke Titus, and I will stop by occasionally to check in with the group. It should be a lot of fun!

No prior Wolfram Language experience is required.

Please feel free to use this thread to collaborate and share ideas, materials and links to other resources with fellow learners.

Dates

August 12- September 6, 2024, 11am-12pm CT (4-5pm GMT)

REGISTER HERE

enter image description here
POSTED BY: Devendra Kapadia
303 Replies
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Tingting Zhao
Tingting Zhao
Posted 2 months ago

hey @Phil ,

I got your book! If you want it back, send 1 million dollars! :D

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POSTED BY: Tingting Zhao

@Phil Earnhardt and @Tingting Zhao ,

Please allow me to apologize for my oversight. As I mentioned in my email to Tingting, I am more then happy to get this fixed.

Cassidy
POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 1 month ago

Hey Cassidy, I haven't got the email you mentioned yet. But please don't worry, I really don't mind :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 1 month ago

@Cassidy You have my permission to fly to Tingting's home and retrieve my book. If you go, please have Tingting to sign another page in it. Thanks, and bon voyage!

@Tingting I hesitate to put any non-calculus messages on here, because I know that people in Devendra's class haven't bothered to turn off e-mail notifications (like me). The thought of sending out a billion notifications for no good reason is a bit troubling. Your hostage-book message was tremendously funny; the Austin Powers reference is well-known and appreciated. I may never get my book, but I did get a picture of the message (forbidden smiley here). I hope Cassidy panicked for about 10 seconds about this and then laughed it off. I've sent her my physical address through two channels; I presume the book is in the mail. If it were the copy signed for you, that would be just fine, foo.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

hey @Phil Earnhardt ,

Have you got your book yet?
POSTED BY: Tingting Zhao

Hi Tingting,

Hope you are doing well, I sent your book overseas last Wednesday, you should receive it soon. Sadly, I have not connected with Phil to send his along yet.

Have a great day and hope to see you in upcoming events!

Cassidy

POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 2 months ago

hey Cassidy!

Thank you so much! I've been studying hard preparing for the Complex Analysis course in November. You will definitely see me soon! Have a great day and I appreciate you immensely!

POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

Hey @Cassidy. I sent you an e-mail back on September 9 at 11pm with my physical address. I noticed that your header had wolfram-u@wolfram.com as the reply-to address. I'll re-send now again to Wolfram U.

--phil
POSTED BY: Phil Earnhardt

Hi Phil, are you using the same email you register with (@floa****)?

POSTED BY: Cassidy Hinkle
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

Yes.
POSTED BY: Phil Earnhardt
Phil Earnhardt
Phil Earnhardt
Posted 1 month ago

Got your book yesterday, @Tingting Zhao. Mine came with a bonus: a note from Cassidy. If we ever have a Wolfram Singularity, we'll exchange books.

Thank you, @Cassidy. All is good.
POSTED BY: Phil Earnhardt
Henry Ward
Henry Ward
Posted 2 months ago

Thanks @Devendra Kapadia and @Luke Titus for a great course.

Is this the completion certificate relating to today's quiz deadline (pictured)? Or is there another certificate?

A screenshot of

POSTED BY: Henry Ward

Hi Henry, the completion certificate within the framework is available when users have watched all the videos, taken all quizzes and passed with the 60% or higher score. For Study Group attendees we offer the certificate for taking quizzes, counting the in session lessons towards the video count. If you went ahead and re-watched the videos in the framework with the quizzes, that's the same certificate you'd get from us.
POSTED BY: Cassidy Hinkle
Henry Ward
Henry Ward
Posted 2 months ago

Great, thanks @Cassidy Hinkle
If it's truly identical I guess I don't need it!
POSTED BY: Henry Ward
Updating Name
Updating Name
Posted 2 months ago

Are there any courses you would recommend taking after this one?

An Elementary Introduction to the Wolfram Language is coming up starting September 16. This 3-week live course covers a great breadth of knowledge into the Wolfram Language. I went through about 95% of the exercises of the interactive course in the last year. In those exercises, you must provide a code sample that performs the specified operations. That grading engine (in the WL, of course) is magical. It requires a tremendous amount of intelligence for the engine to match the correct response while rejecting trivial responses that didn't get to the answer following the correct path. At some point doing the exercises, you'll start to see if you can generate false positives and false negatives by the testing-engine. That's fine, because that work will extend your encompassing knowledge of the language.
POSTED BY: Updating Name
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Hey @Phil Earnhardt,

Congratulations on winning the book! You have been a very inspiring classmate! I think the tensegrity stuff you do is fascinating, I have seen images of tables that look like they are hanging in mid-air like magic using this principle.

Do you remember Louis told you that you couldn't compare complex numbers in the last session? Well, I find it fascinating that complex numbers are an unordered field where they lack the nice Archimedean ordering real numbers have. They are incompatible with multiplication. Specifically, if z=i, the imaginary unit, then any ordering would imply i^2=-1 should be positive (if i is positive), or negative (if i is negative), which leads to contradictions.

I wondered if this problem could be resolved, then I followed the white rabbit and went down the hole, I encountered quaternions, nilpotent elements, and Lexicographic/Magnitude ordering. I can't wait to learn more about complex numbers and check out the new complex analysis course. I hope to see you there! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

Congratulations to you too, @Tingting.

The question was about the difference between two curves, and I asked what it meant if the integrals were complex numbers. Sometimes a question will pop up out of the blue. Since we have this great resource, I'll put those questions in the Q&A as a place to hold them. The second part of the question is what physical significance it was to have two complex curves.

The answer I found was the real and imaginary parts should be managed separately. That makes sense with what I know: in electrical circuits, complex numbers are required to manage the energy in the system: the "real" energy and the "imaginary" energy. "Imaginary" energy is real within the system, and industrial users have to pay for the "imaginary" capacity that they consume. To coin a phrase, it's rather complex! If anyone is curious, ask ChatGPT:

what is imaginary energy in an electrical circuit

In any sort of calculation, it makes sense to keep the real and imaginary parts separate. For billing their customers, the electric company would calculate the real and imaginary components separately and then add them together. Residential customers use a bit of imaginary power, too, but it's so small that they're not billed for it. The YouTube channel "Practical Engineering" has an excellent discussion of the 2003 blackout in Canada, including a thorough discussion about real and imaginary power on the grid.

Well, I find it fascinating that complex numbers are an unordered field where they lack the nice Archimedean ordering real numbers have.

However, if you examine the real and imaginary parts separately for the idea of ordering, everything seems to come out in the wash. That applies for the physical places I know that use complex numbers to model phenomena. That's good enough for me.

I'm grateful the power company doesn't try to measure (then charge) residential customers a surcharge for their imaginary power usage. Trying to explain imaginary numbers to the everyday electrical power customer could get rather ugly. I don't want to try to imagine that.

Electrical impedance is taught in university intro electronics courses. And the Wolfram U course on "signal processing" should be talking about it, but I've never looked at that.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Phil, you are so knowledgeable! I learn a lot from you!

The blackout reminded me of Chernobyl. It was fine until it wasn't.

There's actually an Introduction to Electric Circuits course at Wolfram U, I think we can learn more from it :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 2 months ago

@Devendra Kapadia Sensei,

When I heard you marked yourself F because your students didn't perform well on a certain topic, my heart was heavy with guilt because I made a few of my teachers cry in front of the whole class.

I would like to tell you that please don't be too harsh on yourself, you can lead the horse to the river but you can't make them drink. A lot of children are not mature enough to appreciate what they've got until it's too late.

How I wish I could undo all the ignorance and insolence of my youth and make my teachers smile again. I'm sure your old students wished the same, no matter whether they find the courage to come back and tell you or not. If not, know that we appreciate you! I will always remember your patience, kindness and wisdom! Please mark yourself an A this time and smile! :D

btw, I am reading Mathematical Thought from Ancient to Modern Times
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Wow, Santa Claus came early, hooray!

I am so grateful and happy to have won the book along with Phil! I would like to thank everyone for gifting me this precious present! This will be my first and only proper maths book and I will look after it! I will be sure to write a review too! :D

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POSTED BY: Tingting Zhao

Congratulations Tingting!!! It was a pleasure to work with you these past 4 weeks. Best of luck on all you do in the future!
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Awww, thank you so much, Luke! Thank you for being such a wonderful Senpai! I adore you and I will miss you every day! Buzz me if you need me for anything! You got a friend in me!

To infinity and beyond!
POSTED BY: Tingting Zhao

What’s the best way to remember all the material, given that you covered so many chapters? Are there any courses you would recommend taking after this one?
POSTED BY: Taiboo Song
John P Clark
John P Clark
Posted 2 months ago

IMO, the best way to learn anything, including Calculus, is to teach it. Many years ago, when I first taught Calculus, I thought I was a real hot shot. Turns out I could not have been more mistaken. Teaching gives you the opportunity to look at different concepts in many different ways. I learned a lot from my students. They often came up with ways to approach problem solving that I never thought of. So, I teach for selfish reasons as it affords me the chance to learn. I just have to get up to speed with the Wolfram language.

POSTED BY: John P Clark

I agree with John that teaching is one of the best ways to learn. When that's not possible, working through the exercises helps tremendously to solidify your understanding. You can always review the content and work the exercises on the course framework page at any time.

https://www.wolfram.com/wolfram-u/courses/mathematics/introduction-to-calculus/

As for the next course to take. I would recommend the Multivariable Calculus course through WolframU.

https://www.wolfram.com/wolfram-u/courses/mathematics/introduction-to-multivariable-calculus/
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Hey Taiboo!

I don't usually try too hard to remember stuff, especially if they are tedious. I have saved links to some tables that systematically summarize all the formulas. But as one practises more, these formulas become second nature and the computation part becomes mainly pattern recognition.

I learn the same way Uncle Elon Musk teaches. I encounter a problem and try to solve it with my existing toolkit. If not, I find out if and how others solved it and realize why more tools are needed for this particular problem. If there are none, then I try to find a new way or build a toolkit to solve it.

I recommend the Differential Equation course taught by our sweet Luke Senpai! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

Are there any courses you would recommend taking after this one?

An Elementary Introduction to the Wolfram Language is coming up starting September 16. This 3-week live course covers a great breadth of knowledge into the Wolfram Language. I went through about 95% of the exercises of the interactive course in the last year. In those exercises, you must provide a code sample that performs the specified operations. That grading engine (in the WL, of course) is magical. It requires a tremendous amount of intelligence for the engine to match the correct response while rejecting trivial responses that didn't get to the answer following the correct path. At some point doing the exercises, you'll start to see if you can generate false positives and false negatives by the testing-engine. That's fine, because that work will extend your encompassing knowledge of the language.

What’s the best way to remember all the material, given that you covered so many chapters?

The obvious answer that @Luke didn't mention is to write a Wolfram Language notebook for the course. Have both text and computational lines in the notebook. In the text you write, use the Notebook interface to use appropriate mathematically-correct symbols. Keep an archive of your notebooks.

You can see a different level of conversation/utility with the language from our instructors. Near the end of the class, Luke needed to clear a couple of definitions that he had used earlier in the Notebook session. Did anyone else notice how quickly he did:

Clear[x,y]

in the session? That is one-liner is clearly part of his muscle memory at this point in time. I'm also certain that Luke automatically notices the coloring that the Notebook interface uses. That is also something that would clearly save gobs of time. The first step is noticing how many of the coloring-hints that you miss. I think the EIWL course is a good a place as any to see that stuff.
POSTED BY: Phil Earnhardt
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

What’s the best way to remember all the material, given that you covered so many chapters? Are there any courses you would recommend taking after this one?

I think @Luke Titus provided us with the best possible answer today: write a review notebook. The one that Luke wrote for today's class was excellent. The main problem with using Luke's notebook is that it's far better to write it yourself. Maybe you could have Luke's work in one window and transcribe it into another one.

One of the things I love in the Wolfram Language: all of the calculus rules can be derived in the engine by just typing them in. If you say:

Clear[f, g]

D[f[x]*g[x], x]

you will then get the product rule displayed. That is a deep insight into the Mathematica engine: it works symbolically by applying the rules. IMHO, this is the right way to write the rules down.

I was thinking after the course. I noticed that @Wolfram U never makes an explicit recommendation to take course notes in a notebook. I think they should recommend that. Courses may move too quickly to do that during the live lecture, but it's certainly possible to make some quick notes then and clean them up later. That would be a great thing to have after a course -- maybe to review years later.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

I agree, taking notes is a great way to make sense of what is being taught. Talking about notes...

@Cassidy, Would you mind putting all our study materials in the cloud for easy access? I should think you know how, if not, Arben Senpai knows :D Thanks a lot!
POSTED BY: Tingting Zhao

I really enjoyed this class, and I'm curious about your decision to teach calculus since it's already covered in most high schools and colleges. How did you come up with so many chapters and sequences for the course?
POSTED BY: Taiboo Song

I can't speak for Devendra or the creators of the course, but I believe one of the motivations for making the course was to help students learn the subject before they take Calculus in college or as an AP topic in high school. This is one of the main reasons why we offer the course at the end of summer before most college semesters start. They developed an outline for the content of the chapters by following some of the standard Calculus textbooks that are commonly used when Calculus is taught at universities.

POSTED BY: Luke Titus

You are absolutely right, Luke.The course topics were chosen after looking at standard calculus textbooks and, particularly, the syllabus for the AP Calculus AB exam. One of the major aims of the course is to introduce students learning Calculus for the first time to the subject in an informal way, using the symbolic and graphical capabilities of Wolfram Language.
POSTED BY: Devendra Kapadia
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Sensei! Can't wait to see you later! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 2 months ago

My perception of why this class exists is that although countless courses teach the maths part, this course teaches the Wolfram Language part.

Personally, I think this course is best for people who are already familiar with the maths part and want to have a better understanding of the WL to utilize the computational power to speedily process their data.

Of course, for today's mathematical and computational-minded youth, it's best to learn both at the same time :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 2 months ago

I think someone asked the difference between the disk and washer methods.

I think the disk method is an extension of the area between a curve and the axis while the washer method is an extension of the area between two curves, they both rotate around either axis to form a solid. However, one solid is an integral of disks and one solid is an integral of washers.

A washer is the metal ring used with bolts, see attachment below

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POSTED BY: Tingting Zhao

What are the differences between the disk method and the washer method?
POSTED BY: Taiboo Song
Tingting Zhao
Tingting Zhao
Posted 2 months ago

The washer method is for solids with hollowed centers :D
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 2 months ago

What is being washed in the "washer method"?

I do not get the idea. What is washed? How to intuitively understand the method?
POSTED BY: Artur R Piekosz

The washer refers to a disk with a hole in the middle, such as in the image. enter image description here
POSTED BY: Luke Titus
Henry Ward
Henry Ward
Posted 2 months ago

Are there any plans to run a Study Group for "Introduction to Linear Algebra" in future?

I expect it could be well attended considering the recent increase in interest in Machine Learning/Artificial Intelligence

POSTED BY: Henry Ward

I don't see that on the schedule at the moment, but we'll keep that in mind for future study groups since it would work well with current interest in AI and machine learning.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Those books, they are signed copies kids! I'm so excited!!! :D
POSTED BY: Tingting Zhao

Yes! Devendra has them and will sign them this week.

POSTED BY: Luke Titus

This class is incredible! When I took calculus in college over two semesters, it was challenging to grasp the full scope and benefits of the subject due to the narrow, compartmentalized approach. For the first time, I’m experiencing calculus as a whole, and this comprehensive perspective is invaluable.

Using Mathematica to visualize graphs has been essential for my understanding. I’m curious about how calculus is currently taught at universities and high school. How significant is Mathematica's impact on calculus education today, and how do others teach calculus without it?
POSTED BY: Taiboo Song

Thank you very much for your comments. I really appreciate it. I'd be interested to hear what others say about how calculus is taught at universities today. I think Mathematica is an incredibly useful tool to help students learn Calculus and it should be used much more in university classes to help students understand the material.

POSTED BY: Luke Titus
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

Using Mathematica to visualize graphs has been essential for my understanding. I’m curious about how calculus is currently taught at universities and high school. How significant is Mathematica's impact on calculus education today, and how do others teach calculus without it?

Like Luke, I can't comment on how calculus is taught these days. I do have one alternative perspective. There are 2 independent educators -- Joan Horvath and Rich Cameron -- who have published a series of 3 books: Make:Geometry, Make:Trigonometry, and Make:Calculus. These two are disillusioned with math education as a whole. Their books use a series of 3D-printed (3DP) objects to help visualize -- optically and tactilely -- these subjects. In case you didn't know, Make Magazine is a ~20 year old magazine dedicated to supporting the DIY movement all around the world. Founder Dale Daugherty has a podcast; he has interviewed this pair once for each book: A Better Way to Learn Calculus, A Better Way to Teach Geometry Using 3D Models, and Trig – The Oldest Practical Math. If you use a podcast app, you can get these off of the Make:Cast feed. You can also stream the audio for each of those sessions straight off of those webpages. I have attached a PDF transcript of the Make:Calculus interview to this message.

These two lament that Calculus is taught using the algebraic style of Leibniz and not the graphical style of Newton's Principia Mathematica. They feel this was done to make the process of grading students in Calc classes easier by having "right" and "wrong" answers to questions. Please listen to the interview or read the transcript for details. Joan notes that they frequently (and successfully) teach their calc course to 12-year-olds. Please come to your own conclusions -- or simply appreciate the vastly different point of view these educators have.

BTW: The Wolfram Language does have a set of 3DP functions that package models to be printed. Models could [optionally] be used in Wolfram U courses, but physically getting models to students (or having them 3DP their own) might be complicated. And I fully agree with what you say: plotting/visualizing solutions using the Wolfram Language may be perfectly sufficient for most students -- as long as they give themselves permission to play around. While giving you the "right" answer, Mathematica also gives you fantastic tools for visualizing ideas.

Also, are you aware of Conrad Wolfram's The Maths Fix and his Ted Talk?

Attachments:
A Better Way To ...pdf
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

I thought Conrad is Stephen's kid, lol.
POSTED BY: Tingting Zhao

Not quite, but his brother. Though Stephen's son is also a part of the company, he has done wonderful work in his time here.

POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 2 months ago

hey Cassidy,

Are you working on labor day? I hope you get paid double! :D
POSTED BY: Tingting Zhao

Just checking on a few things so today went smoothly.

POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Love what you do and it will not feel like working at all! We love chatting to you too! :D
POSTED BY: Tingting Zhao

One quick note as I take a peak into this thread—the comparatively new ARPublish may be a "halfway point" between students actually having 3D models and staring at models on a 2D screen (though interactivity is certainly a big deal!).

enter image description here
POSTED BY: Arben Kalziqi
Tingting Zhao
Tingting Zhao
Posted 2 months ago

The legendary Arden!!! Stuff made of pure awesomeness!!! And naughty Mitch!!! I can't wait to see you guys again!!! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

If I converse with Rich and Joan, I'll ask them if ARPublish meets their visualization needs. How important is it to touch the models? In one of their interviews, they noted that their models also work for vision-impaired individuals. There's also questions of structural strength and springiness that would require design to work in a virtual world. Buckminster Fuller was famous for constructing with peas and toothpicks at age 11. He immediately understood the value of constructing with triangles instead of rectangles. Those pea-models were the foundation for his later work with geodesic domes.

Given the popularity of the Raspberry Pi in the Make Community, computing with the Wolfram Language sounds like a natural connection. Mathematica was pretty clunky when it was introduced on the RPi 2, but 64-bit Wolfram Language on an 8GB RPi5 with a 500GB NVMe SSD is a mighty beast!

Make:Geometry/Trig/Calculus should only be a starting point.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

I saw the leaky tank problem a couple of times. It got me thinking: If Archimedes could solve the problem between volume and density, could he have figured out calculus by varying the volume by small amounts? Then I thought, it was very ancient. Did he have the tools to figure it out?

Then I Googled...

Guess what? It blew my mind! He indeed, contemplated integral calculus! Not only this, he did a whole lot more! Wow!!!
POSTED BY: Tingting Zhao

That's a great observation. I never thought about how Archimedes could have figured out Calculus so long ago from the work he was already doing. I feel like we sometimes don't give the mathematicians of ancient times enough credit for their genius.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Luke,

I found a video explaining his method.

Have you watched "Indiana Jones and the Dial of Destiny"? The film is based on the Antikythera mechanism. Indy wanted to go back in time and live with Archimedes, lol :D

The existence of Antikythera shows the development of science and technology is nonlinear, we never truly know what the ancients were capable of due to the lack of surviving evidence. Many of Leonardo da Vinci's stuff were burnt by the church; the Library of Alexandria was destroyed by fire; The House of Wisdom was destroyed by war, tragic...

It also made me contemplate life and the universe. Come to think of it, it's fascinating that we are locked in this spacetime together. I didn't have to read about you from history books nor imagine what you would be like in the future, you are right here replying to me in this forum. I feel very lucky indeed!
POSTED BY: Tingting Zhao

Thanks for the video link, Tingting. I haven't seen Indiana Jones and the Dial of Destiny. I'll check it out.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 2 months ago

I think I wanna borrow the dial from Indy, bring my pet dragon with me and go to ancient Rome to thank Emperor Titus for his super bright and kind offspring! :D
POSTED BY: Tingting Zhao

Some more on the mathematicians of ancient times and their relationship to Modern (1600-) mathematics:

One of the first major advances in mathematical analysis (the name of the field of mathematics to which calculus belongs) was the development of the Method of Exhaustion. One of the problems of appreciating ancient mathematicians is the lack of primary evidence of what they accomplished. Eudoxus (408-355 BCE) is usually credited with developing the method, based on ideas of Antiphon (480-411 BCE). No works by Antiphon have been discovered, and exactly what his method was is unknown. It is sometimes said that Antiphon came up with the idea that a circle is a polygon with infinitely many sides of infinitely small lengths, the root of the idea of infinitesimals. The method of exhaustion was applied in particular to finding areas and volumes and might be considered an early form of integration. However, it was not like a "Riemann integral." It tended to apply one of two processes. Either adding smaller and smaller bits of missing area or volume, as when a kid going off to Plato's Academy packs an amphora, filling in the gaps with smaller and smaller things to get in as much as they can; or by taking away smaller and smaller bits of excess area or volume, as when Phidias sculpted Zeus, first taking away larger chunks of stone, then smaller, finally making the skin smooth by rubbing away bits of dust. In modern terms the processes create infinite sums or sequences that converge to the area or volume.

Antiphon is credited with using the method to determine the area of a circle as equal, in effect, to a triangle with base equal to the circumference and height equal to the radius of the circle. The theory that allows us to define the notion of $\pi$ was not yet available.

Eudoxus is credited by Archimedes with finding the volumes of a pyramid and cone. Archimedes used the method of exhaustion to find several areas and volumes. Alessandra King, Finding Pi with Archimedes's Exhaustion Method presents classroom activities written for use by teachers of middle school (roughly for 13-year-olds). Unfortunately, it's behind a paywall, but the introductory page can be viewed for free on JSTOR.

Another major achieve of Eudoxus in analysis is a theory of proportion that comprises irrational ratios, such as the ratio of a diagonal of square to its side, which had been known to be irrational since Pythagorean times (ca. late 6th cent. BCE). This theory enables Archimedes to express a formula for the area of a circle as a proportion: The area of circle to the square on its radius is as the circumference of the circle is to its diameter. In modern symbols, we represent the ratio by $\pi$, which is also irrational.

The Babylonians sometime between 350 and 50 BCE used the areas of trapezoids to integrate the motion of Jupiter from measurements of its (angular) velocity (in effect, they used what we now call the "trapezoidal rule" of integration). See this news report of this recent 2016 discovery. According to the article, the ancient Greeks never made the conceptual leap that area could be proportional to distance traveled.

To connect to calculus, let's first consider, "What is calculus?" The narrow sense of the word means a system of calculating. In terms of the modern calculus course, this comprises two systems, one for the calculation of derivatives from algebraic formulas and one for integrals. Most or all people would include the connection between these two systems, namely, the Fundamental Theorem(s) of Calculus (some authors split it into two parts). Note that limits and series, standard components of calculus courses, are not included. With this narrow meaning, mathematicians understand the Calculus course to be a course in analysis, using the tools of calculus to solve problems in analysis as well as to understand why the tools work.

One thing to ponder is that most of the algebraic rules comprised by calculus were known before calculus was "invented," including the fundamental theorem of calculus. Calculus, in the coherent algebraic system that is taught today, could not have been invented until after the introduction of modern algebraic notation by Viete (1540-1603) and the introduction of algebra into geometry by Fermat (1606-1655) and Descartes (1596-1650). And perhaps the invention of logarithms (1614). However, analysis goes way back. And problems in analysis occur here and there with more activity at certain periods than at others. In Europe, the late Middle Ages to the beginning of the Modern Era was a time of increasing activity. Napier (1550-1617) constructed his logarithms less than a century before calculus by approximating the solutions to a pair of rate equations, which we might describe in calculus terms as numerically integrating a system of differential equations. Aristotle characterizes the birth of a theory nicely:

The beholding of truth is in one way difficult, but in another way easy. A sign of this is that, while no one happens to be capable of it in an adequate way, neither does anyone miss it, but each one says something about nature, and though one by one they add little or nothing to it, from all of them put together something comes into being with a certain stature. (Aristotle, Metaphysics)

While calculus is the result of many hands over many centuries, just as in Aristotle's characterization, what Newton and Leibniz each put together impressed the world and spurred much research in analysis. It was built on the heavy work of others, including Archimedes who built on the work of Eudoxus and Antiphon. Newton himself famously wrote in a letter, "If I have seen further [than others], it is by standing on the shoulders of giants."

So, did Archimedes do calculus? I'd say he did analysis, geometrical analysis at that, not the modern stuff, real analysis. To my way of thinking, this is a higher compliment than calling it calculus. Nonetheless, to call his work calculus is anachronistic but not totally wrong. You have to be careful not to say, in effect, he did baby Leibniz stuff, though. G.H. Hardy wrote this: "The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or 'scholarship candidates', but 'Fellows of another college'." (A Mathematician's Apology, 1941) The European mathematicians learned analysis from Archimedes among others. They built on this work, and calculus eventually emerged. And then evolved. Thus Archimedes's importance is multiplied by the number of people who in effect became his students simply because we were lucky enough to have some of his books survive. See https://www.archimedespalimpsest.org for an amazing discovery that happened during my career - so recent! - from which lost works of Archimedes were reconstructed after the manuscripts had been erased(!) and written over.

I don't think of calculus as the end point, and somehow Archimedes beat them to it, even in part. Rather he and others laid some of the foundations. More was added over time, some of it was changed, and eventually calculus arose. Even if you think of it as an end point, consider that a hundred years after it was "invented" or "discovered," Lagrange (1736-1813) tried to rebuild it. And half a century later, Cauchy (1789-1857) rebuilt it, more or less. A generation later, Weierstrass (1815-1897) was teaching his students how to fix the imperfections in Cauchy's approach. Weierstrass's is roughly the calculus as it is taught today. But in the 20th century, it was rebuilt in several ways, but some of these are thought to be too abstruse to teach in an introductory course. Going forward, one wonders about the effect of software like Mathematica on the importance of calculus. For instance, now we can Maximize[] without knowing what a derivative is. The journey continues.
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Wow, Mike! You know so much! I love science and humanity in equal measure and listening to a Sensei telling stories that intertwine them is such a treat!

I think you are right. Our ancestors laid the foundation and we continuously evolve and perfect these methods.

My jaw dropped when I saw this:

Antiphon is credited with using the method to determine the area of a circle as equal, in effect, to a triangle with base equal to the circumference and height equal to the radius of the circle. The theory that allows us to define the notion of π was not yet available.

  • area of a circle: Ï€ r^2
  • triangle base: 2 Ï€ r
  • triangle height: r
  • triangle area: 1/2(2 Ï€ r * r) = Ï€ r^2

Antiphon was a wizard! I am so impressed!

In my mind, I sometimes question whether I could stretch or project the curve into a straight line where I can simplify the calculation but then I realized the way we normally do it is simple enough.

Regarding the use of programming language to compute math problems, computer programmers face the same issue. Although programmers come in all levels, many write programs in higher-level languages without knowing how machine code works. I think as humans accumulate knowledge at increasing speed, it takes longer to master a skill thoroughly enough so we specialize more and more in niche fields and optimize with the help of computers and AI. However, we have become easily dispensable single-purpose cogs and it's impossible to be a polymath now, even with all the money and time at hand.

I am checking out the website you recommended about Archimedes' palimpsest. :D
POSTED BY: Tingting Zhao

Thank you so much for all of that information, Michael. The ancient mathematicians deserve more credit than they get.

POSTED BY: Luke Titus
Henry Ward
Henry Ward
Posted 2 months ago

Have I understood correctly that the quizzes have a deadline of 13th September?
But that the final exam has no deadline?
POSTED BY: Henry Ward
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Yeah, I think so.

My quiz answers kept on being wiped but my highest score was recorded I think.
POSTED BY: Tingting Zhao

Hello Henry, you are correct. The quiz deadline is September 13th, but there is no deadline for the final exam.

POSTED BY: Cassidy Hinkle
John P Clark
John P Clark
Posted 2 months ago

Greetings Cassidy!

Is there any way that we can find out which questions we missed on the final exam? That would be useful.
POSTED BY: John P Clark

Hi John, If you email wolfram-u@wolfram.com, we can pulled the exam information and let you know the incorrectly marked questions.

Thank you in advance.

Cassidy Hinkle
POSTED BY: Cassidy Hinkle
John P Clark
John P Clark
Posted 2 months ago

Many thanks, Cassidy. I just emailed them.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi @Cassidy Hinkle and @Luke Titus ,

I passed the final exam.

And I finished the last leg of my notes, there's not much there. Luke can take a look, go through them, throw away my boo-boos and add to his notes.

I was gonna backlog all my posts from the forum, but since Luke has been meticulously taking notes of all my posts, there's no cause to do extra redundant work. Luke is going to pass his notes on to the course development team and if Devendra Sensei would like to check and confirm all the corrections then he's more than welcome to!

I'm so happy I signed up for this course. I learned so much and I am so grateful to be given this opportunity to knowledge and great camaraderie! But I'm gonna stick it out till the end and will always be ready to help and have fun!

To infinity and beyond! :D
POSTED BY: Tingting Zhao

I'm happy to hear that you have passed the final exam! It is a pleasure to have you in the course, Tingting. I can't thank you enough for all of the work you have done to find the typos in the course. We are all very appreciative of your work.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Thank you so much Luke! You are so kind and generous!

I'm very happy to have passed the exam too :D

There's no need to thank me, I only have pre-calc level of maths so helping out with some typos is the least I can do! I wish I could do more! And hopefully, I will be able to as I learn more from other dojos in Wolfram U.

In the last section, although Differential Equations were briefly introduced in previous sessions, I feel the full-on integration of this material is a little sudden. I know you taught the DE course on Wolfram U, I wish I had caught up with your live teaching sessions but I hope I will be able to in the future if you do it again. In the meantime, I'm gonna take a little peek :D
POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 2 months ago

Hi Luke,

Is there any way that we can find out which questions we missed on the final exam? That would be useful. Judging by my score, it appears that I missed three (3) questions.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

In the Sample Exams, there's no way to open the Solution tab

POSTED BY: Tingting Zhao

You're absolutely right. I'm not able to open them either. I'll report that to the developers.

POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Discontinuity

In MathWorld, we read:

A discontinuity is point at which a mathematical object is discontinuous.

But what can be a point? Can the Sun be a point? Or the Moon? Should a point be a member of the class of all sets? To avoid vagueness of the definition, one should state that discontinuities are discussed in the context of partial functions, not usual functions. (Look for discussion about functions there.) One should restrict the set of potential points.
POSTED BY: Artur R Piekosz

There is more discussion about a point here: https://mathworld.wolfram.com/Point.html
POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Still discontinuities are not defined mathematically correctly. And there is no hint (a hyperlink) to the page on points.

If we have a function of one variable, then, maybe, a point in a 423-dimensional space is a discontinuity. Can a point of a complex space be a discontinuity of a function of a real variable? These questions are not answered.
POSTED BY: Artur R Piekosz
Artur R Piekosz
Artur R Piekosz
Posted 2 months ago

So how Wolfram Research defines points? Are they all members of $\mathbb{R}^n$ or $\mathbb{C}^n$?
POSTED BY: Artur R Piekosz

I can file a report with the developers for them to update those pages to be more specific about how they are defining a point. Just let me know how you think a point should be defined so I can forward that to the developers.

POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 2 months ago

In mathematics, points appear in many situations. One should mention affine spaces (containing points), similar to vector spaces (containing vectors). In most situations, points are elements of spaces, for example: topological spaces, metric spaces, uniform spaces, and so on. (There are multitudes of variants of such spaces.) Almost anything in mathematics may be a point, just create some topological or metric space. Mathematicians also speak about spaces of infinite dimension.

I can also see a strange name "hyperspace" that does not agree with its mathematical use.
POSTED BY: Artur R Piekosz

Thank you for the clarification, Artur. I will file a report with the developers and include what you stated in this post in that report.

POSTED BY: Luke Titus
Updating Name
Updating Name
Posted 2 months ago

OK. It is visible from time to time that the definitions in MathWorld were not written by a mathematician. An upgrade is a good idea.

POSTED BY: Updating Name
Artur R Piekosz
Artur R Piekosz
Posted 2 months ago

Will the definition of a discontinuity be changed? Discontinuities are natural notions for partial functions if one wants to have discontunuities outside of the domain.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I was doing my exam per lesson, I resumed my exam today but all my answers were wiped. :(
POSTED BY: Tingting Zhao

Hi Tingting, I am sorry to hear your answers were deleted, sadly we have not been able to code the exams to hold onto answers given in dynamic radio buttons. This is something we are actively looking at for upgrading the framework. What I recommend is downloading the exam or noting your answers in a separate notebook and filling all in at one time.

Thank you as always for your feedback and participation.

Cassidy Hinkle
POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 3 months ago

It's ok. I can take it.

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POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi @Luke Titus , @Michael Rogers ,

I am advancing with my studies and accumulating my notes on this course in a notebook. I will hand the complete version over for Devendra Senei to review at the end of my study.

I will still ask questions but no longer post errors or typos in the forum. I uploaded my notebook online from Wolfram Cloud for easy viewing so you guys can be informed of my progress and for teaching aid if you like.

You can access the notes here. I will be grateful if you can point out the mistakes I made so I don't bother Senei too much with my boo-boos. I appreciate y'all immensely!
POSTED BY: Tingting Zhao

Thank you Tingting. You have been a great help. I really appreciate the work you have done.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Gladly! You are a great role model! I appreciate and respect you immensely!

To infinity and beyond! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Instead of downloading all the notebooks, can we have them in the browser for easy access?

POSTED BY: Tingting Zhao

You can access all of the notebooks in the browser through the online interactive course: https://www.wolfram.com/wolfram-u/courses/mathematics/introduction-to-calculus/
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Including the study group reviews and poll questions? The reason is that Wolfram Player frequently freezes when I try to open the notebooks, and I have to end the tasks manually. Does this happen to anyone else?
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 9, 33 | Exercises: Exponential Functions, Exercise 3—Continuously Compounded Interest

typo: 5 should have been 20

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POSTED BY: Tingting Zhao

Yes, it should say 20 in the text.
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 8, Problem Session 8: Areas between Curves and Volume, Problem 14

The integrate part puzzled me, then I saw Problem 9.

We need to recalculate Problem 14 I think

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POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Ok, Wolfram got 224√2π/15. I checked it by hand, it's correct. :D

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POSTED BY: Tingting Zhao

Glad to hear you got it figured out.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 8, Problem Session 8: Areas between Curves and Volume, Problem 11

I think:

  • the 6 should have been 2
  • dx should have been dy, we are revolving the y-axis, in other words: x0 = 0
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POSTED BY: Tingting Zhao

You are right that the upper bound of the integral should be 2 and the variable of integration should be y. Thank you so much for your help with these typos. I have a notebook where I've collected all of the errors you pointed out that I've been able to confirm. I'll give that notebook to the developers of the course so they can get everything updated.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Wow, the dedication and effectiveness! If I want to assemble a winning team, I will definitely have you on board!

Devendra Sensei never needed my help; he's got you already! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 8, Problem Session 8: Areas between Curves and Volume, Problem 2

I think the 4 should have been 6

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POSTED BY: Tingting Zhao

Good catch. The 4 in the text should definitely be a 6.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 8, Quiz 8, PROBLEM 4

Bro, lying is bad... >_<

In the Scratch Notebook, I used the same command from (Section 8, 28 | Areas between Curves, Enclosed Area)

Solve didn't work but Integrate worked after I plugged in my x results calculated by hand.

Did I miss something?

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POSTED BY: Tingting Zhao
Updating Name
Updating Name
Posted 3 months ago

Hi Tingting. You may have had different definitions for the functions f and g stored on your kernel. It works for me. For example:

In[42]:= f[x_] := 4 - x^2
g[x_] := x^2 + 1
Solve[f[x] == g[x], x]

Out[44]= {{x -> -Sqrt[(3/2)]}, {x -> Sqrt[3/2]}}
POSTED BY: Updating Name
Tingting Zhao
Tingting Zhao
Posted 3 months ago

John, is that you? Or Mike? The "Updating Name" bug is at it again :D

And yes, you are right! I tried your code and it worked this time. I think I saw the clear command before but never used it, so it must be it! I will remember to clear their values next time!

Thank you very much!

POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 8, 28 | Areas between Curves, Timing, Book Text:

typo

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POSTED BY: Tingting Zhao

You are right. The timing of the calculation and the timing stated in the text don't match.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 7, 27 | Exercises: The Substitution Rule, Exercise 5—Definite Integral

Hmmm, not sure where -72 came from. I think it should've been -16, yes?

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POSTED BY: Tingting Zhao

You are right. The lower bound of the integral should be -16. Good catch.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 6, Problem Session 6: Optimization, Antiderivatives and Riemann Sums, Problem 1

Why can't I replicate Sensei's work?

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POSTED BY: Tingting Zhao

It looks like you don't have the area[x_] function defined because it is still in blue. It works for me.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hey it worked! It's black! Maybe because I put the two lines in one bracket this time. I will learn more about the Wolfram language after this course so I will be better prepared for future learning! :D

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POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 6, 21 | Exercises: Optimization, Exercise 4—Poster

The width is 10Sqrt(3)

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POSTED BY: Tingting Zhao

Thanks, Tingting. The text should definitely say 10*Sqrt[3].
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 6, 21| Exercises: Optimization, Exercise 3—Souvenirs

  • I think either weekly or daily doesn't matter but they should be consistent.
  • Regarding the demand function, we are calculating the price -> p(x) so I think the demand equation should be p[x_]:=30-(1/2)*(x-15), and the consequent values need to be adjusted too. My value of p(x) is 23.75, which I think makes more sense as Jake's increasing the price and his demands should be lower than 30 not higher.

Anyway, tell me if I'm wrong.

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POSTED BY: Tingting Zhao

This all makes sense to me. Thanks, Tingting. I would have missed that in the demand function.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are too humble. I don't think you will miss it; you are probably just being kind and gassing me up to give my life some meaning!

Your intellect, knowledge and precision are top-tier! I tried to read your physics papers, it appeared extraterrestrial to me. I feel I will have difficulty understanding them with many extra lifetimes.

I know in real life I will never be able to afford your tutoring so I value this opportunity very much! Thank you for being such a great Senpai! :D
POSTED BY: Tingting Zhao

I really appreciate your comments, Tingting. Thank you so much.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are welcome! Credit should be given when it's due.

Talking about tutoring, look what I found! And an old ad placed by Einstein :D

Private lessons in Mathematics and physics for students and pupils is taught in detail by Albert Einstein, holder of the federal. polyt. Specialist teacher diploma, Rechtheitsgasse 32, 1st floor. The trial session is free. 4977°

I can see it in my mind.

"Hey Doc, can you teach me what a tangent line is?"

"Well, of course, Tingting! After, we can play yo-yo!"

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POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 6, 21 | Exercises: Optimization, Exercise 1—Minimum Product

Minimize

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POSTED BY: Tingting Zhao

Yes, it should be minimize. Thanks for pointing this out.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I appreciate you!

POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 20 | Exercises: Curve Sketching, Exercise 3—Slant

I'm really tired now, anyway, does the concavity intervals make sense to you? I'll take another look tomorrow when I wake up!

I woke up, and yep, this still looks fishy. Why would the author use the maximum and minimum points but not the vertical asymptotes?

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POSTED BY: Tingting Zhao

Hi Tingting. Is the error you are pointing out the intervals that the text says the curve is concave upward and downward?
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Luke :D

By looking at the graph, I would say the graph concave up at intervals (-2, -0.21) and (2, ∞), concave down at (-∞, -2) and (-0.21, 2).

The author used the local minimum at 3.66 instead of 2, and a local maximum at −3.23 instead of -2. Since -1 lies in both (-2, -0.21) and (-3.23, -0.21), 1 lies in both (-0.21, 2) and (-0.2, 3.66), we can take these values of f''(x) for concavity analysis.

The discrepancies lie at intervals (-3.23, -2) and (2, 3.66) since -3 lies between (-3.23, -2) and 3 lies between (2, 3.66), we just need to test f''(x) at these points.

From the concavity test performed by the author at (-3, -1, 1, 3), we can conclude that at -3, the value of f''(x) is negative, so it's concaving down; and at 3, the value of f''(x) is positive, so it's concaving up.

I hope this makes sense.
POSTED BY: Tingting Zhao

Yes, that make sense. Thanks for the clarification, Tingting.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are welcome! Everyone you praise me, I feel like a turbo-charged Looney Tune :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 20 | Curve Sketching, Slant 2

typo

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POSTED BY: Tingting Zhao

Thanks again, Tingting. I'll let the developers of the course know.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Your responsiveness and kindness move me!
POSTED BY: Tingting Zhao

Hi Tingting, thank you for the continued feedback. To save time, would you be able to collect all your noted items in a notebook and send it to wolfram-u@wolfram.com? I can then have Devendra review all in one place, fix errors and send for redeployment in the course framework.

This is very useful, and we really appreciate the time spent providing us the details.

See you in the next session!

Cassidy Hinkle Wolfram U Team
POSTED BY: Cassidy Hinkle
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Cassidy!

Yes, Of course! I will put all the stuff together and send it to you!

See you soon! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 19 | Asymptotes, Special Law

Just a typo, should be x -> −∞

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POSTED BY: Tingting Zhao

Thank you again for catching this error, Tingting. I really appreciate you catching all of these typos in the text.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Luke,

I don't know if I was right or just creating noises until some sharper minds evaluate them, so thank you for tolerating me and I hope you and Devendra Sensei are not annoyed by them.

I feel embarrassed that you thank me for every typo post, so to save you the hassle, just type ok or Roger would do, I understand you are a considerable gentleman and benevolent Senpai, and I appreciate you very much! Thank you so much!

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POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

Question for our staff and other experts: is there a good source of electronic "flash cards" for single-variable differentiation and integration? I'd like to be able to solve these simple problems through muscle memory, and I'm not there now.

Are the Web Apps for Wolfram Alpha a good place to find such things? How about the "study guide" iOS apps? Those iOS apps haven't been mentioned here. I'm guessing that the AIs can generate me a good list of problems.

Does everyone know about the MIT Integration Bee held annually during IAP? Besides the winners, that webpage contains an archive of qualifier and [timed] final questions for past years. How would the WL do -- any unsolvable problems for the engine?

POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Phil,

I'm neither staff nor expert, just passing by. Curious, never heard of the Integration Bee, I'm not an MIT student but I wish you the best of luck! Break a leg!
POSTED BY: Tingting Zhao

Hi Phil,

I have a Manipulate demo for derivatives that we used in class. I just put it on the cloud so you and others can have access:

https://www.wolframcloud.com/obj/135f64fb-b949-45d0-9d85-7167096eada2

Hope it's helpful!

I never did one for integrals. I don't have anything to offer there.
POSTED BY: Michael Rogers
Henry Ward
Henry Ward
Posted 3 months ago

Anki Shared Decks may be useful to you: https://ankiweb.net/shared/decks?search=math
POSTED BY: Henry Ward
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

The first question (Lesson 2) of the Exam

Does anybody understand this question, where the function considered seems to be f(x)=x^3 ?

We are asked the following: For which values of x is f(x) is neither decreasing or increasing?

What does it mean? In a single point "decreasing" or "increasing" does not have sense. So, probably, the authors mean "in a neighborhood" of each of mentioned points. But the function is increasing, so no option is correct and we are forced to choose one of four wrong answers.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Sorry Artur, I can't reveal the answer. I might get banned.

But I think I can tell you this: All the other points on the entire f[x] are increasing, yes, but there is a special point that is neither increasing nor decreasing. You can see this point by looking at the graph, imagining their slopes(one of them is horizontal) or taking a derivative of f[x] and setting the value to 0.
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

But the answer is still wrong. Maybe less wrong than other answers. Derivative does not help with "increasing" or "decreasing" in this case.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Can you please elaborate further on why you think it's wrong?

Tell me if I understand you:

You were saying one can not describe whether a point is increasing or decreasing, correct?

In that case, you are right, a point has no direction.

However, in this case, we are talking about points on a cubic function, a line. Each point on the line has a corresponding (x,y) value which is a related pair.

This question asks: as the value of x increases, do the corresponding y values increase or decrease. Is there a point where the function is neither increasing nor decreasing.

Does this help?
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

And there is no point where in a nbhd the function is decreasing or has a maximum or a minimum. For each point in the domain the function is increasing. So the correct answer is "there is no such a point". But we do not have such an option. I start to understand: physicists teach mathematics.

By the way: a point is zero-dimensional, and a curve is one-dimensional. A plane is two-dimensional.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I also start to understand: you are undoubtedly right about everything.

POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

My advice to the creators of the course is the following: re-formulate the question to

For which values of x is the rate of change of f(x) at x neither negative nor positive?

Speaking (or writing) about rate of change/velocity/derivative will be broadly understood.
POSTED BY: Artur R Piekosz

Hi Artur,

I'm not enrolled in the course, so I don't know anything about an exam. But it seems that the exam might be open notes/book. If so, you might try looking for discussions of increasing/decreasing. Maybe it is defined there.

In teaching calculus, I've encountered increasing/decreasing over an interval but not at a point, just like you. However, definitions might exist, or a definition might be introduced. If it's an innovation, then it might catch on if it's useful.

What I grok from the points you make reminds me of how folks usually talk about instantaneous velocity (= derivative $s'(t)$ of the position function $s(t)$), especially being "instantaneously at rest" when $s'(t)=0$. So if a particle has a position given by $s(t)=t^3$ meters at $t$ seconds, then $s'(0)=0$ and the particle is instantaneously at rest, even though between any two distinct times, the particle moves forward (that is, the position increases).
POSTED BY: Michael Rogers
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

I understand that this course teaches "calculus as seen by a physicist". I thought that Introduction to Calculus, grouped with other mathematical courses, will be treated as a part of mathematics.
Now I understand that in the mind of a physicist one can say "not increasing and not decreasing at a point" meaning "the velocity is zero".
POSTED BY: Artur R Piekosz

this course teaches "calculus as seen by a physicist".

I did not know that. How lucky I made that comment, then! :) :)
POSTED BY: Michael Rogers
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Another very controversial problem is the concept of a function discontinuity. A good mathematical definition is: a function is continuous if it is continuous in each point of its domain. So each rational function is continuous. But they speak about discontinuities outside of the domain! So the notion of a discontinuity of a function is not mathematically precise. What happened? Calculus (seemingly a part of mathematics) is not a part of mathematics, but has been detached. This makes me sad. Where is the American Mathematical Society? Mathematicians gave up? Make calculus mathematical again!
POSTED BY: Artur R Piekosz

Single variable calculus mainly studies relations from $\Bbb R$ to $\Bbb R$ that are functions on their domains. This embeds any calculus function in the real coordinate plane $\Bbb R^2$. It allows us to discuss the closure of the domain, which in turn allows us to describe the points added to close the domain in terms of types of discontinuity. It is important that we be able to discuss, and to teach students to discuss, such points in the context of modeling real-world applications, since they correspond to real-world phenomena. However, for pedagogical reasons, we omit this layer of abstraction, since the embedding and discontinuities are literally obvious when you draw a graph.

That said, I don't define "continuous function" when I teach calculus, nor do the textbooks I've used. We define what it means for a function to be "continuous at a number" and "continuous over an interval." So the vexed question you raise does not enter my work.
POSTED BY: Michael Rogers
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

But teaching calculus should be mathematically correct. I find many sources where a function is told to have discontinuities outside its domain. But a function does not know about points that are in the closure of its domain! If you pass to partial functions, then one can be mathematically correct. We have the true domain and some points in the ambient set.

Do you think that adding one word is too difficult for students?

You use partial functions without telling the students? It is important to speak correctly in the situation when physicists tend to impose physical interpretations on mathematical concepts.

Still defining discontinuities for partial functions needs to be done. One needs to choose among several (or more) natural possibilities.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 18 | Exercises: Derivatives and the Shape of Graphs, Exercise 4—Second Derivative Test

Just a typo, we plug in -5 and -2

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POSTED BY: Tingting Zhao

Thanks for pointing this out. It definitely should say -5 and -2 in the text. I'll report this to the developers.

POSTED BY: Luke Titus
John P Clark
John P Clark
Posted 3 months ago

Intro to Calculus, Section 13. Economics
The following cost function calculates a particular company's total cost of producing x units: cost[x_] := x^2 + 3 x

The marginal cost is found by taking the derivative: cost'[x] = 3 + 2 x The marginal cost is approximately the amount of money it costs to produce an extra unit. In this case, the cost of producing the 1001[Null]^st unit is $2003: cost'[1000] = 2003

I don't agree with this solution. It should be... cost'[1001] = 2005

Unless I'm mistaken, I believe the answer to Section 4, Quiz 4 #5 is wrong.

On the other hand, the cost to produce the 1001st. unit is... cost[1001] - cost[1000] = 2004
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi John,

Marginal Cost is the cost to produce 1 extra unit. if you want to find out the marginal cost to produce the (n+1)th unit, you plug in n

When you plug in cost'[1001], you are looking at the cost of producing the 1002th unit.

I agree, the numbers are skewed, let me have a look.

So, it seems per this function, if you take the derivative first, you lose two decimal place accuracy. I think the reason is: we degraded the integrity of the original quadratic equation by taking its derivative and turning it into a linear equation. I suspect the loss of accuracy happens during this step.

Your method of using total cost[n+1] - cost[n] is correct. But it kept the quadratic form, therefore the result differed from the result of the derived Marginal Cost equation. I think at this point, it's not a calculation error but a rounding-up preference. I hope this helped :D

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POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

hmmmm... In all my microeconomics and math econ textbooks, written by some very notable authors, some of whom are mathematicians as well, suggest otherwise. If one wishes to find the MC of the 250th unit, it's just cost'(250). .... one additional unit beyond the 249th unit having already produced 249 units.

I haven't given much thought to the loss of integrity when calculating the derivative of the original function, However, it does seem compelling, at least on the surface, where the derivative function examines the instantaneous change associated with producing a given unit. Perhaps accuracy can be improved and integrity preserved if we model reality by considering the discrete case as so many introductory econ books do.

At any rate (no pun intended), many thanks for your detailed response.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Woke up in the middle of the night, thinking about marginal cost, bwahahaha.

You know what, I think depending on how the original cost function was formulated, those bigwigs you mentioned may not even be wrong.

But per the equation we are given in this example, I think entering n to get the value of n+1 is correct here. The reason is, if you look at the screenshot(Section 4, 13 | Rates of Change in the Sciences, Book Text), the cost function is cost[x_]:=x^2+3x , it's derivative is marginal cost function cost'[x] = 3+2 x, what does this tell us? I think it tells us, there is a y-intercept at 3, this 3 indicates we have a constant starting value. If we plug in 0 in this marginal cost function, we get 3. This alone tells me to produce the 1st extra unit, it costs 3 dollars. In the method you mentioned earlier, it would not make logical sense, since making 0 units surely should cost 0 dollars, not 3. I encourage you to go over the equations in your book and see if you can find clues to their thinking.

Does this make sense to you? Tell me what you think. I'm going back to sleep :D

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POSTED BY: Tingting Zhao
Updating Name
Updating Name
Posted 3 months ago

Hi Tingting,

Hope you had a good rest! I think a lot depends upon how one formulates the total cost function and interprets it. Suppose the total cost function is modeled by...

C(x) = 1000 + 14x + 12x^2 + 4x^3

How would you interpret C(0) = 1000? Similarly, how would you interpret C'(0) = 14? You claim it doesn't make any sense or, does it? Perhaps there is some fixed cost of production that is dependent upon a specific range of production. Of course, one might argue that if a fixed cost is dependent upon an additional unit of output within some range of production, then it can't be a fixed cost and, therefore, must be a variable cost. But is this always true? I need to think about this some more but now it's time for me to hit the hay. I'm getting dizzy.
POSTED BY: Updating Name
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Madam/Sir,

I'm up! The storm was so strong last night it kept me awake for a while. I thought the wind was gonna take me straight to the Land of Oz. I wish I had a fully functional bunker to have a false sense of security until the zombies come for my brain... >_<

Anyway, I hope I don't offend you. I had difficulty understanding the point you were trying to make in your post, but I think the examples you gave made perfect sense.

Although I really think you need to ask the author of the function what they intended to convey, this is how I would interpret them:

C(0) = 1000 means the business probably needs to invest in the tools to make the product, like a machine of some sort, which costs a lot, I would presume. This is a fixed cost regardless. Say we wanna make Barbie dolls, we need to buy a machine to make them.

As to C'(0) = 14, I think it could be the material to make the first unit of product. Say we need to get some plastic to mould them into the shape of Barbie. This is the fixed cost per unit.

My confusion is about this sentence:

Of course, one might argue that if a fixed cost is dependent upon an additional unit of output within some range of production, then it can't be a fixed cost and, therefore, must be a variable cost.

Would you care to elaborate on it further?

Regarding my conversation with John, what we were discussing was John's different method of using the Marginal Cost function. C'(0) is how we calculate the cost of making the first extra unit, however, John was saying some gigachad directly uses C'(1) to get the marginal cost of making the first extra unit. What I meant by not making logical sense was using C'[0] with that method should yield 0 because we would be making 0 extra units instead of one. I hope I have clarified the matter fully, if not, feel free to reply! :D
POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

I'm John P. Clark and that was my post... Don't know why my name wasn't posted. It said "updating name" whatever that means. EDIT: Now I see that my name has been updated. Geez.

You asked me to elaborate on the following statement I made...

"Of course, one might argue that if a fixed cost is dependent upon an additional unit of output within some range of production, then it can't be a fixed cost and, therefore, must be a variable cost."

I was just thinking out loud and rambling on.

I suppose you could universally define c"(0) = 0 but even that doesn't make sense to me. I just think it's best to restrict the domain to positive integers instead of adding the caveat that c'(O) can't equal zero or c"(0) is undefined. It's just non-sense to characterize an additional unit of output as being zero as I stated in my previous post. .
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Oh hey John,

Thank you for letting me know it was you. This forum is funky at times :D

Not a problem at all, maybe you were just tired, happens to us all. I hope you had a better rest than I did. It's a new day, let's learn and improve together! High five!
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

I'm John P. Clark and that was my post... Don't know why my name wasn't posted. It said "updating name" whatever that means. EDIT: Now I see that my name has been updated. Geez.

John, the "updating name" bug is one of the long-standing glitches in the Wolfram Community Forums software. I see a forum discussion dated >9 years ago where the bug was reported. In that case, the moderation team manually patched the particular message. I presume a formal bug report was filed. It's a jarring bug. FWIW, I looked up your original message and it's still showing up as "Updating name" for me. The intractability of this bug is a mystery.

FWIW, there's a separate discussion launched by @Tingting Zhao discussing the community forums themselves. Anyone interested in the topic should jump over to that discussion.
POSTED BY: Phil Earnhardt
John P Clark
John P Clark
Posted 3 months ago

Phil, Thanks for the information regarding the bug. And, yes, I do see "updating name" in my original message. I must have been looking at another message where my name appears. At least it's appearing except for that one message. Weird.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Wow, it's a bug, I thought it was some kind of mischievous prank :D
POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

Actually, finding c'(0) doesn't make any economic sense now that I think about. Marginal cost is defined as the additional cost incurred triggered by producing an additional unit of output. How can one assign an incremental cost associated with producing an additional unit of zero output (product)? Doesn't make any sense. Therefore, it seems to me that one needs to restrict the domain of the MC function to the set of positive integers. Zero is inadmissible as MC(x) is not defined @x = 0. This seems to make the most sense from an economics perspective. Just my thoughts.

POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

c'(0) makes sense using our method, it means the additional cost of making the first unit.

Regarding the method you mentioned, I haven't read their book nor seen their formulation of the cost function, it's hard to make assumptions without them I think you could try to either take a screenshot of your book or contact the authors and ask them directly :D
POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

Hey Tingting!

With all due respect, I think we'll just have to agree to disagree. c"(0) = 0 just doesn't make any sense as I wouldn't even know how to even address an additional unit of output (product) equal to zero. That would suggest that the additional unit of output immediately preceding the zero level of output is -1. YIKES! I've even googled this issue and the few analyses I've seen agree. I was in the telecommunications industry many years ago (now retired) and used to estimate cost functions. I never came across an MC function with a constant term.

Regarding the calculation of c'(x), Stewart's Calculus- 7th edition provides an example (attached). See what you think. There is a boatload of calculus texts that provide similar examples.

Attachments:
Marginal Cost Ex...pdf
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hey John,

Not a problem at all! We are all here to learn, have fun and make friends. I understand that it may cause some concern regarding the cost function especially when you have been doing this for a long time. But for now, let's enjoy each other's company in good humour, whatever the outcome, finding things out is part of the fun!

And thanks for uploading the photo. I appreciate the effort! From what I read, including the highlighted text. I think there's no conflict between their method and ours.

The additional cost of producing (n+1)th unit is still is by plugging n into the marginal cost function. Therefore, to get the MC of the 501st item, we plug in 500. We can still plug in 0 to get the additional cost of the first item too, which will be 5 dollars.

Anyway, if you see something I didn't, feel free to have a chat. Can't wait to see you guys in class soon! Let's do our best! :D

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POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

Hey Tingting! I have seen the error of my ways and will gladly concede. The answer was under my nose all along and I just couldn't convince myself. You are right on all counts!
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hey John,

Again, no problem at all. I make mistakes all the time too. Don't worry about it. As long as you are having fun then this activity is worthwhile! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Final exam, Lesson 17:

Needs some correction in the options:

  • f(x) attains a minimum in the interval (0,5)
  • f(x)=15 has at least one solution in the interval (0,5)
  • f(x)=10 has at least one solution in the interval (0,5)
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POSTED BY: Tingting Zhao

I agree with you. Thanks for pointing that out. I'll report it to the developers.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 17 | The Mean Value Theorem, Speed Limit

A fun thought, if the driver took a break, the function is not continuous, however, if there were a break time, they would need an even higher instantaneous speed for a period of time to cover the same distance. I know continuity is essential for the Mean Value Theorem but this problem can be solved by comparing the average speed to the speed limit, if the average speed is higher than the speed limit, there should be a period of speeding. If lower, there's no way to tell.
POSTED BY: Tingting Zhao

That's definitely true that if the average were lower than the speed limit there is no way to tell if the driver should get a speeding ticket because it is possible to momentarily speed while overall having an average speed below the speed limit.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 17 | Exercises: The Mean Value Theorem, Exercise 4—Minimum Value

In my mind, the least possible value and the minimum value are different concepts. In this example, I think 23 is the minimum value but not the least possible, any value >=23 is equally probable.

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POSTED BY: Tingting Zhao

The least possible isn't referring to the probability of the value occurring, it refers to the lowest value that the function can have at x=10 given the constraint that g'[x]>=3 for all values of x.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Cool, that's sorted then! Lowest value it is! Thanks Luke! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 17 | The Mean Value Theorem, Roots of a Polynomial

How did the writer naturally know the interval to evaluate the function? I would find the root/roots or draw a graph first and then figure out the appropriate interval to evaluate the number of roots.

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POSTED BY: Tingting Zhao

I feel like the question isn't worded well. Since the function crosses zero inside the interval, you can say that a root exists in that interval by the intermediate value theorem. However, the question is saying to show that only one root exists, but I don't think you can use the intermediate value theorem to say that only one root exists. You can only say that a root does exist in that interval.

POSTED BY: Luke Titus

Correction. I see that the rest of the solution goes on to show that there is only one root. To answer your original question, the interval wasn't naturally known, but is rather obtained by guess and check. The main point is that you want to find an interval where the function crosses zero in that interval. You can use a larger interval to obtain the same conclusion. For example

In[70]:= f[x_] := x^3 + 8 x - 2
f /@ {-2, 5}
Out[71]= {-26, 163}

Since the function crosses zero somewhere between -2 and 5, you can conclude by the intermediate value theorem that a root exists in that interval.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Thanks Luke! Yeah, I suspected the use of interval choice was by trial and error. I just wondered if there was a method of finding the interval.

POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 16 | Exercises: Maxima and Minima, Exercise 5—Friction Revisited

Why did Solve give an interval rather than just 1? How did the writer find 1?

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POSTED BY: Tingting Zhao

The value for mu is dependent on the value of theta. If theta=Pi/4, then mu=1 since Tan[Pi/4]=1. If theta is some other value, then mu will not equal 1.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I thought I had seen this guy somewhere else before so I went to look.

It appeared in Section 3, 10 | Exercises: Derivatives of Trigonometric Functions. In this problem, they got the θ by giving values w=100, μ=0.4, and then plugged in force'[θ] = 0 to solve for θ. It's doable. ________________________________________________________________________________

But in this problem, w and θ are unknown, we get μ in terms of θ.

If you plug in θ = π/4 or μ = 1, force'[θ] = 0 indeed.

How did the writer get either value to start with? We know the value θ is in the range [0,π/2], so trial and error again?
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

My gmail has stopped updating Luke's replies to my posts but not others. Mmmm. I wonder what it's doing under the hood. Jealousy and sabotage protocol triggered, lol. The mail sorting A.I. needs some reeducation :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 5, 16 | Maxima and Minima, Business Example:

I understand we plug in the end numbers 0, 10000 and the critical number, but what does the [1,1,2] mean?

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POSTED BY: Tingting Zhao

The [[1,1,2]] is shorthand notation for the part function and is used here to extract just the x-value that is returned by the Solve function. For example

In[55]:= 
cost[x_] := 7134.23 - 1.396 x + 0.000019647 x^2 + 8.165084*^-9 x^3
crit = Solve[D[cost[x], x] == 0 && 0 <= x <= 10000, x]
Out[56]= {{x -> 6789.63}}

In[57]:= crit[[1]]
Out[57]= {x -> 6789.63}

In[58]:= crit[[1, 1]]
Out[58]= x -> 6789.63

In[59]:= crit[[1, 1, 2]]
Out[59]= 6789.63
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Alright! Thanks Luke! Roger Roger! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi gang,

I wrote some calculations on the Quiz page directly under the quiz question, there's no bracket on the right side to click on, how do I delete them?

I did this because the Scratch Notebook seems to have limited space and once I fill the page, I can't scroll it down for more space. Also, what I wrote on the Scratch Notebook gets carried to the following lessons so I have to delete them every time which is tedious.

But if I write right under the quiz questions, it seems I can extend the space as much as I want. But now I can't delete them when I make mistakes :(

btw, is there a way to mass delete without clicking those brackets? (Note: I'm a pleb, I don't use Mathematica so there are no fancy menu items to choose from, just the plain old browser)
POSTED BY: Tingting Zhao

I'm getting the same behavior if I type code directly into the Quiz. There doesn't seem to be an easy way to delete the code from in the quiz. That may be a limitation of the framework, but I can report that to see if there is a way to fix it. In the scratch notebook, the only way to delete chunks of code is to select the cell brackets and delete them that way. In a Mathematica notebook you can go to the menu Edit -> Select All to select everything in the notebook. You can then delete all the code after it is selected.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 3, 12 | Implicit Differentiation, Implicit Differentiation:

I don't know, is it just me or does it sound weird to use the word "replace"? I think append dy/dx or y' to y sounds more accurate. If we keep the word "replace" then I think we should use the expression y[x] instead, what do you guys think?

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POSTED BY: Tingting Zhao

I think this is the point they are trying to make. Consider an equation defined as eqn=x+y where y depends implicitly on x. If you try to take the derivative of the equation D[eqn,x] you get 1+dy/dx. If y did not depend on x, then D[eqn,x] would equal just 1, since the dy/dx = 0 if y does not depend on x. Therefore, if y depends implicitly on x, upon differentiation, you replace y with dy/dx when taking the derivative.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Ah, of course! The difference between complete and partial differentiation! Now it makes complete sense! Thanks Luke! :D
POSTED BY: Tingting Zhao

Tingting,

While @Luke Titus's math is impeccable, as usual, I'm with you about "replace" sounding weird. Consider ${d \over dx} \cos(y)$, where $y$ is to be considered an implicit function of $x$. Now let's treat $\cos(y)$ as its own expression and replace $y$ by $dy/dx$. What do we get?:

$$ a)\ \ \cos\left({dy \over dx}\right) \quad b)\ \ {-}\sin\left({dy \over dx}\right) \quad c)\ \ {-}\sin(y) \cdot {dy \over dx}$$

Both a) and b) replace $y$ by $dy/dx$. Of course, c) is the correct answer, but $dy/dx$ is not replacing anything. A new factor is being inserted (as is required by the Chain Rule).

So, yes, I think they should make a note to consider revising this description for the next edition.
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Great minds think alike! Highfive Mike! :D

Wow, associated professor of Mathematics! Respect!

Are you the gigachad that teaches Multivariable Calculus here at Wolfram U? That's where I'm heading next after this course!
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

While @Luke Titus's math is impeccable, as usual, I'm with you about "replace" sounding weird.

The concept of matching and replacing [of symbols] is fundamental to the design of the Wolfram Language. Look at the number of functions that have "replace" as part of their name. Then look at the "hold" functions and the "evaluate" functions. Then look at all of the pattern-matching functions. Then look at how all of these operations happen implicitly in the WL's input processing and output generation. Symbolic manipulation is integral to the function of the language. I pay attention when there's some part of the Wolfram Language's symbolic manipulation that I don't understand, and I suspect that there are nuances of that symbolic manipulation that still fly right over my head.

Luke and Devendra have been deeply influenced by the mechanisms and vocabulary of the Wolfram Language. "replace" has subtlety in both calculus and the WL. The use of "replace" in classical calculus does not precisely match with the WL's use of the term. Or maybe Devendra disagrees -- and he is supremely qualified to address this question. I personally don't have a problem with the text, but I don't understand it as well as you two -- Tingting and Michael -- do. IMHO, this is a worthwhile discussion to pay attention to.

Michael has a typo in his last message: "where y is to be considered an implicit function of y". Can you please edit that?
POSTED BY: Phil Earnhardt

Hi Phil,

Thanks for pointing out the typo. I fixed it.

As for "replace," to my mind, the usages of replace in WL and standard mathematics have a shared sense of "take out the old thing and put in the new." I don't think I can find an expression expr such that expr /. y -> y' results in the correct implicit differentiation of Cos[y] (or of x^2+y^2, the example in the textbook).

BTW, if you're interested in people's backgrounds, I've updated my profile. (You mentioned some of Luke's and Devendra's.) I'm teaching a course in Computational Mathematics this fall, and I will use some of the chapters from John and Devendra's book. That's why I check in here so frequently. I really appreciate the contributions of you, Tingting, and other students, since it is helpful to me to see people engage the textbook. Likewise, I appreciate Luke's responses. So thanks to everyone!
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Wow, graphics terminals with a phosphorescent screen! I never knew such a thing existed! Is it like a TV? I will read more about it!

I am so happy to have you around! I admire your dedication to your teaching excellence by doing research. If you want any extra feedback, please feel free to ask! I'm sure everyone here is happy to help!

Thank you for helping us understand our learnings better! I really appreciate it and I wish you great success with your students!

Btw, I had a look at the starting date of your course and I think I will be able to get the course done before then. If not, I will do my best! I think I'm on the same level as your students and I hope you will be armed with maximum information here to aid your teaching!

POSTED BY: Tingting Zhao

Tingting,

Like a TV? Not really. I'm pretty sure it was this, a Tektronix 4010 (images). "Images drawn to the screen remained there until deliberately erased." The drawing process took a small amount of time, so that it looked like someone with a really fast hand drawing with a light pen. We sometimes programmed graphics just to make a light show (college students will do that sort of thing).
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Right! Got it, the mechanism utilizes the Photoelectric Effect, the "write gun" has higher energy than the ones in TV and knocks the electrons from the phosphors causing the written patch to drain electrons from the flood constantly to illuminate the display.

The flashing light pen you saw was the high energy write gun beam moving about the screen. I'm not sure but I think the erasing part is caused by restoring the electrical balance on the display.
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

Michael, can you please write down the exact expression where C is the correct answer? I want to understand why one might naively believe that A and B are correct; I can't quite connect the dots from what you said. I need to see it myself -- computationally -- to completely understand.

Years ago, I was watching a video of a conversation that Stephen Wolfram was having with another scientist. Besides conversing, I believe both of them were updating expressions in real time to Wolfram Language Notebooks -- communicating through both mediums simultaneously. I think that's what they were doing; I didn't note the conversation's location for my notes. It struck me as an awfully cool idea at the time. Stephen notes that WL notebooks from the first revision of Mathematica are still playable today; that's rather remarkable.

About 25 years ago, I became interested in biotensegrity -- the modeling of the human structure as a tensegrity. One of the first popular presentations of this idea was a 1997 Scientific American article "The Architecture of Life". As a simple description, one can say that the bones float in our bodies. Tensegrity is correct, but it's insufficient to model the nuances of our movement. More recently, I've looked at co-contraction of antagonist muscle pairs as a means to precisely control our movement. Professor Neville Hogan's "Adaptive Control of Mechanical Impedance by Coactivation of Antagonist Muscles" (1984) is a great little paper. Hogan is using impedance the same way that the term is used in electrical circuits. For example, I believe "core strength" is mostly about co-activation of muscles in the torso to mediate torsional rigidity and facilitate force transmission. Coactivation gives us "something to push against" -- a similar idea that one finds in electrical impedance. In Mathematica, there are some great tools available here to visualize and analyze anatomical structures.

I've slowly realized how many instructors come here to sharpen their skills to use the Wolfram Language with their students. That's a great thing.

POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Stephen frequently does live CEOing on Twitch during the week and Science and Tech Q&A on Saturday, I chat to him often. You can tell him all your concerns when you catch him live. :D
POSTED BY: Tingting Zhao

Phil,

C is the correct answer to "find ${d \over dx} \cos(y)$". A is the correct answer to "replace $y$ by $dy/dx$ in the expression $\cos(y)$". B is a confused answer that I sometimes see in the work of my students: cosine has been differentiated and the $y$ has been replaced by the derivative.

In Mathematica, we can compute the derivative of $\cos(y)$ as an implicit function of $x$ in the two ways shown in John & Devendra's textbook:

D[Cos[y[x]], x]

(*  -Sin[y[x]] y'[x]  *)

D[Cos[y], x, NonConstants -> {y}]

(*  -D[y, x, NonConstants -> {y}] Sin[y]  *)

This last way was new to me. I usually use Dt[], which is not shown in the textbook. Or at least not in Chapter 12. One can use it in either of these two ways:

Dt[Cos[y], x]

(*  -Dt[y, x] Sin[y]  *)

Dt[Cos[y]]/Dt[x]

(*  -((Dt[y]/Dt[x]) Sin[y])  *)

There is also the function ImplicitD[] but that operates on equations, not on function expressions like Cos[y]. It's not an appropriate way to solve this problem, although one can, if one wants a roundabout way to get to the answer:

Solve[y' == ImplicitD[f[x] == Cos[y], y, x], f'[x]]

(*  {{f'[x] -> -Sin[y] y'}}  *)

You may notice in each case, the derivative $dy/dx$ is represented in a different way. My favorites are y'[x] and Dt[y]/Dt[x].
POSTED BY: Michael Rogers

Thanks for your comments Michael. You and Tingting make a fair point. I'll make a note of that and report it to the developers of the course.

POSTED BY: Luke Titus
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

Submitting my feedback after today's session (August 20) didn't seem to work. That's never happened before. Perhaps I exceeded my quota for comments. cc @Cassidy Hinkle in case you didn't get many (any?) feedback messages at the end of the BigMarker session. I copied my responses from that window and will submit them here:

How would you rate today’s session on a scale of 1 (poor) to 5 (excellent) stars?

5 stars.

What questions would you like covered during any available review time in upcoming sessions?

When Luke was going through the chain rule, he was doing work "by hand". It looks like he had prepared those notes beforehand and was copying/pasting them from another Wolfram Language window into the BigMarker display window. I hadn't seen that done before (or I wasn't aware when someone was doing it). It seems really valuable sometimes to make notes using Wolfram notation sans calculation -- that looks like a great way to perform the exercises. Can Luke show us the mechanics of doing that during the review on Friday? Dumb question: does he ever have Mathematica then evaluate the expressions he has worked through "by hand".

Can you walk through a new example of applying the chain rule by first doing the completely abstract D[f[g[x]],x] and then doing a ReplaceAll with specific functions for f and g? It doesn't have to be a complicated function (functions), I just want to see those steps.

What's on your mind? Share your suggestions and comments.

We don't have the option of clicking on emojis during the BigMarker session (as I've seen in other DSGs). That's nice for participating during Aha! moments. Can you turn them on?

Can we launch a Kickstarter to get Luke a wind screen? :) It's a little bit odd to hear the heavy breathing when we are sweating out a poll question. Otherwise, all is excellent with Luke's voice, pacing, etc.

This was a great session. It's one of the few that I will replay and pause frequently. I think Chain Rule is about the point where I lost all my confidence in calculus, and it's not so bad this time around. So much needless suffering! It's really brutal to attempt to go through a course on Differential Equations if you are not completely comfortable with the chain rule. There's a limit of what you can wing without being fully aware of how to break down the problem. Thank you for this session, and thanks for how you broke down the exercises. I really like your practice of drafting unevaluated Mathematica expressions "by hand". Brilliant.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hahahaha, Phil's hilarious! Ya, Luke sounds like Darth Vader and I was waiting for him to tell us he's our father! :D But I don't mind, I think it's endearing and I got used to it and hardly notice it now.

I think as long as we can understand Luke's calculations, whether he did it by hand is immaterial. If you saw his CV you would know the ability is definitely there. I don't know about you guys but for me, having Luke as a Senpai is overkill for sure.

Also, I think if he prepared materials for us it means he values our time. There's only one hour, we wouldn't want to go completely spontaneous and spend much time debugging, would we?

btw, I love emojis! It's a yes from me! :D

See ya on Thursday!
POSTED BY: Tingting Zhao

Thanks for your comments, Phil. I'll try to include some of your questions into the review on Friday. I'll also try to remember to mute myself during the poll questions so you don't have to hear me breathing. I've always wondered how that sounded on the other end.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

It sounds best with Sith Ambient :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

I'll also try to remember to mute myself during the poll questions so you don't have to hear me breathing. I've always wondered how that sounded on the other end.

Thanks. You didn't have to wonder; you could have referenced the BigMarker recording of a session.

Personally, I wish that WR had a standard audio kit for its professional presenters that automagically provided high-pass filtering and silenced breathing noises. The problem with manually muting is that the speaker will invariably forget to unmute some of the time. As an aside, the physics of deadcat (i.e., fluffy) windscreens is rather fascinating -- definitely worthy of a Mathematica visualization. The fuzzy hairs transduce the wind energy into heat; the lack of a rigid structure means there's no resonant energy (which would generate noise). On the negative side, the deadcats are rather fragile.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 3, 11 | The Chain Rule, Algebraic Function

Nothing wrong but the g[x] is redundant and creates confusion. f[x] = g[x] = Sqrt[x], we can then use g[x] = x^2 - 1, and save a h[x]. But if you guys feel this format is clear, feel free to ignore this post :D

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POSTED BY: Tingting Zhao

It looks like g[x] is used to show that f[x] is a composition of the functions g[h[x]]. It is a little redundant since they could have just used to f[x] function, but I think the g and h functions are introduced just for illustration purposes.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Yes, after going through more learning materials, I realize it's a common occurrence. I had this idea of hierarchy in my mind like f->g->h, but now I see, they borrow more letters to show composition, and sometimes, f(x) is the innermost function. Thanks Luke! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Problem Session 4, Problem 2

Just a typo error. The plus should be minus here.

think the intention was to write +(-4Ï€/180)

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POSTED BY: Tingting Zhao

Thanks for pointing that out, Tingting. I'll let the developers know.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are welcome! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Section 4, 14 | Related Rates, Book Text, Falling Ladder

I solved the problem with simple geometry, does this make sense to you guys?

  1. According to the Pythagorean theorem: we have 5 as hypotenuse, and one of the legs is 4, we know the other leg is going to be 3
  2. The sliding ladder forms two congruent angles.
  3. Since the two variables are related, their change ratio must be the same. Set sliding down ratio as x then 3x = 4*(0.5) -> x = 0.67
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POSTED BY: Tingting Zhao

That looks good to me. Nice job with seeing a geometrical way to solve the problem. Sometimes those are the best way to solve a problem.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You da best! :D
POSTED BY: Tingting Zhao
Henry Ward
Henry Ward
Posted 3 months ago

Limits can be an Interval? That doesn't count as Indeterminate?
POSTED BY: Henry Ward
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Every limit lies on that interval, they are not the interval :D
POSTED BY: Tingting Zhao

There are more sophisticated notions of limit than what is covered in introductory calculus. Limit may return an interval when the option Method -> {"AllowIndeterminateOutput" -> False} specified. This option was introduced around version 13, I think. I surmise the intention was to make Limit[] behave as it is taught in first-year calculus and make the more sophisticated answer optionally available. One can think of the interval as a refinement of the notion of indeterminacy or nonexistence.

For instance:

Limit[Sin[x], x -> \[Infinity], 
 Method -> {"AllowIndeterminateOutput" -> False}]
(*  Interval[{-1, 1}]  *)

The documentation used to read, "Limit[] returns Interval objects to represent ranges of possible values." That is, function values not different possible limits. When an interval is returned, it means Mathematica could prove the limit does not exist. The current documentation clarifies this somewhat: "If an Interval is returned, there are no guarantees that this is the smallest possible interval." What this means is that Limit[] uses (relatively) fast heuristics to bound the values of the function as $x \rightarrow c$. For instance:

Limit[Sin[x] + Sin[x]^2, x -> \[Infinity],
 Method -> {"AllowIndeterminateOutput" -> False}]
(*  Interval[{-1, 2}]  *)

But the actual range of the function is $[-{1\over4},2]$. In version 11.2, MinLimit[] and MaxLimit[] were introduced to provide an easy way to ask Mathematica to compute the lower and upper limits more rigorously:

MinLimit[Sin[x] + Sin[x]^2, x -> \[Infinity]]
(*  -(1/4)  *)

I think I'll stop here, since I believe this topic probably goes beyond the scope of the course. If you want to know more, you can start with this Wikipedia article: https://en.wikipedia.org/wiki/Limit_inferior_and_limit_superior
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Interesting, never knew about all these nuances! Thanks, Mike!

Regarding the context of this question. We were discussing the cyclic nature of the trigonometric functions, taking their higher derivatives can lead to interesting patterns: "Sine", "Cosine", "-Sine", "-Cosine", and cycle back to "Sine", so the Limit here can be the value of the function or the limit of the function, they all lie on the same interval [-1,1].

POSTED BY: Tingting Zhao
Henry Ward
Henry Ward
Posted 3 months ago

Thanks for the comprehensive reply and the Wikipedia link!
POSTED BY: Henry Ward
Henry Ward
Henry Ward
Posted 3 months ago

Will the winner of the book be chosen at random?

Or will it be based on performance/contribution to the Study Group?
POSTED BY: Henry Ward

The choice of the winner will be primarily based on performance/contribution to the Study Group.
POSTED BY: Devendra Kapadia
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Sensei is back from India! Yippee! :D
POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

In lecture 12, Exercise 5, Implicit differentiation, I get (2 a b n^2 - a n^3 V + P V^3)/((b n - V) V^3). Why are the exponents of n reversed? I manually computed the derivative (without non-constants). I treated the right-hand side, nRT (derivative = 0), as a constant.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I did it by hand, the official answer is correct.

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POSTED BY: Tingting Zhao
John P Clark
John P Clark
Posted 3 months ago

Thank you! Got it!
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are very welcome! :D
POSTED BY: Tingting Zhao

Thank you so much for your responses and activity in this discussion, Tingting!
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are welcome Luke! This is like a hobby, I have fun! :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Quiz 3, Problem 3:

When I used D, I still got a Tan'[x] in the result. I know how to do it by hand but is there a function that can get the clean answer from the options?

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POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

@Tingting Zhao , you need to put a space between the "x" and "Tan" when computing the derivative:

D[2x Tan[x],x] will give you the right answer. Better yet, I recommend explicitly putting in a times operator: D[2x*Tan[x],x]

Without the space, Mathematica is trying to take the derivative of "xTan". It doesn't know how to do that, so it just leaves the symbol unevaluated. Mathematica guesses that "2x" actually means "2 times x", but it can't do the same when a bunch of letters are jammed together.

When in doubt, you can always do FullForm[] of an expression to see how Mathematica is evaluating that expression: FullForm[2xTan[x]]

BTW, this is an excellent post. It instructs the rest of us off of our a.. doing the quizzes. Maybe I'm the only one who hasn't started...
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Phil! Wow, I left a blank space between 2x and Tan[x] and it worked like magic! You are a wizard! I think this is the kind of feedback Devendra is looking for. See ya tmw! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

I've become fond of asking such questions to ChatGPT (in a web browser or in the app). It's pretty clever about the Wolfram Language most of the time. It feels safe to ask questions there, and you get an immediately response. Unfortunately, ChatGPT failed the answer to this question correctly. Whenever it boffs a question, I correct it. I can't quite tell if it ever gets any smarter.

I am hardly a wizard. I have learned to try a few tricks to try when things are fouled up. Asking an AI is one of those. If you want a real magician, take a Wolfram U class with Wolfram instructor Arben Kalziqi. For this course, the stuff we saw about needing to use an Evaluate in some Plot calls was really useful. Luke should go over that in some detail on Friday. The answer to the question is in the documentation entry for Plot[], but you have to dive into the little detail-tabs to find it.

Googling on questions works well, especially if you include "wolfram" in your search. It's actually really useful that WR moved from using the name "mathematica" to "wolfram language" because the word "wolfram" gets very few hits outside of the context of this software. Adding the keyword "wolfram" to searches and asking questions to the AIs (while adding "wolfram language") is also a good way to learn.

I also highly recommend the "Elementary Introduction to the Wolfram Language" (EIWL) course after you complete this course. It gives you a great breadth of knowledge about the WL. One of the delights of that course is pondering how it grades exercises. That automated grading engine is a #!$$ work of art, and it's all implemented in the Wolfram Language. You feed it code fragments, and it tells you if your code would generate the correct result. The hard part of that is if you write your answer in a non-traditional fashion, the engine still has to figure out that you did it correctly. And you can't do something dumb like explicitly typing in the answer expression; your code must generate the correct response. It almost always grades the examples right -- even if you try to trick it. Magic!
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Wow, which course does Arben Kalziqi teach? I would like to check it out!

btw, I agree checking with AI is a safe and non-judgmental place to learn. And like you, I don't fully trust them and always go through them by hand on my own to make sure. :D

Can't wait to see you guys in class soon! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

You can see the instructor in the course description when a new course is announced. I'm not sure it's worthwhile to take a course for that reason alone. Arben's very good, but I've also seen some amazing stuff from Theo Gray and Stephen himself. The course designer of An Elementary Introduction to the Wolfram Language is pretty amazing, too. You can see that by doing the EIWL course. That would be a good course to do next. Or you could do the multivariable calculus interactive course. Perhaps both at the same time... :)
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Hey hey Phil!

Nice to hear from you! I was gonna talk to you myself!

Guess what, I talked to Arden yesterday in his tech stack course. It lasted more than 2 hours but it's totally worth it. I learned a lot and Arden is an amazing human being! He is so chillaxed, sharp-minded, and hilarious! He has a mean boss-looking cat called Mitch. Mitch woke up from his nap yesterday and kicked Arden's books off the shelf, hahahaha! Oh, and Mitch was eating plastic while Arden was teaching, lol :D

I signed up for the Complex Analysis course. I'm not sure I have enough time to speed-run through the prerequisites to attend. I need to binge-learn Linear Algebra and Real Analysis.

What are you doing next? Another fun challenging course? I'm amazed you know so many people at Wolfram, they should consider hiring you considering the great rapport you have already built! :D
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 2 months ago

He has a mean boss-looking cat called Mitch. Mitch woke up from his nap yesterday and kicked Arden's books off the shelf, hahahaha! Oh, and Mitch was eating plastic while Arden was teaching, lol :D

Yes, Arben's cat is quite a character. He finds new and fun ways to contribute to the class. And Arben keeps his cool.

I signed up for the Complex Analysis course. I'm not sure I have enough time to speed-run through the prerequisites to attend. I need to binge-learn Linear Algebra and Real Analysis.

Screw the prerequisites; backfill what you need. Follow what calls to you.

During Tuesday's lecture, I asked Oscar Paredes -- someone answering questions in the BigMarker session -- a question about asymptotes and curves. He told me about limit cycles -- a closed loop whose path is approached by some dynamic system. Just like a function will approach an asymptote as the function goes to infinity, you can approach the path of a limit cycle over time. Knowing the name pointed me to this YouTube video by Dr. Shane Ross, Virginia Tech professor. He notes that Limit Cycles happen in all sorts of biological functions: a beating heart, walking, etc. I know this funky pattern of movement with a jump rope called Flow Rope; the fundamental movement from this exercise is called the Dragon Roll. Guess what: the rope's path when moving through a Dragon Roll is a Limit Cycle!

There are perturbations to a Limit Cycle -- some external force that gets you off of the path. Our CNS is master at restoring the system and rapidly returning us to the path. The same thing happens when walking: we have a natural rhythm and cyclic path. When the path is upset, forces coordinated by our CNS know how to restore us to the efficient walking gait. One thing I can tell you: very few anatomists and movement specialists are approaching human oscillating movements as a Limit Cycle.

If you look at the description of that video, you'll see all sorts of frightening terms from Prof. Ross: nonlinear dynamics and chaos, Hamiltonian and Lagrangian systems, Manifolds, and Aeroelasticity. The very name Henri Poincaré will send shivers down your spine -- down my spine. Who cares? I know what I know, and I can fill in what I need to. In today's environment where you have things like Mathworld, Wolfram U courses, YouTube lectures, and patient AIs; one can dive into anything that interests you. And -- by golly -- a jump rope technique whose Limit Cycle follows Viviani's Curve is pretty darn interesting to me.

What am I doing next? I plan to go back to Devendra's multivariable calculus interactive course and do the exercises and quizzes this time.
POSTED BY: Phil Earnhardt
Tingting Zhao
Tingting Zhao
Posted 2 months ago

Amazing! Sergio and Oscar are gigachads! They know so much! I think Luke mentioned phase planes in a question I asked too.

Henri Poincaré's name failed to send a shiver down my spine because I'm ignorant and bold, lol! I don't think I was familiar with his work or connecting his name to known facts. But yeah, he seemed to be an extraordinary individual! How can one achieve so much in one lifetime!

I watched the video by Prof. Ross and it's mighty interesting! I remember the term "resonance", which occurs when a force oscillates at a bridge's natural frequency, can cause a bridge to collapse. I think the CIA did some weird experiments with these concepts, like squeezing a metal ball like liquid. Never knew limit cycle was involved!

I think we did some predator and prey models in this course, or Luke's Differential Equation course? Anyway, it's definitely in Luke's course. I think the Multivariable course is Arden's. He said he designed the exercises and would like some feedback if we encounter any issues :D

Yeah, I do backfilling all the time, lol!

See ya on Tuesday old sport. Have a great weekend!

I wanted to mention I watched the Netflix show "The Three-Body Problem". I think Poincaré had something to do with it. :D
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I'm doing a little preview. In Section 3, 11 | The Chain Rule, Exercise 5—Simple Harmonic Motion:

I understand that in order for s(t) = A, cos(ωt+δ)=1.

I'm a little confused about the notation. Can someone explain to me what C1 is doing here? Is it Complex Number Set, Cycle 1 in Abstract Algebra, a constant, or a variable? Why not just use n ∈ Z instead of C1?

ωt + δ = 2nπ, t = (2nπ−δ)/ω, let n = 1, then t = (2π−δ)/ω ​

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POSTED BY: Tingting Zhao

The c is a general constant used in the Wolfram Language when the solution to an equation, differential equation, integral, etc involves an arbitrary constant. You can see examples of where these constants come up in the following documentation page.

https://reference.wolfram.com/language/ref/C.html
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Cool, thanks Luke! :D
POSTED BY: Tingting Zhao
Soomi Cheong
Soomi Cheong
Posted 3 months ago

How can I type an exponent? e.g., x^2
POSTED BY: Soomi Cheong
Tingting Zhao
Tingting Zhao
Posted 3 months ago

use shift+6, tell me if it worked :D
POSTED BY: Tingting Zhao

You can also use ctrl+6 to enter exponents. In addition, if you go to the file menu Palettes -> Basic Math Assistant, you'll find many options for mathematical typesetting in the Wolfram Language.

POSTED BY: Luke Titus
John P Clark
John P Clark
Posted 3 months ago

Where do I find the quizzes that are due by Sept 6 as well as the final exam? I need a link. The email links are useless.
POSTED BY: John P Clark
Tingting Zhao
Tingting Zhao
Posted 3 months ago

All the quizzes and the final exam are on the course page. The study group ends on September 6th but there's no due date on the quizzes or the exam, you can get the Completion Certificate and Level 1 Certification anytime you finish the required tasks.
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

The quizzes may be solved at any time (until the change of the version of these quizzes) at the course framework

https://www.wolframcloud.com/obj/online-courses/introduction-to-calculus/track-my-progress.html

and the certificates look like the two I am enclosing.

Attachments:
CalcCertificate.pdf
CalcLevel1.pdf
POSTED BY: Artur R Piekosz
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

When plotting the derivative of a function, something counter-intuitive happens. Plot does not evaluate its argument. Is Mathematica too tired of doing two things at the same time? Only using Evaluate makes the correct picture. Without it, we get errors.

[Two first pictures give wrong answers. Only the third is correct.]

Could you explain what happens? How to determine if Evaluate is necessary?

Attachments:
723226457.nb
POSTED BY: Artur R Piekosz

You sometimes need to use Evaluate when using functions that have the HoldAll attribute. Functions like Plot have the HoldAll attribute. 

In[1]:= Attributes[Plot]
Out[1]= {HoldAll, Protected, ReadProtected}

The HoldAll attributes specifies that all arguments to a function are to be maintained in an unevaluated form. This leaves something like D[x^2, x] unevaluated, which gives the errors. You need to use the Evaluate function to override the HoldAll attribute and get D[x^2,x] to evaluate to 2*x so that the Plot function is able to plot the derivative.

POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Thank you Luke. But this provokes new questions:

a) why Plot and similar functions need HoldAll,

b) why Solve, DSolve and similar functions do not need HoldAll,

c) does HoldAll work also on options,

c) why pure functions have priority over named functions like D ?
POSTED BY: Artur R Piekosz

Hi Artur,

I'll give you a long-time user's perspective. Perhaps the developers think the use in WL of HoldAll, HoldFirst, HoldRest, and no holding is perfect. I would say it was good (but not perfect). To keep it simple, I will consider two classes of functions. One will need the HoldAll, and one won't need it. There may be other classes (there is at least one I can think of, those like AppendTo). They would only complicate things. Some choices made in WL do not quite fit my picture. The design is good, but not perfect, as I said.

Before we start, let me say that all functions could have been made HoldAll. Then the arguments would be evaluated inside the function. For efficiency, people writing functions would have to decide whether an argument should be evaluated once and the value reused or reevaluated each time the argument is used (see postscript below). The first is more efficient, when appropriate, but sometimes it's not appropriate. The first happens automatically in functions that are not HoldAll. It's a common case, so that's some reason why one might not want all functions to be HoldAll. It's makes some common programming tasks easier in WL.

Now to the two classes of functions. A typical mathematical operation has the form FUNC[expr, x], where the x indicates that expr is an expression that depends on x. In some cases, the second argument has the form {x, a, b}, where a and b are numbers, but x still has the same role. One class of functions produce an expression that depends on x. For instance, D[x^2, x]. A user who enters this, generally expects an output with an x in it. In that case, x is going to be evaluated in the output. Letting x (and expr) be evaluated in the input is unlikely to cause problems. DSolve is another such function.

For the second class, consider functions FUNC[expr, x] whose output is an expression that does not depend on x. For instance, Plot[x^2, {x, 0, 3}]. Then x is not evaluated on output. By not letting x be evaluated through HoldAll, we extend the usefulness of Plot to the case x = 4; Plot[x^2, {x, 0, 3}]. If the arguments were evaluated first, we would get the nonsense call Plot[16, {4, 0, 3}]. Personally, I'm glad that I don't have to use a variable I haven't used yet generate the plot. Otherwise, I might have to write Module[{x}, Plot[x^2, {x, 0, 3}]] just to quickly see what it looks like.

So those are two cases, one in which no held arguments seems convenient to me, and one in which holding arguments is convenient. You may find sometimes that which way is more convenient, HoldAll or no HoldAll, has reasons for both sides. Well, I guess the developers had to make a choice one way or the other.

There are other reasons held arguments are important, but I think the criterion of whether or not the variable gets evaluated in the output explains the several cases you brought up.

About your first point c): Yes, HoldAll mean all arguments are held, including options. However, options are usually evaluated inside the function (multiple times). Example: Plot[x^2, {x, 0, 3}, Method -> {"Foo" -> 1/0}] versus Plot[x^2, {x, 0, 3}, Evaluate[Method -> {"Foo" -> 1/0}]]. (You can specify almost any nonsense for Method, and Plot just ignores it.)

About your second point c): I didn't understand. I guess it was from class.

P.S. Plot has an old option that has disappeared from the documentation but still exists. I suppose it may go away someday. Recall that x = 4; Plot[x^2, {x, 0, 3}] would fail if the arguments were evaluated. So how can you plot D[x^2, x] in such a case?:

x = 4; Plot[D[x^2, x], {x, 0, 3}, Evaluated -> True] (* succeeds :) *)
x = 4; Plot[Evaluate[D[x^2, x]], {x, 0, 3}]          (* FAILS! *)

Plot creates an environment for evaluating the first argument in which any previous value of x is temporarily cleared, and it can substitute its own values to construct the plot. With Evaluated -> True, Plot evaluates the expression after it has created the environment in which x is cleared, and substitutes values into it to construct the plot.
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

In Section 2, 8 | The Derivative as a Function, Exercise 3—One-Sided Derivatives:

The derivative from below is infinity but from above is complex infinity, what is the difference and what caused the difference?

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POSTED BY: Tingting Zhao

That's a good question. I believe that has to do with the subtlety of calculating cube roots, and how it can sometimes return a result that goes into the complex plane. For example:

In[1]:= N[(-1)^(1/3)]
Out[1]= 0.5 + 0.866025 I

ComplexInfinity represents an infinite quantity, but undetermined complex phase. So approaching the limit from one direction seems to be purely real, so you just get Infinity. However, approaching the limit from the other direction moves into the complex plane due to the root being returned, so the limit picks up a complex phase, which causes ComplexInfinity to be returned.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Fascinating! This is what keeps me up at night! I think as I learn more from the other courses, such as Complex Analysis soon, I will fully understand the subtlety of this one day. Thank you Luke!

POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I tested in Notebook, and they yield the same answers from either side, so it can't tell, maybe Mathematica can but I'm not using it. :P

I also suspect the ComplexInfinity is from below not above, but I don't know enough to be sure. ^_^

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POSTED BY: Tingting Zhao

Hi Luke,

I think the complex Power is the right idea, but it's Limit that senses it when h is negative ("FromBelow") and corrects the limit to a real, positive infinity (DirectedInfinity[1]) in the Piecewise result it returns. However, Limit does not sense the issue when h is positive ("FromAbove"). The resulting piecewise function does not have a special case for x == 10, and we get 1/0 when 10 is substituted for x.

Maybe Limit could do a little better here, but that's a question for the developers.

Here's a way to get a better result. We'll compute the limit for each of the three cases x < 10, x > 10, and x == 10. And then wrap them up in Piecewise.

{Limit[DifferenceQuotient[f[x], {x, h}], h -> 0, 
      Direction -> "FromAbove", Assumptions -> #], #} & /@
   {x < 10, x > 10, x == 10} // Piecewise // PiecewiseExpand

(*
Piecewise[{
  {1/(3*( 10 - x)^(2/3)), x < 10}, 
  {1/(3*(-10 + x)^(2/3)), x > 10}},
 Infinity]
*)
POSTED BY: Michael Rogers
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I'm previewing Section 2: 7 | Derivatives and Rates of Change.

I have questions about Exercise 2. The left side of the slope form like this at point (2,1/4) should be y-1/4, yes?

The screenshot is attached below:

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POSTED BY: Tingting Zhao

Hi Tingting. Thank you for pointing this out. I agree with you that in the text, it should say to plug in 2, not 3. Then, the point slope form y-yo=m(x-xo) with yo=1/4, m=-1/8 and xo=2 should be used. I'll let the developers of the course know about that. Thank you.

POSTED BY: Luke Titus

Actually, the point should be (2,1/2) and not (2,1/4) since the curve goes through the line at (2,1/2).

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are absolutely right! Good catch! :D
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Almost all may be computed using Mathematica! And functions in Wolfram Language are intuitive and natural. But to use those functions effectively, one needs to know the OPTIONS. I am going to learn those options attached below.

Or, maybe, somebody has more options for me?

Attachments:
367711266.nb
POSTED BY: Artur R Piekosz

I think that's a great list. If you look in the Details and Options section of the documentation page for functions like Limit, Plot, or other related functions relevant for Calculus, you will find a complete listing of the options available for that function. Each documentation also has an "Options" section under the Examples section which provides examples of how to use each of the options.

POSTED BY: Luke Titus
Soomi Cheong
Soomi Cheong
Posted 3 months ago

The Laws of Limits; Continuity

Thursday, August 15, 2024 · 11:00 a.m. · Central Time (US & Canada)

-> Poll Question 2.1

Choose the correct statement regarding limit and continuity of the function x/Sin[x] at x = 0.

a. Limit at x = 0 exists.

b. The function is continuous only from the left.

c. The function is continuous only from the right.

d. The function is continuous.

=> enter image description here As the "Plot[x/Sin[x], {x, -1, 1}] " shows, I understand that "Limit at x = 0 exists." Thus, the answer is "a. Limit at x = 0 exists."

enter image description here I also understand that the output of "x / Sin[x] /. x -> 0" is "Indeterminate."

-> However,

my understanding of today's lecture was that;

the output of "Plot[x/Sin[x], {x, -1, 1}] " should be the same as

the output of "x / Sin[x] /. x -> 0"

->

Would you help me understand the discrepancy?

THANK YOU for your consideration of my question, Luke. I am very grateful to you and Wolfram for teaching us the Calculus from the basic, one by one.
POSTED BY: Soomi Cheong
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Hi Soomi,

I hope you don't mind that I have a go in answering your question. This function is discontinuous because x can not be 0, if x = 0, then Sin[x] = 0. But as a denominator, it can not be 0. Therefore, as a function, it does not have a f(x) value at 0. However, this function does have a limit when x -> 0, which is 1. You can see it from the graph where x = 0, it has a global minimum or you can verify this by using Limit[x/Sin[x], x->0].

The discrepancy came from you trying to match an undetermined function value to the graph at x = 0(the graph should really have an exclusion circle at this point, which would actually match the undetermined value). What should be matching is not the value of the function but the limit of the function at this point. Sometimes certain points may not have values or have values at other points parallel to the y-axis away from the curve but they do have limits and the limits we derive match the points on the curve.

The reason why we use Limit x -> 0 but not = 0, is to try to get as close as possible to 0 within the defined domain, but when x reaches the point x = 0, the value disappears due to being out of the defined domain. If this point is still within the domain, we can just plug in the value x but we can't due to the fraction can not have a 0 denominator.

I hope this helps! I attached a couple of screenshots from the book text.

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POSTED BY: Tingting Zhao

Thank you very much for answering the question, Tingting. I agree with everything you said.

POSTED BY: Luke Titus
Soomi Cheong
Soomi Cheong
Posted 3 months ago

THANK YOU, Tingting and Luke.
POSTED BY: Soomi Cheong
Tingting Zhao
Tingting Zhao
Posted 3 months ago

You are very welcome!

POSTED BY: Tingting Zhao
Updating Name
Updating Name
Posted 3 months ago

After this course what is the next course to take? Tensors?
POSTED BY: Updating Name

We offer a number of mathematics courses that can be found in the link below. The "Introduction to Multivariable Calculus" might be a good next course to take.

https://www.wolfram.com/wolfram-u/courses/catalog/?topic=mathematics
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I had something similar on my mind. I think it depends on what problem you want to solve, or which path you want to go forward. The path that led me here was to try to understand physics better. Again, it might be different for others who want to study pure maths. I follow a YouTuber called Zach Star and I think this video may help to give a general idea of what courses are needed for different majors.
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

I've noticed that the examples are not live in the Book tab of the framework of this course. At the start of Chapter 2, the clock does not move and the orbiting planet is still.

I fired up the Elementary Introduction to the Wolfram Language (EIWL) course framework. The examples in the Book tab of the framework seem to be working fine (both frameworks running Safari on a M1 MacBook Pro).

@Luke Titus, do you see the same thing? I noticed when you were showing us the Chapter 2 from the framework on Monday that the examples were not live.
POSTED BY: Phil Earnhardt

Those examples don't appear to be live for me either. If you scroll to the very bottom of the "Book Text" tab you can find a link to download the notebook. Those examples do work in the downloaded notebook.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

The screenshot below is from book text, The Elementary Functions: Power Functions.

I think the last line should be better defined as the even degree functions are mirror symmetrical or it's range is [0, ∞)

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POSTED BY: Tingting Zhao

Thanks for your comments. I agree with you that the description of that plot could be more informative than simply, "The odd degree functions are odd and the even degree functions are even." I will let the developer of the course know about that so they can improve that description.

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Thank you Luke! You are a diamond! Such a friendly, helpful and open-minded teacher! I feel so lucky! Stephen, give him a promotion!
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

The screenshot below is from the book text 4 The Limit of a Function: Piecewise Function.

I talked to Sergio about the removable discontinuity of this graph. I don't think Sergio understood what I was getting at so I thought I would mention it here again:

The function still approaches −0.5 as x approaches −1.

By normal graphing standards, there should be a circle where x approaches -1 and the point moves from -0.5 to -0.75. But as I understand it, the limit is still on the curve and is -0.5. So my question is, should there be a circle to hollow out this point on the curve, or not? . P.S. I don't see any circle where there should be one. Is it because:

  • there shouldn't be a circle
  • it was missed by error
  • Wolfram can not render circles on a graph?
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POSTED BY: Tingting Zhao

It won't draw the point automatically to indicate there is a point missing from the graph, but you can include that by using ExclusionsStyle. For example.

g[x_] := Piecewise[{{-0.75, x == -1}}, (x + 1)/(x^2 - 1)]
Plot[g[x], {x, -2, 0}, GridLines -> {{-1}, {-1/2}}, 
 ExclusionsStyle -> {Automatic, Directive[Red, PointSize[0.02]]}]
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Thank you! You are the best! I recommend the course team put the exclusion points where are needed to reduce confusion and increase accuracy.

POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

The screenshot below is from the book text 4 The Limit of a Function: Rational Function with a Removable Discontinuity. It seems the negative sign on the x-axis did not show and I think the function approaches −0.5 as x approaches -1. Also, is the curve not drawn as a vector? It looks glitchy.

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POSTED BY: Tingting Zhao

The plot has the negative signs and it doesn't look glitchy to me. See the screen shot. I do agree with you that the -2 is not correct. It should be -1. Thank you for pointing this out.

enter image description here

POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I'm sorry! I use a very old laptop, I thought it was my basic graphics card that failed to render but I changed from Chrome to Firefox and it displayed fine, is it Chrome then? Does anyone else have the same experience with the Chrome browser?
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Mathematics is often written in (La)TeX. I was curious if there is a difference between importing with Import[...] a TEX file into a Mathematica notebook as "TeX" or "LaTeX" format. It seems that there is no difference. Checked with a new Diff[...,...] function. I attach the original tex file (shortened to avoid getting $Failed ).

Attachments:
741980266.nb
myfile.tex
POSTED BY: Artur R Piekosz

As far as I'm aware, you can use "TeX" or "LaTeX" interchangeably. I didn't see anything on the documentation pages for those two formats that describes any major difference.

https://reference.wolfram.com/language/ref/format/TeX.html https://reference.wolfram.com/language/ref/format/LaTeX.html
POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

On the other hand, when exporting, there is a difference. "TeX" is accepted as a format (and the exported code compiles in LaTeX to something resembling the original code in LaTeX), while "LaTeX" is not accepted as a format. Why these are two names that are sometimes equivalent and sometimes not?

Attachments:
851063632.nb
POSTED BY: Artur R Piekosz

If you evaluate 

In[1]:= $ImportFormats
Out[1]= {"3DS", "7z", "AC", "ACO", "Affymetrix", "AgilentMicroarray", \
"AIFF", "ApacheLog", "ArcGRID", "ASC", "ASE", "AU", "AVI", "Base64", \
"BDF", "Binary", "BioImageFormat", "Bit", "BLEND", "BMP", "BREP", \
"BSON", "Byte", "BYU", "BZIP2", "CDED", "CDF", "CDX", "CDXML", \
"Character16", "Character32", "Character8", "CIF", "CML", \
"Complex128", "Complex256", "Complex64", "CSV", "Cube", "CUR", "DAE", \
"DBF", "DICOM", "DICOMDIR", "DIF", "DIMACS", "Directory", "DOCX", \
"DOT", "DTA", "DXF", "EDF", "EML", "EPS", "ExpressionJSON", \
"ExpressionML", "FASTA", "FASTQ", "FBX", "FCHK", "FCS", "FITS", \
"FLAC", "FLV", "GaussianLog", "GenBank", "GeoJSON", "GeoTIFF", "GIF", \
"GLTF", "GPX", "Graph6", "Graphlet", "GraphML", "GRIB", "GTOPO30", \
"GXF", "GXL", "GZIP", "HarwellBoeing", "HDF", "HDF5", "HEIF", "HIN", \
"HTML", "HTTPRequest", "HTTPResponse", "ICC", "ICNS", "ICO", "ICS", \
"IFC", "IGES", "Ini", "Integer128", "Integer16", "Integer24", \
"Integer32", "Integer64", "Integer8", "ISO", "JavaProperties", \
"JavaScriptExpression", "JCAMP-DX", "JPEG", "JPEG2000", "JSON", \
"JSONLD", "JVX", "KML", "LaTeX", "LEDA", "List", "LWO", "LXO", "MAT", \
"MathML", "Matroska", "MBOX", "MCTT", "MDB", "MESH", "MGF", "MIDI", \
"MMCIF", "MO", "MOBI", "MOL", "MOL2", "MP3", "MP4", "MPS", "MS3D", \
"MTP", "MTX", "MX", "MXNet", "NASACDF", "NB", "NDK", "NetCDF", \
"NEXUS", "NOFF", "NQuads", "NTriples", "OBJ", "ODS", "OFF", "Ogg", \
"ONNX", "OpenEXR", "OSM", "OWLFunctional", "Pajek", "PBM", "PCAP", \
"PCX", "PDB", "PDF", "PEM", "PGM", "PHPIni", "PLY", "PNG", "PNM", \
"POR", "PPM", "PXR", "PythonExpression", "QuickTime", "RAR", "Raw", \
"RawBitmap", "RawJSON", "RData", "RDFXML", "RDS", "Real128", \
"Real32", "Real64", "RIB", "RLE", "RSS", "RTF", "SAS7BDAT", "SAV", \
"SCT", "SDF", "SDTS", "SDTSDEM", "SFF", "SHP", "SMA", "SME", \
"SMILES", "SND", "SP3", "SPARQLQuery", "SPARQLResultsJSON", \
"SPARQLResultsXML", "SPARQLUpdate", "Sparse6", "STEP", "STL", \
"String", "SurferGrid", "SVG", "SXC", "Table", "TAR", \
"TerminatedString", "TeX", "Text", "TGA", "TGF", "TIFF", "TIGER", \
"TLE", "TriG", "TSV", "Turtle", "UBJSON", "UnsignedInteger128", \
"UnsignedInteger16", "UnsignedInteger24", "UnsignedInteger32", \
"UnsignedInteger64", "UnsignedInteger8", "USD", "USGSDEM", "UUE", \
"VCF", "VCS", "VideoFormat", "VTK", "WARC", "WAV", "Wave64", "WDX", \
"WebP", "WL", "WLNet", "WMLF", "WXF", "X3D", "XBM", "XGL", "XHTML", \
"XHTMLMathML", "XLS", "XLSX", "XML", "XPORT", "XYZ", "ZIP", "ZSTD"}

you'll see that both LaTeX and TeX are accepted import formats, which is why both can be used when importing.

However, if you evaluate

In[2]:= $ExportFormats
Out[2]= {"3DS", "AC", "ACO", "AIFF", "ASE", "AU", "AVI", "Base64", \
"Binary", "Bit", "BLEND", "BMP", "BREP", "BSON", "Byte", "BYU", \
"BZIP2", "C", "CDF", "CDXML", "Character16", "Character32", \
"Character8", "CML", "Complex128", "Complex256", "Complex64", "CSV", \
"Cube", "CUR", "DAE", "DICOM", "DIF", "DIMACS", "DOT", "DTA", "DXF", \
"EPS", "ExpressionJSON", "ExpressionML", "FASTA", "FASTQ", "FBX", \
"FCS", "FITS", "FLAC", "FLV", "FMU", "GeoJSON", "GIF", "GLTF", \
"Graph6", "Graphlet", "GraphML", "GXL", "GZIP", "HarwellBoeing", \
"HDF", "HDF5", "HIN", "HTML", "HTMLFragment", "HTTPRequest", \
"HTTPResponse", "ICNS", "ICO", "IFC", "IGES", "Ini", "Integer128", \
"Integer16", "Integer24", "Integer32", "Integer64", "Integer8", \
"ISO", "JavaProperties", "JavaScriptExpression", "JPEG", "JPEG2000", \
"JSON", "JSONLD", "JVX", "KML", "LEDA", "List", "LWO", "LXO", "MAT", \
"MathML", "Matroska", "Maya", "MCTT", "MGF", "MIDI", "MO", "MOL", \
"MOL2", "MP3", "MP4", "MS3D", "MTX", "MX", "MXNet", "NASACDF", "NB", \
"NetCDF", "NEXUS", "NOFF", "NQuads", "NTriples", "OBJ", "OFF", "Ogg", \
"ONNX", "OpenEXR", "OWLFunctional", "Pajek", "PBM", "PCX", "PDB", \
"PDF", "PGM", "PHPIni", "PLY", "PNG", "PNM", "POR", "POV", "PPM", \
"PXR", "PythonExpression", "QuickTime", "RawBitmap", "RawJSON", \
"RDFXML", "Real128", "Real32", "Real64", "RIB", "RLE", "RTF", \
"SAS7BDAT", "SAV", "SCT", "SDF", "SMA", "SMILES", "SND", \
"SPARQLQuery", "SPARQLResultsJSON", "SPARQLResultsXML", \
"SPARQLUpdate", "Sparse6", "STEP", "STL", "String", "SurferGrid", \
"SVG", "Table", "TAR", "TerminatedString", "TeX", "TeXFragment", \
"Text", "TGA", "TGF", "TIFF", "TriG", "TSV", "Turtle", "UBJSON", \
"UnsignedInteger128", "UnsignedInteger16", "UnsignedInteger24", \
"UnsignedInteger32", "UnsignedInteger64", "UnsignedInteger8", "USD", \
"UUE", "VideoFrames", "VRML", "VTK", "WAV", "Wave64", "WDX", "WebP", \
"WL", "WLNet", "WMLF", "WXF", "X3D", "XBM", "XGL", "XHTML", \
"XHTMLMathML", "XLS", "XLSX", "XML", "XPORT", "XYZ", "ZIP", "ZPR", \
"ZSTD"}

you'll see that only TeX is accepted as an export format, which is why the export to LaTeX failed. LaTeX builds on TeX and abstracts much of the complexity of TeX. Since LaTeX builds on TeX, they can sometimes be interchanged when importing, but exporting to TeX can only be done because it is a low-level typesetting format.
POSTED BY: Luke Titus
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

What is the logic of those choices? Notice that the output file of Export is a LaTeX file like this:

\documentclass{article}
\usepackage{...}
\begin{document}
...
\end{document}

So it is a LaTeX file. Why the export format is called "TeX", not "LaTeX"?
POSTED BY: Artur R Piekosz

I can't comment on the developer's decisions when making that. I would suggest exporting only to formats are supported export formats. You can refer to the section labeled Export in the following documentation page to see details about that export format.

https://reference.wolfram.com/language/ref/format/TeX.html
POSTED BY: Luke Titus
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I understand the vertical rule for checking if a graph is a function. However, why such a definition though? If you have x = y^2, it is not a function, but if you exchange the coordinates or rotate the curve 45 degrees, you will have a function. I know functions are a subset of equations but what's so special about having a single output?
POSTED BY: Tingting Zhao
Artur R Piekosz
Artur R Piekosz
Posted 3 months ago

Single output means you can write f[x] for each x in the domain.
POSTED BY: Artur R Piekosz
Tingting Zhao
Tingting Zhao
Posted 3 months ago

I can do so by switching the coordinates or rotating the graph, I don't need a function
POSTED BY: Tingting Zhao
Tingting Zhao
Tingting Zhao
Posted 3 months ago

Oh my, look what I found: Multivalued Function
POSTED BY: Tingting Zhao
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

Oh my, look what I found: Multivalued Function

As John/Devendra noted in the first chapter of this book/course, Calculus is a tool designed to solve four main problems: the tangent to a line, the area under a curve, finding velocity of a mass (given its position or its acceleration), and optimizing processes by finding a function's maxima and minima. Single-valued functions are what's needed to solve these kinds of problems. Other tools are used to solve different problems.

MathWorld is a fantastic encyclopedia of a vast number of math/science concepts -- a great find. MathWorld entries are not limited to calculus.
POSTED BY: Phil Earnhardt
Henry Ward
Henry Ward
Posted 3 months ago

@Devendra Kapadia It's very nice to hear you speak so highly of your colleagues!
POSTED BY: Henry Ward
Sahra Ali
Sahra Ali
Posted 3 months ago

Hi there! Thank you for hosting this free course.

I am excited to start this course on Monday and just had some questions to get me prepped. I downloaded both the e-book and the Wolfram app player which enables me to see the lessons in the book. I signed in on the Wolfram cloud and am able to view the sections. I also signed up for a free trial of Mathematica and did some brief tutorials. My question is where do we will be doing the practice and exercise, essentially writing the math problems, will it be on the cloud or Mathematica, or elsewhere? I want to make sure I have a checklist complete for what I need to begin the course.

I signed up for the class last week and am currently taking it in tandem with a Full Stack Dev bootcamp course that I am halfway through. I have not taken Calculus since college and need to brush up on math for impending grad school so I am grateful you are all offering this.

Thank you in advance!

peace, Sahra
POSTED BY: Sahra Ali
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

Hi, Sahra. Using Devendra's term, I'm a fellow learner in the course. This course uses the Wolfram Interactive Online Video Course Introduction to Calculus. That framework includes a set of recorded video lectures, lecture transcripts, a copy of the e-book, a group of exercises for the chapter, and a scratch area where you can type and evaluate Wolfram Language expressions.

You can use the framework's scratch area to work through each chapter's exercises. Alternatively, you can fire up the app Mathematica and work through the exercises there. All of the exercises should run fine either place. The advantage of running Mathematica on your local computer is that you can save your work from the exercises. The choice is a personal preference: you can run Wolfram Language code on your computer or in the cloud -- and you can switch whenever you like. There are many ways to view/interact with the course material; the trick is to use the one that makes the most sense to you.

I personally like to access the e-book in a separate computer window. Some students may prefer to use a printed copy of the book. Some may just like to access the book through the course framework. Some people may not even directly read the e-book; they prefer to soak in the course content through the video lectures. So many choices! There are no right answers.

Since you have the Mathematica Free Trial, you don't really need the separate Wolfram Player app. Devendra's e-book should play perfectly in either one. If you've got everything running fine, I wouldn't change anything at this point.

All of the setup is a bit of a distraction. The important thing is to be ready where you can focus on the course. The preliminary chaos reminds me a bit of my favorite snippet from Shakespeare in Love:

Philip Henslowe: Mr. Fennyman, allow me to explain about the theatre business. The natural condition is one of insurmountable obstacles on the road to imminent disaster.

Hugh Fennyman: So what do we do?

Philip Henslowe: Nothing. Strangely enough, it all turns out well.

Hugh Fennyman: How?

Philip Henslowe: I don't know. It's a mystery.

Have fun watching and learning in the theater of this course!
POSTED BY: Phil Earnhardt
Sahra Ali
Sahra Ali
Posted 3 months ago

Hi Phil:

Thank you so much for the thoughtful answer. And I always welcome a Shakespeare antecdote especially to help alleviate the early anxieties of the learning process. The materials are great so far and I will be working through which set-up is most optimal for me. Thanks again for your helpful insights!

peace, S
POSTED BY: Sahra Ali
Phil Earnhardt
Phil Earnhardt
Posted 3 months ago

That's a wonderful downloadable E-book, Devendra. It runs perfectly with the Wolfram Player App on my MacBook. I loved how you were able to have the E-book dynamically load each of the chapters from their separate WL Notebooks. I plan to have the E-book open on my laptop and the interactive course framework on a desktop computer.

Apple promised high interactivity with their Apple Books about 10 years ago; they never delivered on that promise. This Wolfram E-book delivers. It's great to be able to run through the computational demonstrations embedded in the text. This seems a vastly superior method of accessing demonstrations than Wolfram's demonstration project. I fondly wish that we could have a few dozen Wolfram-driven science E-books in the next year. I'm guessing your text looks just as pretty running on an iPad. I'll try that in the next few days.

It might be good to announce to students they have the option of viewing/interacting with the E-book through the [free] Wolfram Player on the first day of class and include a link to the Wolfram Player in the registration materials. Besides helping students, that could also help broaden the reach of your book when students in the course tell their friends/associates about the book -- with or without the accompanying the Wolfram U online course.

Brilliant book. Thank you.
POSTED BY: Phil Earnhardt

Thank you, Phil, for your very positive review comments and excellent suggestions regarding the E-book.

We appreciate them very much!
POSTED BY: Devendra Kapadia

I'm looking forward to leading this Introduction to Calculus study group. Please post any questions you have about the course to this community thread. I will be happy to answer your questions and help you get the most out of this study group.

POSTED BY: Luke Titus
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