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.

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

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.

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

Yeah, I think so.

My quiz answers kept on being wiped but my highest score was recorded I think.

Greetings Cassidy!

Is there any way that we can find out which questions we missed on the final exam? That would be useful.

Many thanks, Cassidy. I just emailed them.

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.

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.

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

There is more discussion about a point here: https://mathworld.wolfram.com/Point.html

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

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.

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.

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

It's ok. I can take it.

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Tingting Zhao

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Tingting Zhao

Tingting Zhao

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Luke Titus, Wolfram Research

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Tingting Zhao

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Section 9, 33 | Exercises: Exponential Functions, Exercise 3—Continuously Compounded Interest typo: 5 should have been 20 0 Tingting Zhao Tingting Zhao Posted 2 years 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 0 Tingting Zhao Tingting Zhao Posted 2 years 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 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 8, Problem Session 8: Areas between Curves and Volume, Problem 2 I think the 4 should have been 6 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 8, 28 | Areas between Curves, Timing, Book Text: typo 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 6, Problem Session 6: Optimization, Antiderivatives and Riemann Sums, Problem 1 Why can't I replicate Sensei's work? 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 6, 21 | Exercises: Optimization, Exercise 4—Poster The width is 10Sqrt(3) 0 Tingting Zhao Tingting Zhao Posted 2 years 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. 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 6, 21 | Exercises: Optimization, Exercise 1—Minimum Product Minimize 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 5, 20 | Curve Sketching, Slant 2 typo 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 5, 19 | Asymptotes, Special Law Just a typo, should be x -> −∞ 1 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Michael Rogers Michael Rogers, Emory University Posted 2 years ago 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 Reply | Flag 1 Henry Ward Henry Ward Posted 2 years ago Anki Shared Decks may be useful to you: https://ankiweb.net/shared/decks?search=math POSTED BY: Henry Ward Reply | Flag 1 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago I also start to understand: you are undoubtedly right about everything. POSTED BY: Tingting Zhao Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Michael Rogers Michael Rogers, Emory University Posted 2 years ago 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 Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 1 Michael Rogers Michael Rogers, Emory University Posted 2 years ago 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 Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Michael Rogers Michael Rogers, Emory University Posted 2 years ago 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 Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 1 John P Clark John P Clark Posted 2 years 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 Reply | Flag 1 Tingting Zhao Tingting Zhao Posted 2 years 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 0 Tingting Zhao Tingting Zhao Posted 2 years 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) 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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. 0 Tingting Zhao Tingting Zhao Posted 2 years 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. 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 1 Tingting Zhao Tingting Zhao Posted 2 years 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? 1 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago It sounds best with Sith Ambient :D POSTED BY: Tingting Zhao Reply | Flag 0 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 0 Tingting Zhao Tingting Zhao Posted 2 years ago Problem Session 4, Problem 2 Just a typo error. The plus should be minus here. think the intention was to write +(-4π/180) 0 Tingting Zhao Tingting Zhao Posted 2 years ago Section 4, 14 | Related Rates, Book Text, Falling Ladder I solved the problem with simple geometry, does this make sense to you guys? 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 The sliding ladder forms two congruent angles. 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 0 Henry Ward Henry Ward Posted 2 years ago Limits can be an Interval? That doesn't count as Indeterminate? POSTED BY: Henry Ward Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago Every limit lies on that interval, they are not the interval :D POSTED BY: Tingting Zhao Reply | Flag 2 Michael Rogers Michael Rogers, Emory University Posted 2 years ago 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 0 Henry Ward Henry Ward Posted 2 years ago Thanks for the comprehensive reply and the Wikipedia link! POSTED BY: Henry Ward Reply | Flag 0 Henry Ward Henry Ward Posted 2 years ago POSTED BY: Henry Ward Reply | Flag 2 Devendra Kapadia Devendra Kapadia, Wolfram Research, Inc. Posted 2 years ago The choice of the winner will be primarily based on performance/contribution to the Study Group. POSTED BY: Devendra Kapadia Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago Sensei is back from India! Yippee! :D POSTED BY: Tingting Zhao Reply | Flag 0 John P Clark John P Clark Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago I did it by hand, the official answer is correct. 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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 ​ 0 Soomi Cheong Soomi Cheong Posted 2 years ago How can I type an exponent? e.g., x^2 POSTED BY: Soomi Cheong Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago use shift+6, tell me if it worked :D POSTED BY: Tingting Zhao Reply | Flag 0 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 0 John P Clark John P Clark Posted 2 years 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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. 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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: 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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? 0 Soomi Cheong Soomi Cheong Posted 2 years 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. => 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." 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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. 1 Updating Name Updating Name Posted 2 years ago After this course what is the next course to take? Tensors? POSTED BY: Updating Name Reply | Flag 1 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 1 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 0 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 0 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years 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, ∞) 0 Tingting Zhao Tingting Zhao Posted 2 years 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? 0 Tingting Zhao Tingting Zhao Posted 2 years 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. 0 Artur R Piekosz Artur R Piekosz Posted 2 years 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 ). 0 Tingting Zhao Tingting Zhao Posted 2 years 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 Reply | Flag 1 Artur R Piekosz Artur R Piekosz Posted 2 years ago Single output means you can write f[x] for each x in the domain. POSTED BY: Artur R Piekosz Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago I can do so by switching the coordinates or rotating the graph, I don't need a function POSTED BY: Tingting Zhao Reply | Flag 0 Tingting Zhao Tingting Zhao Posted 2 years ago Oh my, look what I found: Multivalued Function POSTED BY: Tingting Zhao Reply | Flag 1 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 1 Henry Ward Henry Ward Posted 2 years ago @Devendra Kapadia It's very nice to hear you speak so highly of your colleagues! POSTED BY: Henry Ward Reply | Flag 0 Sahra Ali Sahra Ali Posted 2 years 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 Reply | Flag 0 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 1 Sahra Ali Sahra Ali Posted 2 years 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 Reply | Flag 0 Phil Earnhardt Phil Earnhardt Posted 2 years 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 Reply | Flag 1 Devendra Kapadia Devendra Kapadia, Wolfram Research, Inc. Posted 2 years ago 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 Reply | Flag 4 Luke Titus Luke Titus, Wolfram Research Posted 2 years ago 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 Reply | Flag Reply to this discussion in reply to Add Notebook Community posts can be styled and formatted using the Markdown syntax. Tag limit exceeded Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles. Publish anyway Cancel Reply Preview Attachments Remove Add a file to this post Follow this discussion or Discard Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use Feedback