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

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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It sounds best with Sith Ambient :D

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

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.

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.

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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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.

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.

Problem Session 4, Problem 2

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

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

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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

Cool, thanks Luke! :D

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.

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

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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The choice of the winner will be primarily based on performance/contribution to the Study Group.

Limits can be an Interval? That doesn't count as Indeterminate?

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

You are very welcome! :D

Thank you! Got it!

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.

@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...

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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How can I type an exponent? e.g., x^2

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.

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!

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.

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

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.

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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You are very welcome!

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.

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

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

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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After this course what is the next course to take? Tensors?

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"?

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

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.

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.

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?

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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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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

Oh my, look what I found: Multivalued Function

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 ).

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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?

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

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!

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.