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[WSG23] Daily Study Group: Introduction to Calculus

A Wolfram U Daily Study Group on "Introduction to Calculus" begins on Monday, April 17, 2023.

Join a cohort of fellow mathematics enthusiasts to learn about the fundamentals of calculus along with its applications from video lessons created for the popular Introduction to Calculus course. Participate in live Q&A and review your understanding through interactive in-session polls. Complete quizzes at the end of the study group to get your certificate of program completion.

The study group will be led by expert Wolfram U instructor Luke Titus and should be great fun!

April 17- May 5, 2023, 11am-12pm CT (4-5pm GMT)


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

POSTED BY: Devendra Kapadia
23 Replies

The scratch notebooks in the "Introduction to Calculus" interactive course don't appear to be scrollable. Is there a WL means to enable scrolling on that scratch notebook? I see the WL documentation page on ScrollingOptions, but I don't see how to get a handle on the scratch notebook's notebook.

POSTED BY: Phil Earnhardt

Luke answered this during the chat session during the course today: Format->ScreenStyleEnvironment->"Working" will set the scratch notebook so that scrolling works. Note to the @Wolfram U team: I believe it would be helpful to have this be the default mode for any scratch notebook provided for Wolfram's online courses. I also recommend setting the screen style environment to "Working" for all notebooks published for courses. It took me a couple of months when starting out to realize that the scrolling style was settable in Mathematica.

POSTED BY: Phil Earnhardt

Dear Professor Kapadia,
Great course. I'm a newbie and learning. About Riemann integrability, I read that Lebesgue integrability is also a possibility if the Lebesgue measure is 0.
This then allows Riemann integrability and this justifies a closed interval, but with a Weierstrass covering. The points at which there is no differentiability are allowed f they have Lebesgue measure 0. But can the genus be above 0 ever.
Apologies if my terminology is a bit off.
My basic question is whether Lesbesgue integrability is of use beyond Riemann integrals?
Paul Shapshak, PhD

I took Quiz 5 today; the last issue I had was understanding solving Mean Value Theorem problems in Mathematica. I generally use the transcripts of the video sessions to scrutinize the WL code. The example in Lesson 17 in the "Application" section of the transcript shows the following WL code for the example:


sol = Solve[f[4]-f[0] == f'[x](4-0) && 0<=x<=4,x]

I think this Solve statement is structured in a terrible way. It took me a long time to realize that the "(4-0)" was part of the slope calculation. I found a superior explanation also authored by @Devendra Kapadia : . It used the following WL code fragment -- from ~ 1 minute into the video:

f[x] := x^3 - 3 x

{a, b} {-2, 2};

sol = Solve [ f[b]-f[a] / b - a == f'[c] && a < c < b, c]

The Solve statement is laid out even prettier in the video: the slope calculation is shown with a 2-line fraction with the numerator directly over the denominator. Aha! That way of expressing the code is lucid. It makes a huge difference to show the code this way. I also liked how he used a, b, and c: it was much easier to see that a and b were the secant endpoints and c was the x value at the tangent point -- far better than slinging around constants for the coordinates in the Solve statement.

I just wanted to grumble. In the best of all worlds, that bad example would be edited out of the course and replaced with a lucid one. I think I had an expectation that the code would always be perfect in the examples; I'll be a bit more skeptical in the future. The good news: I don't think I'll ever forget these particular calculus ideas and exemplary WL code fragment now. :)

One other note for @Luke Titus: I noticed that a student in the course asked a question in a new discussion with the title "About a list of Calculus Functions." I thought you might have missed that question. FYI.

POSTED BY: Phil Earnhardt

Hi Phil. Thanks for mentioning the confusion with that code. I can understand why it could be confusing to see where the (4-0) part was coming from, and that the form of f[b]-f[a]/(b - a) is a little more clear about what is going on. I'll keep that in mind when I review the questions and content when the course is over.

Thank you as well for pointing me towards Paul's post about the list of Calculus functions used into course and available in the Wolfram Language. I didn't see that until you brought it up. I mentioned at the beginning of today that I would put together such a document. Once I get that made I'll upload it to the course material and add it to Paul's post.

POSTED BY: Luke Titus

I recently ran across Grant Sanderson's 3Blue1Brown video: The most unexpected answer to a counting puzzle. enter image description here The solution relies on counting small-angle approximations -- something discussed in today's videos. I don't believe Grant's solution uses calculus, but the published paper on the question does. In a podcast ep with Steven Strogratz, Grant noted that this was one of the top 4 videos on the 3Blue1Brown channel. He noted that this knowledge has little practical value for most listeners; it is popular simply for the boundless curiosity that his audience has for interesting problems.

The mind goes to funny places. When I first watched the video, I asked myself why the radix of the answer was 10. After a bit, I lectured myself that the answer is in radix 10 simply because the mass ratios were a factor of 10^x; if the ratios were 16^x, then the value would have been expressed as a hexidecimal fraction. Who is speaking when I lecture myself, and why must I go through that intermediate step? In any case, I'll have to try out those radix 16 values in WL at some point.

I highly recommend this video. It reminded me how fun this stuff can be.

POSTED BY: Phil Earnhardt

Thank you for sharing that video, Phil. That is very interesting. I've seen a lot of ways to calculate Pi in situations where it would seem a circle or Pi shouldn't appear, but this is certainly one of the most intriguing ones.

POSTED BY: Luke Titus

POSTED BY: Zbigniew Kabala

The limit doesn't exist at x=1, so you can't calculate the derivative as that involves a limit at that point.

In[1]:= f[x_] := Piecewise[{{x^2, x <= 1}, {x^2 + 2, x > 1}}]
Limit[f[x], x -> 1]

Out[2]= Indeterminate
POSTED BY: Luke Titus

POSTED BY: Zbigniew Kabala

Hi Zbigniew. Thank you very much for your additional comments and interest in the course

As you pointed out, zero causes problems. IntegerPart[x] is 0 when -1<x<1, so we get a Limit[x/0,x->0] situation, which Mathematica returns as ComplexInfinity. Mathematica returns ComplexInfinity==ComplexInfinity because that is a generic answer. It's similar to how Mathematica will just return a==b when you try to make that comparison if a and b are undefined symbols. That comparison may or may not be true, it all depends on the values for a and b. In the same way, Mathematica won't return Indeterminate for a comparison of two ComplexInfinitys made with ==. Indeterminate is mostly when there is a single numerical quantity whose magnitude cannot be determined, such as if you try to evaluate 0/0.

For all other integer values of n, there is a discrepancy between the directional limits at each integer. For example:

In[1]:= Limit[x/IntegerPart[x], x -> n, Direction -> "FromAbove",  Assumptions -> n \[Element] Integers && n > 0]
Limit[x/IntegerPart[x], x -> n, Direction -> "FromBelow",  Assumptions -> n \[Element] Integers && n > 0]

Out[1]= 1

Out[2]= n/(-1 + n)

In[3]:= Limit[x/IntegerPart[x], x -> n, Direction -> "FromAbove",  Assumptions -> n \[Element] Integers && n < 0]
Limit[x/IntegerPart[x], x -> n, Direction -> "FromBelow",  Assumptions -> n \[Element] Integers && n < 0]

Out[3]= n/(1 + n)

Out[4]= 1

For your second question. The limit of Sin[x]/x and x/Sin[x] are defined, finite, and are consistent from both directions. However, the value of those functions at x=0 is not defined. You get Indeterminate when trying to get the value at x=0.

In[5]:= (Sin[x]/x) /. x -> 0

(x/Sin[x]) /. x -> 0

During evaluation of In[5]:= Power::infy: Infinite expression 1/0 encountered.

During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

Out[5]= Indeterminate

During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

Out[6]= Indeterminate

The issue at x=0 is why Mathematica has the Sinc function. Sinc[x]=Sin[x]/x for all nonzero values of x, and equals 1 at x=0 to avoid the Indeterminate problem at that point.

POSTED BY: Luke Titus

POSTED BY: Zbigniew Kabala

You need to include the assumptions, otherwise Mathematica will treat n like a generic variable. For example, consider the following:

In[1]:= Limit[1/x, x -> a]
Out[1]= 1/a

That result is true for all values of a except for 0.

In[4]:= Limit[1/x, x -> 0]
Out[4]= Indeterminate
POSTED BY: Luke Titus

POSTED BY: Zbigniew Kabala

Hi Zbigniew. Thank you very much for all of your comments and interest in the course content. I'm always happy to have these kinds of discussions.

POSTED BY: Luke Titus

In the following expression given in the exercises, the function IS defined at x=-3 because the piecewise condition says x greater than or equal to -3. Yet the Limit function says it is indeterminate at x=-3. If it had said x>-3 I get it. The function was never defined at x=-3.

What gives?

POSTED BY: Carl Hahn

The Limit[f[x],x->=-3] is indeterminate, because different values are derived when the limit is approached from above or from below.

However, you can do




to specify the limits from each side.

POSTED BY: Phil Earnhardt

But the function is defined at x=-3 as being equal to 6/(x+4). Therefore it's limit at x=-3 is given by that equation and should be equal to 6. In fact if you evaluate the function at x=-3 that's the answer you get.

POSTED BY: Carl Hahn

The limit to f[x], approached from above, is 6

The limit to f[x], approached from below, is -8

You can see those 2 limits computed by using that option with the Limit[] function. If you do not specify fromabove/frombelow , my understanding is that the limit is indeterminate if it doesn't approach the same value from the 2 sides of the limit.

This example is showing the difference between evaluating at the value at a location, and determining the limit as it approaches that value. That is a difference -- the crucial difference -- between those two things. I believe that's the case, but I don't see in the Wolfram Language documentation where the behavior of Limit[] is formally stated.

POSTED BY: Phil Earnhardt

Carl, if you go to the online "Introduction to Calculus" course, select , select Lesson #4, and jump to 8:25 in the video, you'll see the Devendra discussing one-sided limits. At the end of that section @9:06, Devendra notes,

Because you don't have the same behavior on the left and right, you get back indeterminate as the answer for the limit.

POSTED BY: Phil Earnhardt

In general, the limit of a function at 'a' has nothing to do with its value at 'a', unless the function is continuous at 'a', which happens when the value at 'a' coincides with its limit at 'a'. In the limit, you only care about values AROUND 'a', different than 'a'. The limit may NOT exist at 'a', even if f(a) is defined and the limit at 'a' may exist even if f(a) is not defined.

Hello, Devendra and Luke. I participated in the Wolfram Language study group with Arben earlier this year; I discovered that the material using trig and calculus was more enjoyable than I remember from school. I'm interested in principles of mechanical impedance and impedance matching. As an EE/CS, I learned about electrical impedance. Many years later; I've figured out that most of those lessons are applicable to mechanical systems. Professors talked about the "impedance analogy", but they never dived into the delightful applications of the principles to mechanical systems.

I'm particularly interested in musculosksletal impedance: impedance applied to mechanical systems. OTOH, very few professionals in manual and movement therapies speak that way. I've found repeatedly that industry leaders think in this way -- they simply lack the formal context. I believe Wolfram Notebooks, CDFs, and cloud executables can provide a powerful way to understand and visualize those concepts. Impedance is about the interface; the Wolfram Language can help provide impedance matching (so to speak).

It's awesome and a bit intimidating to have such highly-qualified instructors.

POSTED BY: Phil Earnhardt

Hi Phil! It's great to have you in the study group. I'm glad you enjoyed some of the earlier ones that Arben taught. What you are working on with impedance sounds very interesting. I hope you enjoy the study group.

POSTED BY: Luke Titus
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