[WSG24] Daily Study Group: Introduction to Calculus

[dkapadia (Devendra Kapadia)]

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

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

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

No prior Wolfram Language experience is required.

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

Dates

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

REGISTER HERE

[luket (Luke Titus)]

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.

[phil (Phil Earnhardt)]

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.

[sahra (Sahra Ali)]

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

[phil (Phil Earnhardt)]

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!

[dkapadia (Devendra Kapadia)]

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

We appreciate them very much!

[sahra (Sahra Ali)]

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

[henryawarduk (Henry Ward)]

@Devendra Kapadia It’s very nice to hear you speak so highly of your colleagues!

[titaniumdavinci (Tingting Zhao)]

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?

[pupiekos1 (Artur R Piekosz)]

Single output means you can write f[x] for each x in the domain.

[pupiekos1 (Artur R Piekosz)]

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 ).

741980266.nb (33.5 KB)
myfile.tex (593 Bytes)

[titaniumdavinci (Tingting Zhao)]

I can do so by switching the coordinates or rotating the graph, I don’t need a function

[titaniumdavinci (Tingting Zhao)]

Oh my, look what I found: Multivalued Function

[phil (Phil Earnhardt)]

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.

[luket (Luke Titus)]

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.

TeX—Wolfram Documentation

LaTeX—Wolfram Documentation

[titaniumdavinci (Tingting Zhao)]

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.

[titaniumdavinci (Tingting Zhao)]

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?

[titaniumdavinci (Tingting Zhao)]

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, ∞)

[pupiekos1 (Artur R Piekosz)]

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?

851063632.nb (28.4 KB)

[luket (Luke Titus)]

If you evaluate

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

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

However, if you evaluate

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

you’ll see that only TeX is accepted as an export format, which is why the export to LaTeX failed. LaTeX builds on TeX and abstracts much of the complexity of TeX. Since LaTeX builds on TeX, they can sometimes be interchanged when importing, but exporting to TeX can only be done because it is a low-level typesetting format.