{
"nodes": {
"000": {
"id": "000",
"label": "Logical Statements\nand Operations",
"meta": " SLL01",
"content": "Logic is the foundation to formulate proofs and to understand the language of mathematics.",
"notes": "000-snippet.html",
"video": "https://www.youtube.com/embed/DU4wKBDm2Z4?start=7",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/SDSU/Discrete/Logic/ttcontratautB4.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto000?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"001": {
"id": "001",
"label": "Sets",
"meta": " SLS01 + SLS02 + SLS03 + SLS04 ",
"content": "Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets.",
"notes": "001-snippet.html",
"video": "https://www.youtube.com/embed/iA-Dtf7529M?start=14",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Rochester/setSetTheory1/ur_st_1_5.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto001?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/4dhxy6YLYN7ItlWa0QgsG1?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"002": {
"id": "002",
"label": "Real Numbers",
"meta": " SLR03 ",
"content": "In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line.",
"notes": "002-snippet.html",
"video": "https://www.youtube.com/embed/E2MAvASTcg4?start=19",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Mizzou/Intermediate_Algebra/Compound_Inequalities/P_02.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto002?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/1i62R3ZdwhjASp1TNJ2Bcy?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"003": {
"id": "003",
"label": "Maps",
"meta": " SLS04 ",
"content": "Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed.",
"notes": "003-snippet.html",
"video": "https://www.youtube.com/embed/JoLDwNh1lZ8?start=260",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Mizzou/Intermediate_Algebra/Functions_Evaluating/Quad_No_Constant.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto003?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"004": {
"id": "004",
"label": "Natural Numbers\nand Induction",
"meta": " SLN01 + SLN02 + SLN03 + SLN04 ",
"content": "Using natural numbers is our first mathematical abstraction as children. Mathematical induction is an important technique of proof.",
"notes": "004-snippet.html",
"video": "https://www.youtube.com/embed/WMgiYh7tac0?start=13",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/MC/Proofs/EssayProofs/InductionDivisibility01.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto004?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/30dINCx7oSXMkCcSIOIBrv?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"005": {
"id": "005",
"label": "Image and\nPreimage",
"meta": " SLS05 ",
"content": "Via images and preimages we describe how functions work on sets.",
"notes": "005-snippet.html",
"video": "https://www.youtube.com/embed/twBYYTJdcjc?start=14",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Mizzou/Algebra/functions_domain_range/fun_dom_19.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto005?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"006": {
"id": "006",
"label": "Injectivity, Surjectivity,\nBijectivity",
"meta": " SLS06 ",
"content": "These are important notions for maps.",
"notes": "006-snippet.html",
"video": "https://www.youtube.com/embed/CSzJchEvfpE?start=9",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UMass-Amherst/Abstract-Algebra/PS-Functions/Functions2.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto006?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"007": {
"id": "007",
"label": "Composition",
"meta": " SLS07 ",
"content": "The composition for maps is just applying two maps in a row.",
"notes": "007-snippet.html",
"video": "https://www.youtube.com/embed/NiJ1yWKM9CU?start=13",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UMN/algebraKaufmannSchwitters/ks_8_6_18.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto007?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"008": {
"id": "008",
"label": "Logical Deduction",
"meta": " SLL03",
"content": "How to get new true proposition from other true propositions.",
"notes": "008-snippet.html",
"video": "https://www.youtube.com/embed/AjdIPOXRgoQ?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setDiscrete/katie5.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto008?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"009": {
"id": "009",
"label": "Predicates and Quantifiers",
"meta": " SLS01 + SLS02 + SLS03 + SLS04 ",
"content": "Formal mathematical statements are often built by predicates.",
"notes": "009-snippet.html",
"video": "https://www.youtube.com/embed/yIdECC6QExY?start=11",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/SDSU/Discrete/Predicates/predicateB8.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto009?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"010": {
"id": "010",
"label": "Operations on Sets",
"meta": " SLL03",
"content": "Sets can be joined or intersected in order to create new sets.",
"notes": "010-snippet.html",
"video": "https://www.youtube.com/embed/",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UMN/logicAndSetTheory/prob03.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto010?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"011": {
"id": "011",
"label": "Sums and Products",
"meta": " SLL03",
"content": "An important shorthand notation for calculations.",
"notes": "011-snippet.html",
"video": "https://www.youtube.com/embed/S5DdXfxl3ac?start=3",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Union/setIntSigmaNotation/an6_4_02.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto011?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"012": {
"id": "012",
"label": "Countable Sets",
"meta": "",
"content": "A notion of cardinality that covers finite sets and thos that can be enumerated via the natural numbers.",
"notes": "012-snippet.html",
"video": "https://www.youtube.com/embed/",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Rochester/setSetTheory1/ur_st_1_6.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto012?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/6JdbJSYkihEiLJQijcPkYO?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"013": {
"id": "013",
"label": "Bounded Sets,\nMaxima and Minima",
"meta": "",
"content": "The values inside a set of real numbers can be bounded.",
"notes": "013-snippet.html",
"video": "https://www.youtube.com/embed/",
"webwork": "",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto013?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"100": {
"id": "100",
"label": "Sequences",
"meta": " RA02 ",
"content": "These object are needed to define limits later on.",
"notes": "100-snippet.html",
"video": "https://www.youtube.com/embed/1SguKALJji8?start=17",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UMN/calculusStewartCCC/s_11_1_5.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto100?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"101": {
"id": "101",
"label": "Convergence",
"meta": " RA02 ",
"content": "Convergent sequences have a well-defined limit.",
"notes": "101-snippet.html",
"video": "https://www.youtube.com/embed/1SguKALJji8?start=367",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.1_Sequences/10.1.31.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto101?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"102": {
"id": "102",
"label": "Bounded\nSequences",
"meta": " RA03 ",
"content": "Sequences can be bounded from above and from below.",
"notes": "102-snippet.html",
"video": "https://www.youtube.com/embed/k-Wm6gJYfrw?start=221",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WinonaState/StewartCalcEarlyTran7ed/Chap11Section01/SCalcET7-11-1-037.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto102?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"103": {
"id": "103",
"label": "Limit Theorems",
"meta": " RA04 ",
"content": "Combining limits is a useful tool.",
"notes": "103-snippet.html",
"video": "https://www.youtube.com/embed/237VMLNVtQs?start=18",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Dartmouth/setStewartCh12S1/problem_7.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto103?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"104": {
"id": "104",
"label": "Monotonicity and\nSandwich Theorem",
"meta": " RA05 ",
"content": "Sandwich a sequence by two converging sequences to get its limit.",
"notes": "104-snippet.html",
"video": "https://www.youtube.com/embed/Y6rRSip3QN4?start=14",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Hope/Calc2/APEX_08_01_Sequences/Q_17.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto104?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"105": {
"id": "105",
"label": "Supremum and\nInfimum of Sets",
"meta": " RA06 ",
"content": "Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum.",
"notes": "105-snippet.html",
"video": "https://www.youtube.com/embed/8Cyvdv7Sm2s?start=16",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=%%%&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto105?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"106": {
"id": "106",
"label": "Completeness",
"meta": " RA07 ",
"content": "Completeness says that Cauchy sequences must converge.",
"notes": "106-snippet.html",
"video": "https://www.youtube.com/embed/R2AFZD0jiKQ?start=14",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest07/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto106?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/2oh22JbFBcqSdfBJHEYUGo?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"107": {
"id": "107",
"label": "Subsequences and\nAccumulation Values",
"meta": " RA09 ",
"content": "A sequence that does not converge may still have converging subsequences.",
"notes": "107-snippet.html",
"video": "https://www.youtube.com/embed/xZ5vjdZzTUI?start=14",
"webwork": "https://bright.jp-g.de/bsom/real_analysis/ratest09.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto107?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"108": {
"id": "108",
"label": "Bolzano-\nWeierstrass",
"meta": "RA10 ",
"content": "Every bounded sequence has at least one converging subsequence.",
"notes": "108-snippet.html",
"video": "https://www.youtube.com/embed/e2QaNklYZGg?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest10/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto108?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/4MKVAhO2q57xmMn1P7ONKg?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"109": {
"id": "109",
"label": "Limit Inferior and\nLimit Superior",
"meta": "RA11+RA12 ",
"content": "The largest and smallest limit of all convergent subsequences.",
"notes": "109-snippet.html",
"video": "https://www.youtube.com/embed/-y0v2V0-_8E?start=21",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest11/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto109?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"110": {
"id": "110",
"label": "Open, Closed,\nCompact Sets",
"meta": " RA13 ",
"content": "Important notions for subsets of real numbers.",
"notes": "110-snippet.html",
"video": "https://www.youtube.com/embed/Wqo4Svs4erw?start=12",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest13/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto110?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"111": {
"id": "111",
"label": "Heine-Borel\nTheorem",
"meta": " RA14 ",
"content": "The theorem connecting the concept of compactness with boundedness and closedness.",
"notes": "111-snippet.html",
"video": "https://www.youtube.com/embed/vjOefDHOVIg?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest14/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto111?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"112": {
"id": "112",
"label": "Interior, Closure,\nBoundary",
"meta": " ",
"content": "Topological operations on sets.",
"notes": "112-snippet.html",
"video": "https://www.youtube.com/embed/",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Union/setAlgebraIntervals/ur_ab_10_1.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto112?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"113": {
"id": "113",
"label": "Uncountability\nof the Reals",
"meta": " ",
"content": "The real numbers cannot be enumerated.",
"notes": "113-snippet.html",
"video": "https://www.youtube.com/embed/jCiIsigwaBE",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto113?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/7ElhBgZebKwOXqMsWMGgGJ?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"114": {
"id": "114",
"label": "Convergence of Bounded Monotonic Sequences",
"meta": " ",
"content": "If a sequence of real numbers is bounded and monotonic, then it is convergent.",
"notes": "114-snippet.html",
"video": "",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WinonaState/StewartCalcEarlyTran7ed/Chap11Section01/SCalcET7-11-1-027.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto114?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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"115": {
"id": "115",
"label": "Cauchy\nSequences",
"meta": " RA07 ",
"content": "The sequence members of a Cauchy Sequence eventually become arbitrarily close to each other.",
"notes": "115-snippet.html",
"video": "https://www.youtube.com/embed/R2AFZD0jiKQ?start=14",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest07/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto106?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/2oh22JbFBcqSdfBJHEYUGo?utm_source=generator&theme=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"200": {
"id": "200",
"label": "Partial Sums",
"meta": "RA15",
"content": "A series is a sequence of partial sums.",
"notes": "200-snippet.html",
"video": "https://www.youtube.com/embed/BgfP3riDcrc?start=13",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.2_Summing_an_Infinite_Series/10.2.3.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto200?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe style=\"border-radius:12px\" src=\"https://open.spotify.com/embed/episode/4rf4zO7V8UJErJxgzmZFgh?utm_source=generator&theme=0&t=0\" width=\"100%\" height=\"232\" frameBorder=\"0\" allowfullscreen=\"\" allow=\"autoplay; clipboard-write; encrypted-media; fullscreen; picture-in-picture\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"201": {
"id": "201",
"label": "Geometric and\nHarmonic Series",
"meta": "RA16",
"content": "The most important examples of series.",
"notes": "201-snippet.html",
"video": "https://www.youtube.com/embed/Y4yRcz-b17A?start=12",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setSequenceandSeries/jj14.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto201?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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},
"203": {
"id": "203",
"label": "Convergent Series and\nLimit Theorems",
"meta": "RA17",
"content": "Basic operations with convergent series.",
"notes": "203-snippet.html",
"video": "https://www.youtube.com/embed/lr12HwXQ3uw?start=20",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Rochester/setSeries4Geometric/ns8_2_23.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto203?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"204": {
"id": "204",
"label": "Cauchy Criterion",
"meta": "RA17",
"content": "A series convergence if its partial sums form a Cauchy sequence.",
"notes": "204-snippet.html",
"video": "https://www.youtube.com/embed/lr12HwXQ3uw?start=130",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/AlfredUniv/anton8e/chapter10/10.4/prob1.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto204?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"205": {
"id": "205",
"label": "Leibniz Criterion",
"meta": "RA18",
"content": "A convergence criterion for sums based on an alternating sequence.",
"notes": "205-snippet.html",
"video": "https://www.youtube.com/embed/MjjMwQ6zFko?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ma123DB/set11/s11_5_5.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto205?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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},
"206": {
"id": "206",
"label": "Absolute Convergence",
"meta": "RA19",
"content": "A strong concept of convergence of series.",
"notes": "206-snippet.html",
"video": "https://www.youtube.com/embed/mI40-tAtP58?start=21",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ma123DB/set11/s11_6_15.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto206?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"207": {
"id": "207",
"label": "Comparison Test",
"meta": "RA19",
"content": "If a series converges can be checked with different tests.",
"notes": "207-snippet.html",
"video": "https://www.youtube.com/embed/mI40-tAtP58?start=160",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WinonaState/StewartCalcEarlyTran7ed/Chap11Section04/SCalcET7-11-4-002a.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto207?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"208": {
"id": "208",
"label": "Quotient Criterion",
"meta": "RA20",
"content": "An important criterion to prove absolute convergence by means of ratios of the underlying sequence's terms.",
"notes": "208-snippet.html",
"video": "https://www.youtube.com/embed/yLbgdL9HAeg?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/FortLewis/Calc2/9-4-Ratio-test/ratio-05.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto208?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"209": {
"id": "209",
"label": "Root Criterion",
"meta": "RA20",
"content": "An important criterion to prove absolute convergence by means of the behavior of the n-th roots of the underlying sequence's terms.",
"notes": "209-snippet.html",
"video": "https://www.youtube.com/embed/yLbgdL9HAeg?start=420",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/10_Infinite_Series/10.5_The_Ratio_and_Root_Tests/10.5.37.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto209?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"210": {
"id": "210",
"label": "Reordering",
"meta": "RA21",
"content": "Series can be reordered.",
"notes": "210-snippet.html",
"video": "https://www.youtube.com/embed/GADre0hHc4c?start=9",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest21/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto210?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe src=\"https://anchor.fm/profmoppi/embed/episodes/Rearrangement-of-Series-with-Fabian-Gabel-e1iq2sr/a-a7vb2vp\" height=\"102px\" width=\"100%\" frameborder=\"0\" scrolling=\"no\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"211": {
"id": "211",
"label": "Cauchy Product",
"meta": "RA22",
"content": "A special way to multiply two absolutely convergent sequences.",
"notes": "211-snippet.html",
"video": "https://www.youtube.com/embed/tRa0Ex_0Yfo?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest22/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto211?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"300": {
"id": "300",
"label": "Sequences of\nBounded Functions",
"meta": "RA23",
"content": "The concept of sequences but for functions instead of real numbers.",
"notes": "300-snippet.html",
"video": "https://www.youtube.com/embed/RM2hytsyMpc?start=20",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest23/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto300?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"301": {
"id": "301",
"label": "Pointwise Convergence",
"meta": "RA24",
"content": "A notion of convergence for sequences functions that reduces the question of convergence to convergence of sequences of real numbers.",
"notes": "301-snippet.html",
"video": "https://www.youtube.com/embed/Kq_KZpljeXo?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest24/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto301?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"302": {
"id": "302",
"label": "Uniform Convergence",
"meta": "RA25",
"content": "A strong notion of convergence for sequences of functions that helps to preserve favorable properties like continuity in the limit.",
"notes": "302-snippet.html",
"video": "https://www.youtube.com/embed/O2HKxNcom7g?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest25/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto302?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"303": {
"id": "303",
"label": "Limits of Functions",
"meta": "RA26",
"content": "How function evaluations change when the argument approaches a certain point.",
"notes": "303-snippet.html",
"video": "https://www.youtube.com/embed/QoLlvvro6rE?start=15",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setRateChange/3-2-72.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto303?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"304": {
"id": "304",
"label": "Continuity",
"meta": "RA27",
"content": "The concept that relates functions with convergent sequences.",
"notes": "304-snippet.html",
"video": "https://www.youtube.com/embed/8VTG6EsyJh4?start=8",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setContinuity/4-1-57.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto304?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"305": {
"id": "305",
"label": "Epsilon-Delta\nDefinition",
"meta": "RA28",
"content": "A different notion of continuity using open intervals.",
"notes": "305-snippet.html",
"video": "https://www.youtube.com/embed/",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Berkeley/StewCalcET7e/2.4/2-4-03.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto305?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"306": {
"id": "306",
"label": "Continuity for Sums,\nProducts, Quotients,\nand Compositions",
"meta": "RA29",
"content": "How combination of continuous functions leads to new continuous functions.",
"notes": "306-snippet.html",
"video": "https://www.youtube.com/embed/W-E4LqZyEHA?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest29/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto306?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"307": {
"id": "307",
"label": "Continuous Images\nof Compact Sets\nAre Compact",
"meta": "RA30",
"content": "A mapping property for continuous functions.",
"notes": "307-snippet.html",
"video": "https://www.youtube.com/embed/6VWTG4wlRoA?start=11",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest30/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto307?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"308": {
"id": "308",
"label": "Uniform Limits of \n Continuous Functions",
"meta": "RA31",
"content": "How to preserve continuity in the limit.",
"notes": "308-snippet.html",
"video": "https://www.youtube.com/embed/llJruZnO-t4?start=11",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest31/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto308?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"309": {
"id": "309",
"label": "Intermediate Value\nTheorem",
"meta": "RA32",
"content": "This theorem tells us that continuous functions don't jump. They have to attain every value between two values in their image.",
"notes": "309-snippet.html",
"video": "https://www.youtube.com/embed/BNLu4_3Okuk?start=9",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Valdosta/APEX_Calculus/1.5/APEX_1.5_34.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto309?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<audio controls src=\"/e-10/pontifex/podcast/imvt.mp3\">Your browser does not support the <code>audio</code> element.</audio>"},
"400": {
"id": "400",
"label": "Exponential\nFunction",
"meta": "RA33",
"content": "A special function that can be defined via a power series.",
"notes": "400-snippet.html",
"video": "https://www.youtube.com/embed/onmh9nzkfDA?start=25",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UCSB/Stewart5_1_5/Stewart5_1_5_15.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto400?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<iframe src=\"https://anchor.fm/profmoppi/embed/episodes/Exponential-Series-with-Fabian-Gabel-e1iq43j\" height=\"102px\" width=\"100%\" frameborder=\"0\" scrolling=\"no\"></iframe><p>Courtesy of Marcus Waurick. <i>Well-defined & Wonderful podcast</i>, <a href=\"https://www.marcus-waurick.de/teaching\">marcus-waurick.de</a>.</p>"
},
"401": {
"id": "401",
"label": "Logarithm Function",
"meta": "RA33",
"content": "The inverse of the exponential function.",
"notes": "401-snippet.html",
"video": "https://www.youtube.com/embed/onmh9nzkfDA?start=213",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setLogarithmicFunctions/srw4_3_43.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto401?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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},
"402": {
"id": "402",
"label": "Polynomials",
"meta": "RA33",
"content": "A basic class of functions that consists a linear combinations of monomials.",
"notes": "402-snippet.html",
"video": "https://www.youtube.com/embed/onmh9nzkfDA?start=297",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/CollegeOfIdaho/setAlgebra_05_01_IntroPolynomials/51IntAlg_03_Polynomial.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto402?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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},
"403": {
"id": "403",
"label": "Power Series",
"meta": "RA33",
"content": "A sequence of partial sums of polynomial functions.",
"notes": "403-snippet.html",
"video": "https://www.youtube.com/embed/onmh9nzkfDA?start=332",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Indiana/Indiana_setSeries8Power/eva8_5a_2.pg&problemSeed=123567890&displayMode=MathJax&courseID=pontifex&userID=pontifexuser&course_password=ki(JH7j3m4)k_46)&outputformat=simple",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto403?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
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"500": {
"id": "500",
"label": "Differentiability",
"meta": "RA34",
"content": "How to quantify the rate of change of a function.",
"notes": "500-snippet.html",
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"edges": {
"000-001": {
"source": "000",
"target": "001",
"label": "Logical statements usually contain sets and elements"
},
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"target": "008",
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"001-002": {
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"001-004": {
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"label": "Convergent sequences are bounded."
},
"002-102": {
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"label": "The statement of boundedness involves a comparison of real numbers."
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"001-013": {
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"target": "013",
"label": "Bounded sets are sets of real numbers that don't get arbitrarily small or large."
},
"005-013": {
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"target": "013",
"label": "If the image or preimage of a real valued map is bounded, one also calls the map bounded."
},
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"source": "013",
"target": "300",
"label": "If the image of a map is a bounded set, one also calls the map bounded."
},
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"target": "300",
"label": "One bound to a map is given by the finite supremum of the set of the absolute function values."
},
"005-102": {
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"target": "102",
"label": "Boundedness of a sequence means that the image of the sequence is a bounded set."
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"target": "102",
"label": "Boundedness is a property of sequences."
},
"003-005": {
"source": "003",
"target": "005",
"label": "A mapping induces images and preimages."
},
"005-006": {
"source": "005",
"target": "006",
"label": "Injectivity and surjectivity can be expressed in terms of images and preimages."
},
"005-305": {
"source": "005",
"target": "305",
"label": "Characterize continuity in terms of preimages of open balls."
},
"005-307": {
"source": "005",
"target": "307",
"label": "Topological properties of images under continuous functions."
},
"110-307": {
"source": "110",
"target": "307",
"label": "Continuous functions preserve compactness of sets in their image."
},
"111-307": {
"source": "111",
"target": "307",
"label": "The Heine-Borel theorem gives the existence of maxima and minima for continuous functions defined on compact sets."
},
"307-506": {
"source": "307",
"target": "506",
"label": "The existence of minima and maxima for continuous functions on compact sets is necessary for the proof of Rolle's theorem."
},
"000-006": {
"source": "000",
"target": "006",
"label": "Image and Preimage are characterized via logical statements involving quantifiers."
},
"003-007": {
"source": "003",
"target": "007",
"label": "Composition is an operation on maps."
},
"001-105": {
"source": "001",
"target": "105",
"label": "Supremum and infimum are numbers associated to sets of real numbers."
},
"002-105": {
"source": "002",
"target": "105",
"label": "Supremum and infimum are numbers associated to sets of real numbers."
},
"100-115": {
"source": "100",
"target": "115",
"label": "Being Cauchy is a property of a sequence."
},
"115-106": {
"source": "115",
"target": "106",
"label": "Completeness can be defined by stating that every Cauchy sequences converges."
},
"105-106": {
"source": "105",
"target": "106",
"label": "Existence of suprema of bounded sets is a characterization of completeness."
},
"001-110": {
"source": "001",
"target": "110",
"label": "Open, closed and compact are properties of sets of numbers."
},
"001-112": {
"source": "001",
"target": "112",
"label": "The interior, the closure or the boundary of a set is a set again."
},
"002-100": {
"source": "002",
"target": "100",
"label": "A sequence is a map having the real numbers as codomain "
},
"002-101": {
"source": "002",
"target": "101",
"label": "The definition of convergence involves the absolute value of a difference of real numbers and a quantitative comparison with another real number."
},
"002-106": {
"source": "002",
"target": "106",
"label": "Real numbers are complete by the completeness axiom."
},
"003-100": {
"source": "003",
"target": "100",
"label": "A sequence is a map that assigns to each natural number a value."
},
"003-300": {
"source": "003",
"target": "300",
"label": "Function is just another name for map."
},
"003-304": {
"source": "003",
"target": "304",
"label": "Continuity is a central notion for maps on the real numbers."
},
"004-011": {
"source": "004",
"target": "011",
"label": "A lot of identities for sums and products are proved via induction."
},
"004-100": {
"source": "004",
"target": "100",
"label": "A sequence is a map having the natural numbers as domain."
},
"006-400": {
"source": "006",
"target": "400",
"label": "The exponential function is a bijective function onto the positive reals."
},
"400-401": {
"source": "400",
"target": "401",
"label": "The bijectivity of the exponential gives the logarithm function as inverse."
},
"206-400": {
"source": "206",
"target": "400",
"label": "The exponential series is absolutely convergent."
},
"211-400": {
"source": "211",
"target": "400",
"label": "The functional equation of the exponential is a consequence of the Cauchy product."
},
"006-505": {
"source": "006",
"target": "505",
"label": "Bijective differentiable functions can be differentiated using the inversion formula."
},
"006-401": {
"source": "006",
"target": "401",
"label": "The natural logarithm is the inverse of the exponential function."
},
"304-401": {
"source": "304",
"target": "401",
"label": "The natural logarithem is the inverse of a continuous function."
},
"007-306": {
"source": "007",
"target": "306",
"label": "Composition of continuous functions gives a continuous function."
},
"007-502": {
"source": "007",
"target": "502",
"label": "How to differentiate a composition of functions."
},
"500-502": {
"source": "500",
"target": "502",
"label": "Break down differentiability of compositions to the differentiability of simpler functions."
},
"100-101": {
"source": "100",
"target": "101",
"label": "Having a limit is a property of a sequence."
},
"101-109": {
"source": "101",
"target": "109",
"label": "Limit inferior and limit superior are in particular limits."
},
"107-109": {
"source": "107",
"target": "109",
"label": "Limit inferior and limit superior are the largest and smallest accumulation points of a subsequence."
},
"101-103": {
"source": "101",
"target": "103",
"label": "Calculating limits of sums, products or quotients of convergent sequences."
},
"101-104": {
"source": "101",
"target": "104",
"label": "Monotonicity and boundedness imply convergence."
},
"101-117": {
"source": "101",
"target": "115",
"label": "Every convergent sequence is also a Cauchy sequence."
},
"101-110": {
"source": "101",
"target": "110",
"label": "Closedness is characterized by convergence of sequences."
},
"107-110": {
"source": "107",
"target": "110",
"label": "Compactness is characterized by existence of converging subsequences."
},
"101-107": {
"source": "101",
"target": "107",
"label": "A sequence may have a convergent subsequence and this limit is then an accumulation value of the original sequence."
},
"101-108": {
"source": "101",
"target": "108",
"label": "Bolzano Weierstrass guarantees convergence of certain sequences."
},
"102-104": {
"source": "102",
"target": "104",
"label": "A bound for the sequence is also a bound for the limit by the monotonicity of limits."
},
"102-108": {
"source": "102",
"target": "108",
"label": "The Bolzano Weierstrass Theorem guarantees the existence of accumulation points for bounded sequences."
},
"103-203": {
"source": "103",
"target": "203",
"label": "Considering series as limits of sequences of partial sums leads to limit theorems for sequences."
},
"000-108": {
"source": "000",
"target": "108",
"label": "Bolzano Weierstrass theorem is a logical statement."
},
"108-109": {
"source": "108",
"target": "109",
"label": "Bolzano Weierstrass theorem guarantees the existence of a limit inferior and superior."
},
"110-111": {
"source": "110",
"target": "111",
"label": "Heine-Borel theorem characterizes bounded and closed sets."
},
"110-112": {
"source": "110",
"target": "112",
"label": "Interior is an open set, closure and boundary are closed sets."
},
"111-112": {
"source": "111",
"target": "112",
"label": "The closure of a bounded set is always compact by Heine-Borel theorem."
},
"104-111": {
"source": "104",
"target": "111",
"label": "Sandwiching gives rise to the proof using nested intervals in the Heine-Borel theorem."
},
"104-309": {
"source": "104",
"target": "309",
"label": "The convergence of the intervals is based on monotonicity of the underlying sequences."
},
"108-309": {
"source": "108",
"target": "309",
"label": "The nested intervals technique used in the proof of the Bolzano-Weierstrass theorem is also used for the proof of the intermediate value theorem."
},
"107-108": {
"source": "107",
"target": "108",
"label": "Bolzano Weierstrass guarantees existence of accumulation values."
},
"100-200": {
"source": "100",
"target": "200",
"label": "A series is a sequence of partial sums."
},
"101-200": {
"source": "101",
"target": "200",
"label": "A convergent series is a convergent sequence."
},
"011-200": {
"source": "011",
"target": "200",
"label": "Partial sums are described using the formal sum notation."
},
"200-201": {
"source": "200",
"target": "201",
"label": " "
},
"200-203": {
"source": "200",
"target": "203",
"label": " "
},
"200-204": {
"source": "200",
"target": "204",
"label": "The Cauchy criterion works for sequences of partial sums that are Cauchy."
},
"106-204": {
"source": "106",
"target": "204",
"label": "Cauchy sequences converge, therefore the Cauchy criterion for series gives convergence of a series."
},
"115-204": {
"source": "115",
"target": "204",
"label": "If the sequence of partial sums is a Cauchy sequence, then it converges."
},
"102-204": {
"source": "102",
"target": "204",
"label": "The Cauchy criterion involves bounds on certain parts of the partial sum."
},
"008-204": {
"source": "008",
"target": "204",
"label": "The necessary criterion for series convergence is an implication but no equivalence."
},
"203-205": {
"source": "203",
"target": "205",
"label": "A sufficient criterion for convergence of a series."
},
"008-205": {
"source": "008",
"target": "205",
"label": "The Leibniz criterion gives the sufficient condition for convergence of a series but no equivalence."
},
"200-205": {
"source": "200",
"target": "205",
"label": "The Leibniz criterion makes a statement about partial sums of an alternating sequence."
},
"104-205": {
"source": "104",
"target": "205",
"label": "The Leibniz criterion makes a statement about partial sums of an monotonic sequence."
},
"200-206": {
"source": "200",
"target": "206",
"label": "A strong concept of convergence for sequences of partial sums."
},
"201-206": {
"source": "201",
"target": "206",
"label": "The geometric series is an example of an absolutely convergent sequence."
},
"205-206": {
"source": "205",
"target": "206",
"label": "The Leibniz theorem shows that the alternating harmonic series is convergent but not absolutely convergent."
},
"200-207": {
"source": "200",
"target": "207",
"label": "The comparison tests compare the terms of the partial sums to the ones of other series."
},
"200-208": {
"source": "200",
"target": "208",
"label": " "
},
"102-209": {
"source": "102",
"target": "209",
"label": "If the nth root of the underlying sequence is bounded by a number striclty smaller than one, the series converges by the root criterion."
},
"200-210": {
"source": "200",
"target": "210",
"label": "Reordering a series leads to a new sequence of partial sums."
},
"206-210": {
"source": "206",
"target": "210",
"label": "Reordering an absolutely convergent sequence does not change the limit."
},
"203-210": {
"source": "203",
"target": "210",
"label": "Reordering of a series may change the limit of the series."
},
"006-210": {
"source": "006",
"target": "210",
"label": "Reordering of a series is defined via an bijective mapping of natural numbers."
},
"200-211": {
"source": "200",
"target": "211",
"label": " "
},
"109-209": {
"source": "109",
"target": "209",
"label": "The root criterion in its limit form gives a condition for the limsup of the underlying sequence."
},
"209-403": {
"source": "209",
"target": "403",
"label": "The root criterion is useful for determining domains of convergence."
},
"201-209": {
"source": "201",
"target": "209",
"label": "The geometric series is used in the proof of the root criterion as a majorant."
},
"201-207": {
"source": "201",
"target": "207",
"label": "The geometric series is often used as a majorant in the comparison test."
},
"201-208": {
"source": "201",
"target": "208",
"label": "The geometric series is used in the proof of the quotient criterion as a majorant."
},
"207-208": {
"source": "207",
"target": "208",
"label": "The proof of the quotient criterion relies on majorant criterion."
},
"208-400": {
"source": "208",
"target": "400",
"label": "By the quotient criterion on can show that the exponential series converges absolutely."
},
"207-209": {
"source": "207",
"target": "209",
"label": "The proof of the root criterion relies on majorant criterion."
},
"206-208": {
"source": "206",
"target": "208",
"label": " The quotient criterion is a statement about absolute convergence of a series."
},
"206-209": {
"source": "206",
"target": "209",
"label": " The root criterion is a statement about absolute convergence of a series."
},
"206-211": {
"source": "206",
"target": "211",
"label": "Taking the Cauchy product is an operation on two absolute convergent series."
},
"005-300": {
"source": "005",
"target": "300",
"label": "Boundedness of a function can be expressed in terms of the image or range."
},
"100-300": {
"source": "100",
"target": "300",
"label": "A sequence of functions gives rise to a sequence of real numbers via point evaluations."
},
"101-303": {
"source": "101",
"target": "303",
"label": "Limits of functions are limits of sequences of real numbers."
},
"001-300": {
"source": "001",
"target": "300",
"label": "Bounded functions on an interval form a set."
},
"300-301": {
"source": "300",
"target": "301",
"label": "Pointwise convergence is a notion of convergence for sequences of functions."
},
"301-302": {
"source": "301",
"target": "302",
"label": "Pointwise convergence is a weaker notion of convergence for sequences of functions than uniform convergence."
},
"300-308": {
"source": "300",
"target": "308",
"label": "Properties of the limit of a sequence of continuous functions."
},
"301-308": {
"source": "301",
"target": "308",
"label": "The pointwise limit of a sequences of continuous functions does not have to be continuous."
},
"300-400": {
"source": "300",
"target": "400",
"label": "The exponential series can be considered as a sequence of partial sums of functions"
},
"200-400": {
"source": "200",
"target": "400",
"label": "The exponential series can be considered as a sequence of partial sums of functions"
},
"300-403": {
"source": "300",
"target": "403",
"label": "A power series is essentially a sequence of partial sums of functions."
},
"001-010": {
"source": "001",
"target": "010",
"label": "We use different operations to work with multiple sets."
},
"010-110": {
"source": "010",
"target": "110",
"label": "Set operations may be used to modify set properties like opennes, closedness or compactness."
},
"304-500": {
"source": "304",
"target": "500",
"label": "Differentiability of a function is a stronger property than continuity."
},
"500-501": {
"source": "500",
"target": "501",
"label": "How to differentiate a sum of differentiable functions."
},
"500-506": {
"source": "500",
"target": "506",
"label": "Rolle's theorem makes a statement about zeros of the first derivative of a differentiable function."
},
"506-507": {
"source": "506",
"target": "507",
"label": "A notion stronger than pointwise convergence is uniform convergence."
},
"300-302": {
"source": "300",
"target": "302",
"label": "Special notion of convergence for sequences of functions."
},
"101-304": {
"source": "101",
"target": "304",
"label": "Continuous functions map convergent sequences to convergent sequences."
},
"304-400": {
"source": "304",
"target": "400",
"label": "The exponential function is continuous."
},
"103-306": {
"source": "103",
"target": "306",
"label": "Continuity of sums, products, and quotients is preserved as a consequence of the limit theorems for sequences."
},
"304-306": {
"source": "304",
"target": "306",
"label": "Continuity is preserved under certain operations with functions."
},
"502-609": {
"source": "502",
"target": "609",
"label": "Integration by substitution is loosely speaking the inverse operation to calculating the derivative via the chain rule."
},
"304-307": {
"source": "304",
"target": "307",
"label": "Continuity preverves compactness of sets."
},
"304-309": {
"source": "304",
"target": "309",
"label": "Continuous functions on intervals don't have jumps. Instead they attain every value in between to elements of their range."
},
"303-304": {
"source": "303",
"target": "304",
"label": "If a function is continuous at a point, the the value coincides with the function limit at this point."
},
"302-304": {
"source": "302",
"target": "304",
"label": "Uniform convergence of continuous functions leads to a continuous limit function."
},
"305-308": {
"source": "305",
"target": "308",
"label": "The epsilon-delta criterion allows for an elegant proof of the continuity of the uniform limit of a sequence of continuous functions."
},
"103-303": {
"source": "103",
"target": "303",
"label": "Calculating limits of functions builds on the limit theorems of sequences."
},
"402-403": {
"source": "402",
"target": "403",
"label": "Each partial sum of a power series is a polynomial."
},
"110-305": {
"source": "110",
"target": "305",
"label": "Characterize continuity using open intervals."
},
"402-511": {
"source": "402",
"target": "511",
"label": "Use polynomials to approximate differentiable functions."
},
"303-508": {
"source": "303",
"target": "508",
"label": "Calculate limits of functions using l'Hospital's rule."
},
"403-511": {
"source": "403",
"target": "511",
"label": "A Taylor series is a particular power series."
},
"508-509": {
"source": "508",
"target": "509",
"label": "Further situations where limits can be calculated using l'Hospital's rule."
},
"403-503": {
"source": "403",
"target": "503",
"label": "Uniform convergence within the domain of convergence of a power series."
},
"300-503": {
"source": "300",
"target": "503",
"label": "Differentiability of sequences of functions."
},
"500-503": {
"source": "500",
"target": "503",
"label": "Differentiability of sequences of functions."
},
"301-503": {
"source": "301",
"target": "503",
"label": "Proving pointwise convergence of a sequence of differentiable functions is part of proving the differentiability of the limit function."
},
"302-503": {
"source": "302",
"target": "503",
"label": "Proving uniform convergence of the derivatives of a function is part of proving the differentiability of the limit function."
},
"510-511": {
"source": "510",
"target": "511",
"label": "Evaluations of higher derivatives form the coefficients of a Taylor polynomial."
},
"511-512": {
"source": "511",
"target": "512",
"label": "Application of Taylor's theorem to a particular example."
},
"512-513": {
"source": "512",
"target": "513",
"label": "Proof of Taylor's theorem."
},
"304-305": {
"source": "304",
"target": "305",
"label": "A different notion of continuity using open balls."
},
"002-600": {
"source": "002",
"target": "600",
"label": "Functions that are constant on intervals of real numbers."
},
"003-600": {
"source": "003",
"target": "600",
"label": "Step functions are a particular type of piecewise defined mapping."
},
"600-601": {
"source": "600",
"target": "601",
"label": "A first notion of area under a function graph for step functions."
},
"601-603": {
"source": "601",
"target": "603",
"label": "Generalizing the concept of an integral to bounded functions via approximation."
},
"500-606": {
"source": "500",
"target": "606",
"label": "Differentiation is in some sense the opposite of integration."
},
"500-606": {
"source": "500",
"target": "606",
"label": "Differentiation is in some sense the opposite of integration."
},
"601-602": {
"source": "601",
"target": "602",
"label": "The definition of the Riemann integral gives rise to nice properties."
},
"603-609": {
"source": "603",
"target": "609",
"label": "A tool to calculate the integral of a bounded function."
},
"006-609": {
"source": "006",
"target": "609",
"label": "Transformations of the domain of integration need to be invertible or at least injective."
},
"500-609": {
"source": "500",
"target": "609",
"label": "Transformations of the domain of integration need to be differentiable."
},
"400-609": {
"source": "400",
"target": "609",
"label": "Using the substitution rule may help to calculate integrals of exponentials."
},
"007-609": {
"source": "007",
"target": "609",
"label": "Some integrands in the substitution rule involve compositions of functions."
},
"207-613": {
"source": "207",
"target": "613",
"label": "Comparison principles aim at inheriting a nice property from a well-known object to a new object of study."
},
"606-607": {
"source": "606",
"target": "607",
"label": "The second fundamental theorem tells us how antiderivatives differ."
},
"500-607": {
"source": "500",
"target": "607",
"label": "The second fundamental theorem connects integration and differentiation."
},
"601-604": {
"source": "601",
"target": "604",
"label": "By definition the Riemann integral consists of limits of integrals of step functions."
},
"603-605": {
"source": "603",
"target": "605",
"label": "Properties of the Riemann integral."
},
"605-609": {
"source": "605",
"target": "609",
"label": "The Riemann integral comes with an order of the integral boundaries."
},
"605-608": {
"source": "605",
"target": "608",
"label": "A further property of the Riemann integral is the mean-value theorem."
},
"501-610": {
"source": "501",
"target": "610",
"label": "This integration rule can be seen as an inversion of the product rule of differentiation."
},
"603-610": {
"source": "603",
"target": "610",
"label": "A further important rule for calculating integrals."
},
"507-608": {
"source": "507",
"target": "608",
"label": "A further mean value property for integrals instead of derivatives."
},
"603-604": {
"source": "603",
"target": "604",
"label": "Compute integrals according to the definition."
},
"603-607": {
"source": "603",
"target": "607",
"label": "The second fundamental theorem connects integration and differentiation."
},
"203-613": {
"source": "203",
"target": "613",
"label": "Characterize the convergence of an integral by the convergence of a suitable series."
},
"612-613": {
"source": "612",
"target": "613",
"label": "Estimate improper Riemann integrals via series."
},
"604-606": {
"source": "604",
"target": "606",
"label": "The fundamental theorem of calculus allows to reuse examples of integrals and derivatives in order to calculate new integrals."
},
"603-612": {
"source": "603",
"target": "612",
"label": "An improper integral is defined via a limit of Rimeann integrals on bounded sets."
},
"003-606": {
"source": "003",
"target": "606",
"label": "An antiderivative is also a map."
},
"303-612": {
"source": "303",
"target": "612",
"label": "An improper integral is defined as the limit of a function (the antiderivative)."
},
"003-402": {
"source": "003",
"target": "402",
"label": "Polynomials can be interpreted as real valued functions."
},
"609-611": {
"source": "609",
"target": "611",
"label": "The substitution rule is often used for calculating integrals of rational functions."
},
"108-106": {
"source": "108",
"target": "106",
"label": "The fact that every Cauchy sequence converges can be derived from the Bolzano-Weierstraß theorem."
},
"102-117": {
"source": "102",
"target": "115",
"label": "Cauchy sequences are bounded."
},
"402-611": {
"source": "402",
"target": "611",
"label": "How to integrate rational functions."
},
"008-009": {
"source": "008",
"target": "009",
"label": "How to construct logical statements."
},
"004-012": {
"source": "004",
"target": "012",
"label": "Countable sets can be bijectively mapped to the natural numbers or a finite subset of the natural numbers."
},
"012-113": {
"source": "012",
"target": "113",
"label": "Uncountability of the real numbers in particular means that the reals are not a countable set."
},
"104-113": {
"source": "104",
"target": "113",
"label": "The uncountability of the reals can be shown via a interval nesting which converges do to the monotonicity of the underlying sequences."
},
"106-113": {
"source": "106",
"target": "113",
"label": "Uncountability of the real numbers can be proven as a consequence of the completeness of the real numbers."
},
"006-012": {
"source": "006",
"target": "012",
"label": "Countable sets can be bijectively mapped to the natural numbers or a finite subset of the natural numbers."
},
"203-207": {
"source": "203",
"target": "207",
"label": "Characterize the convergence/divergence of a series by the convergence/divergence of a majorant/minorant."
}
}
}