Newer
Older
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
{
"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 Preimage",
"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, Maxima 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": "Boundedness",
"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-Weierstrass",
"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",
"podcast": ""
},
"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",
"podcast": ""
},
"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",
"podcast": ""
},
"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",
"podcast": ""
},
"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",
"podcast": ""
},
"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",
"podcast": ""
},
"500": {
"id": "500",
"label": "Differentiability",
"meta": "RA34",
"content": "How to quantify the rate of change of a function.",
"notes": "500-snippet.html",
"video": "https://www.youtube.com/embed/TLdBLqPTsYc?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/3_Differentiation/3.1_Definition_of_the_Derivative/3.1.51.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/discussingbridgesfromandto500?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"501": {
"id": "501",
"label": "Sum and Product Rule",
"meta": "RA35",
"content": "Useful rules for differentiating sums or products of differentiable functions.",
"notes": "501-snippet.html",
"video": "https://www.youtube.com/embed/wp-s9c1IKhI?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Union/setDervProductQuotientRule/s2_2_13.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/discussingbridgesfromandto501?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"502": {
"id": "502",
"label": "Chain Rule",
"meta": "RA36",
"content": "How to differentiate compositions of functions.",
"notes": "502-snippet.html",
"video": "https://www.youtube.com/embed/g57hlenwvis?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ASU-topics/setChainRuleExpLogFunctions/5-3-65.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/discussingbridgesfromandto502?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"503": {
"id": "503",
"label": "Uniform Convergence \n for Differentiable Functions",
"meta": "RA37",
"content": "Compatibility of differentiability with uniform limits.",
"notes": "503-snippet.html",
"video": "https://www.youtube.com/embed/PwkaPb_sCVQ?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest37/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto503?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"504": {
"id": "504",
"label": "Examples of \nDifferentiable Functions",
"meta": "RA38",
"content": "Calculation of derivatives by example.",
"notes": "504-snippet.html",
"video": "https://www.youtube.com/embed/93i7uKScVvc?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/ma122DB/set4b/s3_4_10.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/discussingbridgesfromandto504?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"505": {
"id": "505",
"label": "Inversion Formula",
"meta": "RA39",
"content": "How to calculate the derivative of the inverse function.",
"notes": "505-snippet.html",
"video": "https://www.youtube.com/embed/h0nBAMhdSMk?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/3_Differentiation/3.8_Derivatives_of_Inverse_Functions/3.8.7.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/discussingbridgesfromandto505?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"506": {
"id": "506",
"label": "Rolle's Theorem",
"meta": "RA40",
"content": "The derivatives of functions with equal boundary conditions always have at least one zero.",
"notes": "506-snippet.html",
"video": "https://www.youtube.com/embed/5Scawd2WLLA?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest40/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto506?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"507": {
"id": "507",
"label": "Mean Value Theorem",
"meta": "RA41",
"content": "This theorem helps us to link monotonicity of a function with values of its derivative.",
"notes": "507-snippet.html",
"video": "https://www.youtube.com/embed/FQo9OYku5aY?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/4_Applications_of_the_Derivative/4.3_The_Mean_Value_Theorem_and_Monotonicity/4.3.1.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/discussingbridgesfromandto507?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": "<audio controls src=\"/e-10/pontifex/podcast/mvt.mp3\">Your browser does not support the <code>audio</code> element.</audio>"
},
"508": {
"id": "508",
"label": "Theorem of l'Hospital",
"meta": "RA42",
"content": "An important tool for calculating limits.",
"notes": "508-snippet.html",
"video": "https://www.youtube.com/embed/KbS_cRToPFA?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/4_Applications_of_the_Derivative/4.5_LHopitals_Rule/4.5.40.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/discussingbridgesfromandto508?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"509": {
"id": "509",
"label": "Generalisations of\nl'Hospital's Rule",
"meta": "RA43",
"content": "Further scenarios in which limits can be calculated by l'Hospital's rule.",
"notes": "509-snippet.html",
"video": "https://www.youtube.com/embed/KuF0JRsWhBk?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Valdosta/APEX_Calculus/6.7/APEX_6.7_27.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/discussingbridgesfromandto509?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"510": {
"id": "510",
"label": "Higher Derivatives",
"meta": "RA44",
"content": "Taking derivatives of derivatives of differentiable functions.",
"notes": "510-snippet.html",
"video": "https://www.youtube.com/embed/vyZ5ESoqsxw?start=11",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=brary/Valdosta/APEX_Calculus/3.4/APEX_3.4_13.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/discussingbridgesfromandto510?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"511": {
"id": "511",
"label": "Taylor's Theorem",
"meta": "RA45",
"content": "An approximation method for differentiable functions.",
"notes": "511-snippet.html",
"video": "https://www.youtube.com/embed/Pb390hRaLrw?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest45/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto511?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"512": {
"id": "512",
"label": "Application for\nTaylor's Theorem",
"meta": "RA46",
"content": "Calculate an approximation via Taylor's Theorem",
"notes": "512-snippet.html",
"video": "https://www.youtube.com/embed/zRoyHrMNOO8?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/FortLewis/Calc2/10-2-Taylor-series/Taylor-series-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/discussingbridgesfromandto512?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"513": {
"id": "513",
"label": "Proof of\nTaylor's Theorem",
"meta": "RA47",
"content": "Derive an approximation result from the generalised mean value theorem.",
"notes": "513-snippet.html",
"video": "https://www.youtube.com/embed/oZZrwKsqVro?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest47/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto513?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"600": {
"id": "600",
"label": "Partitions and\nStep Functions",
"meta": "RA48",
"content": "Splitting up an innterval in subintervals and defining functions that are constant on them.",
"notes": "600-snippet.html",
"video": "https://www.youtube.com/embed/joXBmJ1KInU?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest48/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto600?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"601": {
"id": "601",
"label": "Riemann Integral for\nStep Functions",
"meta": "RA49",
"content": "A notion of integral for a simple class of functions.",
"notes": "601-snippet.html",
"video": "https://www.youtube.com/embed/VdXLenrjIao?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest49/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto601?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"602": {
"id": "602",
"label": "Linearity and Monotonicity\nof the Riemann Integral",
"meta": "RA49",
"content": "Important properties of the Riemann integral.",
"notes": "602-snippet.html",
"video": "https://www.youtube.com/embed/6Pb97_7huwI?start=34",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest50/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto602?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"603": {
"id": "603",
"label": "Riemann Integral\nfor Bounded Functions",
"meta": "RA51",
"content": "Notion of integral for a large class of functions.",
"notes": "603-snippet.html",
"video": "https://www.youtube.com/embed/t8Hh73HxP1o?start=12",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest51/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto603?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"604": {
"id": "604",
"label": "Examples for\nCalculating the\nRiemann Integral",
"meta": "RA52",
"content": "Use the approximation by step functions to calculate integrals.",
"notes": "604-snippet.html",
"video": "https://www.youtube.com/embed/J9qXHzxeDN4?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest52/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto604?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"605": {
"id": "605",
"label": "Properties of\nthe Riemann Integral",
"meta": "RA53",
"content": "Important properties of the Riemann integral of bounded functions.",
"notes": "605-snippet.html",
"video": "https://www.youtube.com/embed/h4XohuM2iK4?start=10",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest53/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto605?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"606": {
"id": "606",
"label": "First Fundamental\nTheorem of Calculus",
"meta": "RA54",
"content": "Linking differentiation and integration.",
"notes": "606-snippet.html",
"video": "https://www.youtube.com/embed/AKhlP6IHDLk?start=13",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Michigan/Chap6Sec2/Q21.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/discussingbridgesfromandto606?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"607": {
"id": "607",
"label": "Second Fundamental\nTheorem of Calculus",
"meta": "RA55",
"content": "Characterization of all antiderivatives",
"notes": "607-snippet.html",
"video": "https://www.youtube.com/embed/SJC4DGuyg4c?start=12",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Rochester/setIntegrals4FTC/osu_in_4_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/discussingbridgesfromandto607?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"608": {
"id": "608",
"label": "Mean Value Theorem\nof Integration",
"meta": "RA56",
"content": "Continuous functions attain their mean value.",
"notes": "608-snippet.html",
"video": "https://www.youtube.com/embed/E4zieCbfdcs?start=12",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest56/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto608?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"609": {
"id": "609",
"label": "Substitution Rule\nfor Integration",
"meta": "RA57",
"content": "An important integration rule.",
"notes": "609-snippet.html",
"video": "https://www.youtube.com/embed/wmZCoV6Y0_c?start=12",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Valdosta/APEX_Calculus/6.1/APEX_6.1_3-6.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/discussingbridgesfromandto609?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"610": {
"id": "610",
"label": "Integration by Parts",
"meta": "RA58",
"content": "An important integration rule.",
"notes": "610-snippet.html",
"video": "https://www.youtube.com/embed/wmZCoV6Y0_c?start=12",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Rochester/setIntegrals15ByParts/sc5_6_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/discussingbridgesfromandto610?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"611": {
"id": "611",
"label": "Partial Fraction\nDecomposition",
"meta": "RA59",
"content": "How to integrate rational functions.",
"notes": "611-snippet.html",
"video": "https://www.youtube.com/embed/dLYIijHqH9U?start=10",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/UMN/calculusStewartET/s_7_4_prob04.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/discussingbridgesfromandto611?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"612": {
"id": "612",
"label": "Improper Riemann Integrals",
"meta": "RA60",
"content": "How to integrate\non unbounded domains.",
"notes": "612-snippet.html",
"video": "https://www.youtube.com/embed/Q7qdknAF-ME?start=15",
"webwork": "https://jupyterhub.mat.tu-harburg.de/webwork2/html2xml?&answersSubmitted=0&sourceFilePath=Library/Michigan/Chap7Sec8/Q32.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/discussingbridgesfromandto612?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
},
"613": {
"id": "613",
"label": "Comparison Test",
"meta": "RA61",
"content": "Compare improper Riemann integrals to infinite series.",
"notes": "613-snippet.html",
"video": "https://www.youtube.com/embed/yEp9BTDgOjk?start=13",
"webwork": "https://bright.jp-g.de/bsom/ra/ratest61/quiz.html",
"discussion": "https://etherpad.studiumdigitale.uni-frankfurt.de/p/discussingbridgesfromandto613?showControls=true&showChat=true&showLineNumbers=true&useMonospaceFont=false",
"podcast": ""
}
},
"edges": {
"000-001": {
"source": "000",
"target": "001",
"label": "Logical statements usually contain sets and elements"
},
"000-008": {
"source": "000",
"target": "008",
"label": "Logical deduction techniques are used to derive new logical statements from old ones"
},
"000-101": {
"source": "000",
"target": "101",
"label": "Logical Statement, Quantifiers"
},
"000-104": {
"source": "000",
"target": "104",
"label": "Logical Statement, Quantifiers"
},
"000-111": {
"source": "000",
"target": "111",
"label": "Logical Statement, Quantifiers"
},
"001-003": {
"source": "001",
"target": "003",
"label": "A mapping operates between two sets."
},
"001-002": {
"source": "001",
"target": "002",
"label": "Real numbers are a set of numbers."
},
"001-004": {
"source": "001",
"target": "004",
"label": "Natural numbers are a set of numbers."
},
"001-005": {
"source": "001",
"target": "005",
"label": "Image and Preimage are special sets related to a mapping."
},
"002-102": {
"source": "002",
"target": "102",
"label": "The statement of boundedness involves a comparison of real numbers."
},
"001-102": {
"source": "001",
"target": "102",
"label": "Boundedness is a property of sets."
},
"005-102": {
"source": "005",
"target": "102",
"label": "Boundedness of a sequence means that the image of the sequence is a bounded set."
},
"100-102": {
"source": "100",
"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."
},