dr. vogel's gallery of calculus pathologies
TRANSCRIPT
-
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
1/34
Dr. Vogel's Gallery of Calculus Pathologies http://www.math.tamu.edu/~tvogel/gallery/gallery.html
1 of 2 22.02.2011 23:00
http://www.math.tamu.edu/~tvogel/gallery/gallery.htmlhttp://www.math.tamu.edu/~tvogel/gallery/gallery.html -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
2/34
Dr. Vogel's Gallery of Calculus Pathologies http://www.math.tamu.edu/~tvogel/gallery/gallery.html
2 of 2 22.02.2011 23:00
http://www.math.tamu.edu/~tvogel/gallery/gallery.htmlhttp://www.math.tamu.edu/~tvogel/gallery/gallery.html -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
3/34
Introduction http://www.math.tamu.edu/~tvogel/gallery/node1.html#SECTION00010000000000000000
1 of 1 22.02.2011 23:01
http://www.math.tamu.edu/~tvogel/gallery/node1.html#SECTION00010000000000000000http://www.math.tamu.edu/~tvogel/gallery/node1.html#SECTION00010000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
4/34
A function which is continuous at only one point. http://www.math.tamu.edu/~tvogel/gallery/node3.html#SECTION00021000000000000000
1 of 1 22.02.2011 23:02
http://www.math.tamu.edu/~tvogel/gallery/node3.html#SECTION00021000000000000000http://www.math.tamu.edu/~tvogel/gallery/node3.html#SECTION00021000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
5/34
A function which is differentiable at a single point, and discontinuous everywhere else. http://www.math.tamu.edu/~tvogel/gallery/node4.html#SECTION00022000000000000000
1 of 1 22.02.2011 23:02
http://www.math.tamu.edu/~tvogel/gallery/node4.html#SECTION00022000000000000000http://www.math.tamu.edu/~tvogel/gallery/node4.html#SECTION00022000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
6/34
A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000
1 of 3 22.02.2011 23:02
http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
7/34
A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000
2 of 3 22.02.2011 23:02
http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
8/34
A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000
3 of 3 22.02.2011 23:02
http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
9/34
A function continuous at all irrationals, discontinuous at all rationals. http://www.math.tamu.edu/~tvogel/gallery/node6.html#SECTION00024000000000000000
1 of 1 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node6.html#SECTION00024000000000000000http://www.math.tamu.edu/~tvogel/gallery/node6.html#SECTION00024000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
10/34
A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000
1 of 5 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
11/34
A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000
2 of 5 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
12/34
A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000
3 of 5 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
13/34
A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000
4 of 5 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
14/34
A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000
5 of 5 22.02.2011 23:03
http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
15/34
A divergent alternating series whose terms go to zero http://www.math.tamu.edu/~tvogel/gallery/node9.html#SECTION00031000000000000000
1 of 1 22.02.2011 23:04
http://www.math.tamu.edu/~tvogel/gallery/node9.html#SECTION00031000000000000000http://www.math.tamu.edu/~tvogel/gallery/node9.html#SECTION00031000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
16/34
The commutative law doesn't hold for certain series http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000
1 of 2 22.02.2011 23:04
http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
17/34
The commutative law doesn't hold for certain series http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000
2 of 2 22.02.2011 23:04
http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
18/34
A function which doesn't equal its Taylor series, part 1 http://www.math.tamu.edu/~tvogel/gallery/node11.html#SECTION00033000000000000000
1 of 1 22.02.2011 23:05
http://www.math.tamu.edu/~tvogel/gallery/node11.html#SECTION00033000000000000000http://www.math.tamu.edu/~tvogel/gallery/node11.html#SECTION00033000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
19/34
A function which doesn't equal its Taylor series, part 2 http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000
1 of 2 22.02.2011 23:05
http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
20/34
A function which doesn't equal its Taylor series, part 2 http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000
2 of 2 22.02.2011 23:05
http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
21/34
A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000
1 of 3 22.02.2011 23:06
http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
22/34
A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000
2 of 3 22.02.2011 23:06
http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
23/34
A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000
3 of 3 22.02.2011 23:06
http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
24/34
A function without a limit, although limits exist along all lines http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000
1 of 2 22.02.2011 23:06
http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
25/34
A function without a limit, although limits exist along all lines http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000
2 of 2 22.02.2011 23:06
http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
26/34
A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000
1 of 3 22.02.2011 23:07
http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
27/34
A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000
2 of 3 22.02.2011 23:07
http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
28/34
A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000
3 of 3 22.02.2011 23:07
A diff i bl f i i h l l di i l d i i h // h d / l/ ll / d 17 h l#SECTION00044000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
29/34
A non-differentiable function with al l di rectional derivatives http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000
1 of 2 22.02.2011 23:07
A diff ti bl f ti ith l l di ti l d i ti htt // th t d / t l/ ll / d 17 ht l#SECTION00044000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
30/34
A non-differentiable function with al l di rectional derivatives http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000
2 of 2 22.02.2011 23:07
A f ti h i d ti l l htt // th t d / t l/ ll / d 18 ht l#SECTION00045000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
31/34
A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000
1 of 3 22.02.2011 23:08
A function whose mixed partials are unequal http://www math tamu edu/ tvogel/gallery/node18 html#SECTION00045000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
32/34
A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000
2 of 3 22.02.2011 23:08
A function whose mixed partials are unequal http://www math tamu edu/ tvogel/gallery/node18 html#SECTION00045000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
33/34
A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000
3 of 3 22.02.2011 23:08
About this document http://www math tamu edu/~tvogel/gallery/node19 html#SECTION00050000000000000000
http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000 -
7/28/2019 Dr.Vogel's Gallery of Calculus Pathologies
34/34
About this document ... http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000
1 f 1 22 02 2011 23 08
http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000