{"id":1602,"date":"2017-11-30T02:38:52","date_gmt":"2017-11-30T07:38:52","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1602"},"modified":"2017-11-30T02:38:52","modified_gmt":"2017-11-30T07:38:52","slug":"what-happened-1128","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/30\/what-happened-1128\/","title":{"rendered":"What Happened 11\/28"},"content":{"rendered":"

Class on Tuesday 11\/28 began with student presentations of solutions to the Homework 9 challenge problems.<\/p>\n

We then made the proposition that if \"f:[a,b]\to is non-decreasing (i.e. \"f(x)\leq, then \"\int_a^b exists.<\/p>\n

We then produced an outline of how to prove it, which essentially consisted of 3 steps:<\/p>\n

a) choose arbitrary \"\epsilon>0\"<\/p>\n

b) choose \"P_\epsilon\in, with a condition that we determined to be \"(x_k-x_{k-1}<\frac{\epsilon}{f(b)-f(a)}\"<\/p>\n

c)\"U(f,P_\epsilon)-L(f,P_\epsilon)\"<\/p>\n

We then produced a partial proof of the following:<\/p>\n

If \"g:[0,1]\to[0,1]\" is bijective and increasing, then \"\int_0^1g+\int_0^1g^{-1}=1\"<\/p>\n

We first looked at a geometric argument, by drawing an example graph, to show that the sum was in fact 1.<\/p>\n

Then, we proved that the integrals on the left side exist.<\/p>\n

\"\int_0^1g\" followed directly from the proposition we provide the proof outline for<\/p>\n

We then showed that \"g^{-1}\" must be increasing on the interval, which then allows us to apply the proposition<\/p>\n

Finally, we went over the statement of the Fundamental Theorem of Calculus.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Class on Tuesday 11\/28 began with student presentations of solutions to the Homework 9 challenge problems. We then made the proposition that if is non-decreasing (i.e. , then exists. We then produced an outline of how to prove it, which essentially consisted of 3 steps: a) choose arbitrary b) choose , with a condition that […]<\/p>\n","protected":false},"author":3527,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[58820,58821],"tags":[],"class_list":["post-1602","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-what-happened-today"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-pQ","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1602","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/users\/3527"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1602"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1602\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1602"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1602"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}