{"id":1540,"date":"2017-11-20T22:12:01","date_gmt":"2017-11-21T03:12:01","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1540"},"modified":"2017-11-20T22:12:01","modified_gmt":"2017-11-21T03:12:01","slug":"what-happened-1116","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/20\/what-happened-1116\/","title":{"rendered":"What Happened 11\/16"},"content":{"rendered":"

On Thursday 11\/16, Dr. Kerckhove lectured in Dr. LeCrone’s absence. The topic of the day was Riemann Integration.<\/p>\n

Dr. K began with a definition of Riemann Integration and then went through an outline of what we would cover:<\/p>\n

    \n
  1. Computing Riemann sums to approximate Riemann Integral<\/li>\n
  2. Controlling error<\/li>\n
  3. Systematic way to improve a given estimate<\/li>\n
  4. If all sequences of improved values converge to the same number, then that number is the Riemann Integration result.<\/li>\n
  5. A theorem that characterizes properties of the function f that guarantee steps 1-4 will work to produce exactly one result<\/li>\n<\/ol>\n

    We then covered several definitions:<\/p>\n

      \n
    1. f is bounded on [a,b]<\/li>\n
    2. a partition\u00a0P of [a,b], as well as the set of all possible partitions of [a,b], \"\mathcal{P}_{[a,b]}\"<\/li>\n
    3. A refinement Q of a given partition P\n
        \n
      • We looked at the example \"f(x)=x\; and after examining the right end point sum produced a lemma<\/li>\n
      • Lemma: Under certain conditions, if Q is a refinement of P, then \"|Q_{error}|\leq<\/li>\n<\/ul>\n<\/li>\n
      • From a given f,[a,b], P we can construct\n
          \n
        1. The Lower Sum for f over [a,b] using P (denoted \"L(f,[a,b],P)\")<\/li>\n
        2. The Upper Sum\u00a0(denoted \"U(f,[a,b],P)\")<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

          We then produced the following Lemma:<\/p>\n

          For a lower sum built from P if we take a refinement Q of P the the lower sum for the refinement satisfies $latex\u00a0L(f,[a,b],Q)\\geq\u00a0L(f,[a,b],P)$ and \"U(f,[a,b],Q)\leq. Moreover $s=latex \\Sigma_{k=1}^n f(x)(x_k-x_{k-1})$ is any Riemann sum built from P \"x\in[x_{k-1},x_k]\", then \"L(f,[a,b],P)\leq.<\/p>\n

          A proof of the moreover section of the lemma was then produced.<\/p>\n

          We made two observations:<\/p>\n

            \n
          1. \u00a0For any estimate \"|P_{error}|\leq<\/li>\n
          2. If Q is a refinement of P then \"|U(f,[a,b],Q)-L(f,[a,b],Q)|\leq<\/li>\n<\/ol>\n

            This was followed by defining the notation \"L(f,[a,b])=L(f)\" and \"U(f,[a,b])=U(f)\"<\/p>\n

            We then defined what it meant to be Riemann integrable (\"L(f)=U(f)\").<\/p>\n

            We finished class with an example of a non-Riemann integrable function: The Dirichlet function on [0,1].<\/p>\n","protected":false},"excerpt":{"rendered":"

            On Thursday 11\/16, Dr. Kerckhove lectured in Dr. LeCrone’s absence. The topic of the day was Riemann Integration. Dr. K began with a definition of Riemann Integration and then went through an outline of what we would cover: Computing Riemann sums to approximate Riemann Integral Controlling error Systematic way to improve a given estimate If […]<\/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-1540","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-oQ","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1540","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=1540"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1540\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}