{"id":1345,"date":"2017-10-29T15:39:51","date_gmt":"2017-10-29T19:39:51","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1345"},"modified":"2017-10-29T15:43:11","modified_gmt":"2017-10-29T19:43:11","slug":"definitions-102617","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/10\/29\/definitions-102617\/","title":{"rendered":"Definitions 10\/26\/17"},"content":{"rendered":"

The Intermediate Value Theorem:<\/p>\n

In the book, we have the formal definition stating: Let f : [a, b] \u2192 R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c \u2208 (a, b) where f(c) = L.<\/p>\n

To prove this theorem, we mainly have three possible approaches:<\/p>\n

    \n
  1. Using the axiom of completeness.<\/li>\n
  2. Using the nested interval property.<\/li>\n<\/ol>\n

    Details are required in the written homework.<\/p>\n

    Then, we moved on to chapter 5 where we discussed differentiation. The definition of differentiability is stated as:<\/p>\n

    Let g : A \u2192 R be a function defined on an interval A. Given c \u2208 A, the derivative of g at c is defined by \"g'(c) provided this\u00a0limit exists. In this case we say g is differentiable at c. If \"g'\" exists for all points c \u2208 A, we say that g is differentiable on A.<\/p>\n

    Then, Theorem 5.2.3 states that\u00a0If g : A \u2192 R is differentiable at a point c \u2208 A, then g is continuous at c as well.<\/p>\n

    To prove this,\u00a0Algebraic Limit Theorem for functional limits allows us to write \"\lim_{x\rightarrow. Therefore, \"\lim_{x\rightarrow.<\/p>\n

    Algebraic Differentiability Theorem:<\/p>\n

    Let f and g be functions defined on an interval A, and assume both are differentiable at some point c \u2208 A. Then:<\/p>\n

      \n
    1. \"(f+g)'(c)<\/li>\n
    2. \"(kf)'(c)<\/li>\n
    3. \"(fg)'(c)<\/li>\n
    4. \"(f\/g)'(c)<\/li>\n<\/ol>\n

       <\/p>\n

      Reference:<\/p>\n

      Abbott, Stephen. \u201cChapter 4 & Chapter 5.\u201d\u00a0Understanding Analysis<\/i>, Springer, 2015.<\/p>\n

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

      The Intermediate Value Theorem: In the book, we have the formal definition stating: Let f : [a, b] \u2192 R be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c \u2208 (a, b) where f(c) = L. To prove […]<\/p>\n","protected":false},"author":3537,"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":[58823],"tags":[],"class_list":["post-1345","post","type-post","status-publish","format-standard","hentry","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-lH","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1345","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\/3537"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1345"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1345\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}