{"id":1676,"date":"2017-12-07T13:31:43","date_gmt":"2017-12-07T18:31:43","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1676"},"modified":"2017-12-07T13:31:43","modified_gmt":"2017-12-07T18:31:43","slug":"daily-definitions-125","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/12\/07\/daily-definitions-125\/","title":{"rendered":"Daily Definitions (12\/5)"},"content":{"rendered":"<p>The last subject we covered in class was The Fundamental Theorem of Calculus which brings this course full circle with a proof that proves all the basic things we used in every Calculus class we have been in. I have stated the definition:<\/p>\n<p><strong>Fundamental Theorem of Calculus:\u00a0<\/strong><\/p>\n<p>(i) If <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=f%3A+%5Ba%2Cb%5D+%5Crightarrow+%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"f: [a,b] &#92;rightarrow &#92;mathbb{R}\" class=\"latex\" \/> is integrable and<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=F%27%28x%29+%3D+f%28x%29+%5Cforall+x+%5Cin+%5Ba%2Cb%5D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"F&#039;(x) = f(x) &#92;forall x &#92;in [a,b]\" class=\"latex\" \/> then <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cint_a%5Eb+f+%3D+F%28b%29-F%28a%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;int_a^b f = F(b)-F(a)\" class=\"latex\" \/><\/p>\n<p>(ii) If <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=g%3A+%5Ba%2Cb%5D+%5Crightarrow+%5Cmathbb%7BR%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"g: [a,b] &#92;rightarrow &#92;mathbb{R}\" class=\"latex\" \/> is integrable, define <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=G%28x%29+%3D+%5Cint_a%5Ex+g&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"G(x) = &#92;int_a^x g\" class=\"latex\" \/> for x in [a,b]. Then, G is continuous on [a,b]. Furthermore, if g is continuous at a point c in our domain, then <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=G%27%28c%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"G&#039;(c)\" class=\"latex\" \/> exists.<\/p>\n<p>&nbsp;<\/p>\n<p>Notice that with some work, we can prove that if we have a continuous function, then the integral of that function is:\u00a0<img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cint_a%5Eb+f+%3D+h%28b%29-h%28a%29&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;int_a^b f = h(b)-h(a)\" class=\"latex\" \/> for some <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=h&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"h\" class=\"latex\" \/> where <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=h%27+%3D+f&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"h&#039; = f\" class=\"latex\" \/>.<\/p>\n<p>Thus, proving the computation of a proof if the function is continuous has now become 100x.<\/p>\n<p>Note that this proof also allows us to prove many of the properties of integrals.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The last subject we covered in class was The Fundamental Theorem of Calculus which brings this course full circle with a proof that proves all the basic things we used in every Calculus class we have been in. I have stated the definition: Fundamental Theorem of Calculus:\u00a0 (i) If is integrable and then (ii) If [&hellip;]<\/p>\n","protected":false},"author":3525,"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":[58818],"tags":[],"class_list":["post-1676","post","type-post","status-publish","format-standard","hentry","category-class-blogs"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-r2","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1676","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\/3525"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1676"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1676\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1676"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1676"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1676"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}