{"id":1585,"date":"2017-11-29T19:24:26","date_gmt":"2017-11-30T00:24:26","guid":{"rendered":"http:\/\/blog.richmond.edu\/math320\/?p=1585"},"modified":"2017-12-06T12:33:49","modified_gmt":"2017-12-06T17:33:49","slug":"daily-definition-1128","status":"publish","type":"post","link":"https:\/\/blog.richmond.edu\/math320\/2017\/11\/29\/daily-definition-1128\/","title":{"rendered":"Daily Definition (11\/28)"},"content":{"rendered":"

The problem from the proof outline from 11\/28 covered the definition of a nondecreasing function. We learned about the nondecreasing sequences but not nondecreasing functions in this course. In this post, I would like to review the definition of a non-decreasing function and further explore relative definitions of a nonincreasing function, an increasing function, a decreasing function.<\/p>\n

Nondecreasing function: A function f(x) is said to be nondecreasing on an interval I if \"f(b) for all\u00a0b>a, where \"a,b\in<\/p>\n

Nonincreasing function: a function f(x) is said to be nonincreasing on an\u00a0interval\u00a0I\u00a0if \"f(b) for all b>a with\u00a0\"a,b\in.<\/p>\n

Increasing function:A function\u00a0f(x) increases on an\u00a0interval\u00a0I\u00a0if \"f(b) for all\u00a0b>a, where\u00a0\"a,b\in. If\u00a0,\u00a0f(b)>f(a), for all\u00a0b>a, the function is said to be\u00a0strictly increasing.<\/p>\n

Decreasing function: a function\u00a0f(x) decreases on an\u00a0interval\u00a0I\u00a0if \"f(b) for all\u00a0b>a\u00a0with\u00a0\"a,b\in. If\u00a0f(b)<f(a), for all\u00a0b>a, the function is said to be\u00a0strictly decreasing.<\/p>\n

 <\/p>\n

Further thoughts:<\/p>\n

If the\u00a0derivative\u00a0f'(x) of a\u00a0continuous function\u00a0f(x)\u00a0satisfies ,f'(x)>0 on an\u00a0open interval (a,b), then f(x) is increasing on (a,b). However, a function may increase on an interval without having a derivative defined at all points. For example, the function \"x^\frac{1}{3}\" (see the graph below)is increasing everywhere, including the origin\u00a0x=0, despite the fact that the\u00a0derivative\u00a0is not defined at that point.<\/p>\n


\n\"\"<\/a><\/p>\n

 <\/p>\n

This graph also demonstrates the features of nondecreasing functions, nonincreasing functions, increasing functions and decreasing functions.<\/p>\n

\"\"<\/a><\/p>\n

Reference:<\/p>\n

Jeffreys, H. and Jeffreys, B.\u00a0S. “Increasing and Decreasing Functions.” \u00a71.065 in\u00a0Methods of Mathematical Physics, 3rd ed.<\/i>\u00a0Cambridge, England: Cambridge University Press, p.\u00a022, 1988.<\/p>\n","protected":false},"excerpt":{"rendered":"

The problem from the proof outline from 11\/28 covered the definition of a nondecreasing function. We learned about the nondecreasing sequences but not nondecreasing functions in this course. In this post, I would like to review the definition of a non-decreasing function and further explore relative definitions of a nonincreasing function, an increasing function, a […]<\/p>\n","protected":false},"author":2107,"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":[58820,58823],"tags":[],"class_list":["post-1585","post","type-post","status-publish","format-standard","hentry","category-daily-blogs","category-definitions"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7L4E1-pz","jetpack-related-posts":[],"_links":{"self":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1585","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\/2107"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/comments?post=1585"}],"version-history":[{"count":0,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/posts\/1585\/revisions"}],"wp:attachment":[{"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/media?parent=1585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/categories?post=1585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.richmond.edu\/math320\/wp-json\/wp\/v2\/tags?post=1585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}