{"id":2265,"date":"2014-02-16T20:40:43","date_gmt":"2014-02-17T01:40:43","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/groisser\/?page_id=2265"},"modified":"2026-03-19T08:13:08","modified_gmt":"2026-03-19T12:13:08","slug":"hw4","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/groisser\/classes\/4212_s14\/hw\/hw4\/","title":{"rendered":"MAA 4212 Assignment 4"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h2 class=\"wp-block-heading\">Due date: Monday, 3\/17\/14<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><i>Last updated Mar 10 17:48 EDT 2014 <\/i><\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">\u00a0<br>\nYou are required to do <strong> all <\/strong> of the problems below. You will not be required<br>\nto hand them all in. I&#8217;ve indicated below which ones you do have to hand in. <!-- I will announce --><\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">Don&#8217;t make the mistake of thinking that I&#8217;m collecting only the problems I<br>\nthink are important. The &#8220;due date&#8221; above is the date that your written-up<br>\nproblems should be handed in, but don&#8217;t wait to get started on the assignment.<br>\nYou should always get started on problems as soon as we cover the relevant<br>\nmaterial in class.<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> Rosenlicht Chap. VI (pp. 132-135)\/ 7, 10, 11, 15, 16 (it&#8217;s implicit in the<br>\nnotation &#8220;<i>C<\/i>([<i>a,b<\/i>])&#8221; that the uniform metric is intended here), 17-20, 22.<br>\n<strong>Of these, hand in only 7, 10, 11, 15, 16, 20, 22.<\/strong><!--PARAGRAPH_SEPARATOR--><p><strong><i>Note<\/i><\/strong>: #20 is not easy. If you have what you think is a quick proof, you<br>\nare probably overlooking something, making an implicit assumption, etc.<\/p><!--PARAGRAPH_SEPARATOR--><p><strong><i>Note on #7<\/i><\/strong>: A function <i>f<\/i> : [<i>a,b<\/i>] \u2192 <strong>R<\/strong> (or, more generally, [<i>a,b<\/i>] \u2192 <i>X<\/i>,<br>\nwhere <i>X<\/i> is any metric space) is called <i>piecewise continuous<\/i> if there<br>\nexists a partition {<i>x<sub>i<\/sub><\/i>}<sub><i>i<\/i>=0<\/sub><sup><i>N<\/i><\/sup> such that (i) the restriction of <i>f<\/i> to each open<br>\ninterval (<i>x<sub>i-1<\/sub>, x<sub>i<\/sub><\/i>) is continuous, 1 \u2264 <i>i<\/i> \u2264 <i>N<\/i>, (ii) lim<i><sub>x \u2192 c\u2013<\/sub> f(x)<\/i> and lim<i><sub>x \u2192 c+<\/sub> f(x)<\/i><br>\nexist for all <i>c \u2208 (a,b)<\/i>, and (iii) lim<i><sub>x \u2192 a+<\/sub> f(x)<\/i> and lim<i><sub>x \u2192 b\u2013<\/sub> f(x)<\/i> exist.<br>\n(Note that condition (ii) is effectively a statement about the one-sided limits<br>\nof <i>f<\/i> just at the partition points <i>x<sub>i<\/sub><\/i>, 1 \u2264 <i>i<\/i> \u2264 <i>N-1<\/i>, since condition (i) guarantees<br>\nthat both one-sided limits exist at all other points of (<i>a,b<\/i>).) The function <i>f<\/i><br>\nin problem 7 is <i>not<\/i> assumed piecewise-continuous; condition (i) is assumed,<br>\nbut not conditions (ii) and (iii). If you&#8217;re able to do this problem only under the<br>\nstronger assumption that <i>f<\/i> is piecewise continuous, that&#8217;s still worth something,<br>\nbut recognize that you&#8217;re adding an assumption that&#8217;s not stated in the problem. <\/p><!--PARAGRAPH_SEPARATOR--><p><strong><i>Note on #18<\/i><\/strong>: No trig functions or their inverses allowed. The point of the problem<br>\nis to show directly that the two integrals are equal to each other, not that they<br>\nare equal because they both yield tan<sup>\u20131<\/sup><i>x<\/i>.<\/p><!--PARAGRAPH_SEPARATOR--><\/li><li>Read this <a href=\"https:\/\/people.clas.ufl.edu\/groisser\/content-removed\/\">handout on improper integrals<\/a>. (This handout was updated Mar. 10.<br>\nNotes to the 2009 class have been deleted, and a note to myself in one of the<br>\nexercises has been acted upon.) This is good spring-break reading.  Exercises<br>\nin the handout will be postponed to a later homework assignment. <!--PARAGRAPH_SEPARATOR-->\n<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"..\/..\/hw\"> General homework page <\/a><\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">\n<a href=\"..\/homepage\"> Back to class home page <\/a><\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":245,"featured_media":0,"parent":1993,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-2265","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/2265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/users\/245"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/comments?post=2265"}],"version-history":[{"count":10,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/2265\/revisions"}],"predecessor-version":[{"id":5301,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/2265\/revisions\/5301"}],"up":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/pages\/1993"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/groisser\/wp-json\/wp\/v2\/media?parent=2265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}