If $f$ is a bounded function on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$ such that $U(P_n) \\\\to \\\\ell$ and $L(P_n) \\\\to \\\\ell$ as $n \\\\to \\\\infty$, then $f$ is Riemann integrable on $[a,b]$ and $\\\\int_a^b f(x) dx = \\\\ell$.
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\"", "description": "", "name": "f1"}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,if(u=3,tr6, if(u=4, tr7,tr8))))", "description": "", "name": "ch2"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "u"}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a bounded function $f$ is Riemann integrable on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$, then $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$.
\"", "description": "", "name": "f7"}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$ such that $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr7"}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$ with $Q$ a refinement of $P$, then $L(P) \\\\geq U(Q)$.
\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $L(P) \\\\leq L(Q)$.
\"", "description": "", "name": "f4"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and increasing on $[a,b]$.
\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$ with $Q$ a refinement of $P$, then $L(P) \\\\leq L(Q)$.
\"", "description": "", "name": "tr5"}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "description": "", "name": "ch1"}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $U(P \\\\cup Q) \\\\leq U(P)$.
\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is bounded and decreasing on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr2"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and decreasing on $[a,b]$.
\"", "description": "", "name": "f2"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "f"}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr3"}, "f8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is continuous on $[a,b]$.
\"", "description": "", "name": "f8"}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P_n$ ($n \\\\in \\\\mathbb{N}$) are partitions of $[a,b]$, then $U(P_n)-L(P_n) \\\\to 0$ as $n \\\\to \\\\infty$.
\"", "description": "", "name": "f6"}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,f1,if(v=2,f2,if(v=3,f3,f8)))", "description": "", "name": "ch3"}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f4,if(f=2,f5,if(f=3,f6, f7)))", "description": "", "name": "ch4"}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $L(P) \\\\leq U(Q)$.
\"", "description": "", "name": "tr4"}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is bounded and increasing on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
\"", "description": "", "name": "tr1"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "g"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "t"}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "t", "tr1", "u", "tr2", "tr3", "tr4", "tr5", "tr6", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "v", "tr7", "tr8", "f7", "f8"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "True/false statements about Riemann integration, ", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"displayType": "checkbox", "layout": {"expression": ""}, "marks": 0, "choices": ["{Ch1}
", "{Ch2}
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\n \n \n \n", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Answer the following question on continuity and differentiability. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.
", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "MAS1601", "MAS2224", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "17/04/15:
\n(OK) new question adapting the format of an older question
", "licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.
"}, "advice": "You should be able to work out the correct answers from your notes.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}