You should be able to work out the correct answers from your notes.
", "preamble": {"js": "", "css": ""}, "name": "Luis's copy of True/false statements about continuity and differentiability,", "variable_groups": [], "extensions": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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.
", "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "sortAnswers": false, "type": "gapfill", "customMarkingAlgorithm": "", "marks": 0, "prompt": "\n \n \n[[0]]
\n \n \n \n", "variableReplacementStrategy": "originalfirst", "unitTests": [], "extendBaseMarkingAlgorithm": true, "gaps": [{"showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "marks": 0, "layout": {"expression": ""}, "maxMarks": 0, "answers": ["True", "False"], "unitTests": [], "extendBaseMarkingAlgorithm": true, "shuffleAnswers": false, "shuffleChoices": true, "variableReplacements": [], "maxAnswers": 0, "warningType": "none", "matrix": [[1, -1], [1, -1], ["-1", "1"], [-1, 1]], "type": "m_n_x", "choices": ["{Ch1}
", "{Ch2}
", "{Ch3}
", "{Ch4}
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\nCan choose true and false for each option. Also in one test run the second choice was incorrectly entered, rest correct, but the feedback indicates that the third was wrong.
", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"u": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "u", "definition": "random(1..3)"}, "f2": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f2", "definition": "\"If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous on $(a,b)$ and $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\""}, "f6": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f6", "definition": "\"If a function $f$ is continuous and differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\""}, "f4": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f4", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.
\""}, "ch2": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ch2", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))"}, "f": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "f", "definition": "random(1..3)"}, "tr3": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr3", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\""}, "ch1": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ch1", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))"}, "h": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "h", "definition": "random(1..4)"}, "t": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "t", "definition": "random(1..3)"}, "f5": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f5", "definition": "\"If a function $f$ is differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\""}, "tr4": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr4", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) = f(b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.
\""}, "ch3": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ch3", "definition": "if(v=1,f1,if(v=2,f2,f3))"}, "g": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "g", "definition": "random(1..3)"}, "tr1": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr1", "definition": "\"If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is differentiable at $c \\\\in \\\\mathbb{R}$, then it is continuous at $c$.
\""}, "f3": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f3", "definition": "\"Given any function defined on $[a,b]$ with $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\""}, "tr5": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr5", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.
\""}, "f1": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "f1", "definition": "\"If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous at $c \\\\in \\\\mathbb{R}$, then it is differentiable at $c$.
\""}, "tr2": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr2", "definition": "\"If a function $f$ is continuous on $[a,b]$ and $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.
\""}, "v": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "v", "definition": "random(1..3)"}, "ch4": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "ch4", "definition": "if(f=1,f4,if(f=2,f5,f6))"}, "tr6": {"description": "", "group": "Ungrouped variables", "templateType": "long string", "name": "tr6", "definition": "\"If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f\\'(x) >0$ for all $x \\\\in (a,b)$, then $f(b)>f(a)$.
\""}}, "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}