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\$S = \\left\\{\\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})} \\; : \\; n \\in \\mathbb{Z} \\right\\}\$

\n

Greatest lower bound = [[0]]

\n

Does this lie in the set? [[1]]

\n

Least upper bound = [[2]]

\n

Does this lie in the set? [[3]]

\n

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\$S = \\left\\{ x \\in \\mathbb{R} \\; : \\; \\simplify[std]{{c}x^{2m+1} < {d}x^{2m}} \\right\\}\$

\n

Greatest lower bound = [[0]]

\n

Does this lie in the set? [[1]]

\n

Least upper bound = [[2]]

\n

Does this lie in the set? [[3]]

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\$S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}\\;\\;:\\;\\;n \\in \\mathbb{N} \\right\\}\$

\n

Greatest lower bound = [[0]]

\n

Does this lie in the set? [[1]]

\n

Least upper bound = [[2]]

\n

Does this lie in the set? [[3]]

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\$S = \\left\\{\\simplify[std]{{a4}x+{b4}/x}\\;\\;:\\;\\;x \\in \\mathbb{R},\\;\\;x \\gt 0 \\right\\}\$

\n

Greatest lower bound = [[0]]

\n

Does this lie in the set? [[1]]

\n

Least upper bound = [[2]]

\n

Does this lie in the set? [[3]]

", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

For each of the following sets $S$ , state the least upper bound (lub) and the greatest lower bound (glb), where appropriate.

\n

Enter each number as a fraction or integer, not a decimal.

\n

If the set is not bounded above, write infinity.

\n

If the set is not bounded below, write -infinity.

\n

$\\mathbb{N}$ denotes the set of natural numbers, $\\mathbb{Z}$ the set of integers and $\\mathbb{R}$ the set of real numbers.

\n

Also state if the lub or glb belong to each set.

\n

There are four parts to this question, so you may need to scroll down to answer all parts.

", "tags": ["bounded above", "bounded below", "bounded set", "bounds", "checked2015", "cr1", "glb", "greatest lower bound", "least upper bound", "limit", "limits", "lower bound", "lub", "MAS2224", "max value", "maximum value", "min value", "minimum value", "not bounded", "query", "sets", "tested1", "upper bound"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

4/07/2012:

\n\t\t

\n\t\t

Corrected mistake in answer to first part (minus sign missing).

\n\t\t

5/07/2012:

\n\t\t

There is an issue with the MCQs - this has been reported on Github.

\n\t\t

Also an issue with recognising infinity as an answer - also reported on Github.

\n\t\t

Changed to Match Text Pattern, but Correct Answer not properly displayed for $\\pm \\infty$

\n\t\t

Also an issue with reordering gaps in a gapfill - wishlist item on Github

\n\t\t

\n\t\t

21/07/2012:

\n\t\t

Error in part c first MCQ. Corrected.

\n\t\t

Instructions about using fractions and integers included.

\n\t\t

\n\t\t

Have used Matching Expressions question typefor identifying $\\pm \\infty$ as answers.

\n\t\t

27/7/2012:

\n\t\t

\n\t\t

\n\t\t

24/12/2012:

\n\t\t

\n\t\t

Question now accepts infinity and -infinity as possible answers. Query raised as could use html code for infinity to display correct answer for the Word Match rather than the input strings. However, these are not the same as the student would input. Added query tag.

\n\t\t

Tested rounding, OK. Added cr1 tag.

\n\t\t

C

\n\t\t

20/01/2014:

\n\t\t

Got rid of last four parts.

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Four questions on finding least upper bounds and greatest lower bounds of various sets.

#### a)

\n

\\begin{align}
\\simplify[std]{({a}n^2+{a1})/({b}n^2+{b1})} &= \\simplify[std]{(({a} / {b}) * ({b} * n ^ 2 + {b1}) + {a1} -({a * b1} / {b})) / ({b} * n ^ 2 + {b1})} \\\\
&= \\simplify[std]{{a} / {b} -({( -a1) * b + a * b1} / ({b} * ({b} * n ^ 2 + {b1})))}
\\end{align}

\n

Note that 1) the values for positive and negative values of $n$ are the same and 2) as $n$ increases this expression increases.

\n

The greatest lower bound occurs when $n=0$ and the value is $\\displaystyle \\simplify[std]{{a1}/{b1}}$.

\n

As $n$ increases, the value of the expression approaches as close as we like to $\\displaystyle \\simplify[std]{{a}/{b}}$ , but is always less than $\\displaystyle \\simplify[std]{{a}/{b}}$.

\n

Hence the least upper bound is $\\simplify[std]{{a}/{b}}$.

\n

#### b)

\n

\\begin{align}
&&\\simplify[std]{{c} * x ^ {2 * m + 1}} &\\lt \\simplify[std]{ {d} * x ^ {2 * m}} \\\\
\\iff && \\simplify[std]{x ^ {2 * m} * ({c} * x -{d})} &\\lt 0 \\\\
\\iff &&\\simplify[std]{{c}x-{d}} &\\lt 0 \\text{ as } x^{\\var{2*m}} \\geq 0
\\end{align}

\n

Hence this set is the same as the set

\n

\$\\left \\{x \\in \\mathbb{R} \\; : \\; x \\lt \\simplify[std]{{d}/{c}}\\right\\}\$

\n

This set does not have a greatest lower bound so you enter -infinity.

\n

It does have a least upper bound and this is $\\simplify[std]{{d}/{c}}$

\n

#### c)

\n

\$S = \\left\\{\\simplify[std]{{a2}+{s1}*{b2}/n^{r}} \\; : \\; n \\in \\mathbb{N} \\right\\}\$

\n

Let $\\displaystyle a_n=\\simplify[std]{{a2}+{s1}*{b2}/n^{r}}$

\n

As $n$ increases we see that $a_n$ {if(s1=1,'decreases','increases')} and converges to the limit $\\var{a2}$.

\n

Hence, greatest lower bound = $\\var{glb3}$ and least upper bound = $\\var{lub3}$

\n

#### d)

\n

\$S = \\left\\{\\simplify[std]{{a4}x+{b4}/x} \\; : \\; x \\in \\mathbb{R}, \\; x \\gt 0 \\right\\}\$

\n

This set does not have a least upper bound, so you enter infinity.

\n

However, it does have a lower bound because $\\displaystyle \\var{a4}x+\\frac{\\var{b4}}{x} \\gt 0$, $\\forall x \\gt 0$.

\n

To find the greatest lower bound we find the minimum value of $\\displaystyle g(x)=\\var{a4}x+\\frac{\\var{b4}}{x}$, $x \\gt 0$.

\n

Now $\\displaystyle g'(x)=\\var{a4}-\\frac{\\var{b4}}{x^2}$ and $g'(x)=0$ when $\\displaystyle x=\\sqrt{\\frac{\\var{b4}}{\\var{a4}}} = \\var{r5}$.

\n

(We take the positive square root, since $x \\gt 0$)

\n

This gives a minimum value for $g(x)$ at $x = \\var{r5}$, and $g(\\var{r5})=\\var{ans4}$.

\n

Hence the greatest lower bound is $\\var{ans4}$ as we have shown that $g(x) \\geq \\var{ans4}$, $\\forall x \\gt 0$.

"}, {"name": "Find point when a sequence becomes strictly decreasing", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(c)", "name": "n", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t^(1/k)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a/b", "name": "c", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1-a", "name": "b", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.5..0.95 # 0.05)", "name": "t", "description": ""}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "k", "description": ""}}, "ungrouped_variables": ["a", "b", "c", "n", "t", "k"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "variable_groups": [], "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "integerAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "{n}", "integerPartialCredit": 0, "marks": "8", "maxValue": "{n}"}], "type": "gapfill", "prompt": "

Find the smallest integer $N$ such that $x_{m+1} \\leq x_m$ for all $m \\geq N$.

\n

The smallest integer is [[0]]

", "showCorrectAnswer": true, "marks": 0}], "statement": "

Let $\\{x_n\\}$ be the sequence given by

\n

\$x_n=n^\\var{k} \\var{t}^n\$

", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "limits", "MAS2224", "query", "sequences", "taking the limit", "tested1", "udf"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

$x_n=n^k t^n$ where $k$ is a positive integer and $t$ a real number with $0 < t<1$. Find the smallest integer $N$ such that $(m+1)^k t^{m+1} \\leq m^k t^m$ for all $m \\geq N$.

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The condition $x_{m+1} \\leq x_m$ is $(m+1)^\\var{k} \\var{t}^{m+1} \\leq m^\\var{k} \\var{t}^m$. This can be shown to be equivalent to $m \\geq \\dfrac{\\var{t}^{1/\\var{k}}}{1-\\var{t}^{1/\\var{k}}}=\\var{c}$. We take $N$ to be the smallest integer $\\geq \\var{c}$, so $N=\\var{n}$. Then $m \\geq \\var{c}$ for all $m \\geq \\var{n}$ and therefore $x_{m+1} \\leq x_m$ for all $m \\geq \\var{n}$.

"}, {"name": "True/false statements about limits of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "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/"}], "variable_groups": [], "variables": {"tr8": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A divergent sequence can have a convergent subsequence.

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A bounded sequence is convergent.

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If {$\\{x_n\\}$} converges, then {$\\{x_{n+i}\\}$} could diverge for some natural number $i$.

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A convergent sequence is either increasing or decreasing.

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If {$\\{x_{n+i}\\}$} diverges for some natural number $i$, then {$\\{x_n\\}$} diverges.

\"", "name": "tr2", "description": ""}, "f9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'There exists a sequence that is not bounded but which converges.'", "name": "f9", "description": "

There exists a sequence that is not bounded but which converges.

"}, "tr11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A finite set is bounded.', 'A set that is not bounded has an infinite number of elements.')", "name": "tr11", "description": ""}, "tr3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

If a sequence is not bounded, then it does not converge.

\"", "name": "tr3", "description": ""}, "tr6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A sequence with only a finite number of non zero terms converges to 0.'", "name": "tr6", "description": ""}, "ch8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(x=1,f11,if(x=2,f12,if(x=3,f13,f14)))", "name": "ch8", "description": ""}, "f13": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set never has a maximum element.', 'A bounded set never has a minimum element.')", "name": "f13", "description": ""}, "f14": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A set with an infinite number of elements cannot be bounded.'", "name": "f14", "description": ""}, "tr1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A convergent sequence is bounded.

\"", "name": "tr1", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "g", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "t", "description": ""}, "w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "w", "description": ""}, "f11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A set with a maximum element is necessarily bounded.', 'A set with a minimum element is necessarily bounded.')", "name": "f11", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "v", "description": ""}, "ch1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "name": "ch1", "description": ""}, "tr10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A set with a maximum element is necessarily bounded above.', 'A set with a minimum element is necessarily bounded below.')", "name": "tr10", "description": ""}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "u", "description": ""}, "tr12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A bounded set has both a least upper bound and a greatest lower bound.'", "name": "tr12", "description": ""}, "ch6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(h=1,f7,if(h=2,f8,if(h=3,f9,f10)))", "name": "ch6", "description": ""}, "f7": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

All convergent sequences of positive terms converge to a value $> 0$.

\"", "name": "f7", "description": ""}, "f12": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set necessarily has a maximum element.', 'A bounded set necessarily has a minimum element.')", "name": "f12", "description": ""}, "tr9": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

There exists a sequence with all terms greater than zero and with limit 0.

\"", "name": "tr9", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

Every divergent sequence is unbounded.

\"", "name": "f2", "description": ""}, "tr20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'If a sequence has the subsequence given by the even terms converges to the same limit as the subsequence of odd terms, then the sequence also converges to that limit.'", "name": "tr20", "description": ""}, "f10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for a sequence to be both increasing and decreasing.'", "name": "f10", "description": ""}, "tr5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

It is possible for a sequence to be both increasing and decreasing.

\"", "name": "tr5", "description": ""}, "f5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

There exists a convergent sequence {$\\{x_n\\}$} with $x_n >0$ for all $n \\\\in \\\\mathbb{N}$ and limit $\\\\ell <0$.

\"", "name": "f5", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "f", "description": ""}, "ch7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=1,tr10,if(w=2,tr11,tr12))", "name": "ch7", "description": ""}, "f8": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

If {$\\{x_n\\}$} diverges, then {$\\{x_{n+i}\\}$} could converge for some natural number $i$.

\"", "name": "f8", "description": ""}, "tr7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded increasing sequence converges.','A bounded decreasing sequence converges.')", "name": "tr7", "description": ""}, "f6": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

There exists a bounded increasing sequences that does not converge.

\"", "name": "f6", "description": ""}, "ch3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(v=1,tr7,if(v=2,tr8,tr9))", "name": "ch3", "description": ""}, "ch4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(f=1,f1,if(f=2,f2,f3))", "name": "ch4", "description": ""}, "tr4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

In a convergent sequence, all subsequences converge to the same limit.

\"", "name": "tr4", "description": ""}, "ch5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(g=1,f4,if(g=2,f5,f6))", "name": "ch5", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "t", "tr9", "tr8", "tr1", "u", "tr3", "tr2", "tr5", "tr4", "tr7", "tr6", "tr20", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "v", "f10", "f20", "w", "x", "f11", "f12", "f13", "f14", "tr10", "tr11", "tr12", "ch7", "ch8"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.

"}, "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"displayType": "radiogroup", "showCellAnswerState": true, "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "minAnswers": 0, "minMarks": 0, "shuffleAnswers": false, "variableReplacements": [], "layout": {"type": "all", "expression": ""}, "choices": ["

{Ch1}

", "

{Ch2}

", "

{Ch3}

", "

{Ch4}

", "

{Ch5}

", "

{Ch6}

", "

{Ch7}

", "

{Ch8}

"], "extendBaseMarkingAlgorithm": true, "matrix": [[1, -1], [1, -1], [1, -1], [-1, 1], [-1, 1], [-1, 1], ["1", "-1"], ["-1", "1"]], "customMarkingAlgorithm": "", "unitTests": [], "marks": 0, "scripts": {}, "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "answers": [true, false], "warningType": "none"}], "prompt": "\n \n \n

[[0]]

\n \n \n \n", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "

Answer the following question on sequences and sets. Note that a sequence is said to be unbounded if it is not bounded.

\n

Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

", "tags": ["bounded sequences", "bounded sets", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "tested1", "unbounded sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

You should be able to work out the correct answers from your notes.

"}, {"name": "When does a sequence get within $d$ of its limit?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t2,t1)", "description": "

", "name": "u2"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(c,c)", "description": "", "name": "d"}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u1=t1,v1,v2)", "description": "", "name": "w1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1*c=a*d,b1+1,b1)", "description": "", "name": "b"}, "t2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Decreasing\"", "description": "", "name": "t2"}, "t3": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Neither\"", "description": "", "name": "t3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..20)", "description": "", "name": "a"}, "mono": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c>b/d,1,2)", "description": "

If a/c > b/d, the sequence is increasing. If a/c < b/d, the sequence is decreasing. a,b,c,d are chosen so that $\\dfrac{a}{c} \\neq \\dfrac{b}{d}$.

", "name": "mono"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "b1"}, "t1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"Increasing\"", "description": "", "name": "t1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(isint(tval), tval +1,ceil(tval))", "description": "", "name": "n"}, "v2": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"decreasing\"", "description": "", "name": "v2"}, "v1": {"templateType": "string", "group": "Ungrouped variables", "definition": "\"increasing\"", "description": "", "name": "v1"}, "u1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a/c > b/d,t1,t2)", "description": "

", "name": "u1"}, "tval": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt((1 / c) * ((10 ^ r * abs(b * c -(a * d))) / c -d))", "description": "", "name": "tval"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "chcop(a,a)", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "s1", "b1", "d", "r", "n", "tval", "mono", "t1", "t2", "t3", "u1", "u2", "v1", "v2", "w1"], "functions": {"chcop": {"type": "number", "language": "jme", "definition": "if(gcd(a,b)=1,b,chcop(a,random(1..20)))", "parameters": [["a", "number"], ["b", "number"]]}}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

Find the limit $\\ell$ of $\\{x_n\\}$. Input as a fraction or an integer.

\n

Limit $\\ell=$ [[0]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{a}/{c}", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "marks": 2, "type": "jme", "unitTests": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "

Find the least integer $N$ such that

\n

\$\\left|{x_n -\\ell}\\right| < 10 ^ { -\\var{r}}, \\quad \\text{for } n \\geq N\$

\n

Least $N=$ [[0]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "showCorrectAnswer": true, "minValue": "{n}", "maxValue": "{n}", "unitTests": [], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "correctAnswerFraction": false, "variableReplacements": [], "marks": "8", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"displayType": "radiogroup", "choices": ["

{u1}

", "

{u2}

", "

{t3}

"], "showCorrectAnswer": true, "displayColumns": "3", "unitTests": [], "prompt": "

Which one of the following describes $\\{x_n\\}$?

", "customMarkingAlgorithm": "", "distractors": ["", "", ""], "variableReplacements": [], "shuffleChoices": true, "showFeedbackIcon": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "matrix": ["2", 0, 0], "marks": 0}], "statement": "

Let

\n

\$x_n=\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})}, \\quad n=1,2,3, \\ldots\$

", "tags": ["checked2015", "convergence of a sequence", "limit", "limit of a sequence", "Limits", "limits", "query", "sequences", "taking the limit", "tested1", "udf"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

$x_n=\\frac{an^2+b}{cn^2+d}$. Find the least integer $N$ such that $\\left|x_n -\\frac{a}{c}\\right| < 10 ^{-r},\\;n\\geq N$, $2\\leq r \\leq 6$. Determine whether the sequence is increasing, decreasing or neither.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

#### a)

\n

To begin with, the limit $\\ell$ is obtained by dividing top and bottom by $n^2$:

\n

\$\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}} = \\simplify[std]{({a}+{b}/n^2) /({c}+{d}/n^2)} \\to \\simplify[std]{({a})/({c})}\$ as $n \\to \\infty$, so $\\displaystyle \\ell= \\simplify[std]{{a}/{c}}$.

\n

\n

#### To find the least $N$ such that all terms from the $N$th are less than $10^{\\var{-r}}$ from the limit we proceed as follows:

\n

\\begin{align}
\\left|\\simplify[std]{x_n -({a} / {c})}\\right| < 10 ^ { -\\var{r}} &\\iff \\left|\\simplify[std]{({a}n^2+{b})/({c}n^2+{d})-{a}/{c}}\\right| < 10 ^ { -\\var{r}} \\\\
&\\iff \\simplify[std]{abs({b*c-a*d})/({c^2}n^2+{c*d})} <10 ^ { -\\var{r}}
\\end{align}

\n

(We can get rid of the absolute value in the denominator as $\\simplify[std]{{c^2}n^2+{c*d}} \\gt 0$, $\\forall n=1,2,3,\\ldots$)

\n

Rearranging this last inequality by multiplying both sides by $(\\simplify[std]{{c^2}n^2+{c*d}})10^{\\var{r}}$ (this is positive and so the inequality does not reverse), we get:

\n

\$\\simplify[std]{{c^2}n^2+{c*d}} > \\var{10^r*abs(b*c-a*d)} \\iff n^2 > \\frac{1}{\\var{c^2}}\\left(\\simplify[std]{{10^r*abs(b*c-a*d)}-{c*d}}\\right)=\\var{tval^2} \\iff n> \\var{tval}\$

\n

Hence the least integer value is given by $N=\\var{N}$.

\n

#### c)

\n

Given $x_n = \\dfrac{an^2+b}{cn^2+d}, c \\gt 0, d\\gt 0$ it can be shown that $x_n \\leq x_{n+1} \\iff \\dfrac{b}{d} \\leq \\dfrac{a}{c}$. Here $\\dfrac{b}{d}=\\dfrac{\\var{b}}{\\var{d}}$ and $\\dfrac{a}{c}=\\dfrac{\\var{a}}{\\var{c}}$. Therefore the sequence will be increasing if $\\dfrac{\\var{b}}{\\var{d}} \\leq \\dfrac{\\var{a}}{\\var{c}}$ and decreasing if $\\dfrac{\\var{b}}{\\var{d}} \\geq \\dfrac{\\var{a}}{\\var{c}}$. Hence the sequence is $\\var{w1}$.

"}, {"name": "True/false statements about convergent and divergent series, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tr8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\neq 0$  for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell$ with $|\\\\ell | <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "tr8", "description": ""}, "x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

\"", "name": "f1", "description": ""}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr5,if(u=2,tr6,if(u=3,tr7,tr8)))", "name": "ch2", "description": ""}, "f20": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "name": "f20", "description": ""}, "tr13": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for all $x$ with $|x|<R$.

\"", "name": "tr13", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.

\"", "name": "tr3", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$.

\"", "name": "f4", "description": ""}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If  $a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.

\"", "name": "f3", "description": ""}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr2", "description": ""}, "f15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0<R<\\\\infty$, then $a_n \\\\neq 0$ and $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.

\"", "name": "f15", "description": ""}, "f9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.

\"", "name": "f9", "description": ""}, "tr11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ does not converge to $0$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr11", "description": ""}, "f10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.

\"", "name": "f10", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell >1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr6", "description": ""}, "tr16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ diverges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\leq |X|$

\"", "name": "tr16", "description": ""}, "ch8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(x=1,f13,if(x=2,f14,if(f=x,f15,f16)))", "name": "ch8", "description": ""}, "f13": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for $x=R$.

\"", "name": "f13", "description": ""}, "f14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for $x=R$

\"", "name": "f14", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

\"", "name": "tr1", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "g", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "u", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "w", "description": ""}, "f11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is not decreasing, then $\\\\Sigma a_n$ diverges.

\"", "name": "f11", "description": ""}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "v", "description": ""}, "tr14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for all $x$ with $|x|>R$.

\"", "name": "tr14", "description": ""}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,if(t=3,tr3,tr4)))", "name": "ch1", "description": ""}, "tr10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.

\"", "name": "tr10", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "t", "description": ""}, "tr12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr12", "description": ""}, "tr20": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.

\"", "name": "tr20", "description": ""}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$  $\\\\ell >0$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f7", "description": ""}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\infty$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr7", "description": ""}, "tr9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.

\"", "name": "tr9", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

\"", "name": "f2", "description": ""}, "tr15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ converges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\geq |X|$.

\"", "name": "tr15", "description": ""}, "ch6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(g=1,f5,if(g=6,f2,if(g=3,f7,f8)))", "name": "ch6", "description": ""}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "tr5", "description": ""}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell =1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f5", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "ch7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(h=1,f9,if(h=2,f10,if(h=3,f11,f12)))", "name": "ch7", "description": ""}, "f8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell<1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f8", "description": ""}, "f12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n \\\\geq 0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.

\"", "name": "f12", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell=1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

\"", "name": "f6", "description": ""}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,tr9,if(v=2,tr10,if(v=3,tr11,tr12)))", "name": "ch3", "description": ""}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,tr13,if(w=2,tr14,if(w=3,tr15,tr16)))", "name": "ch4", "description": ""}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If the series $\\\\Sigma a_n$ converges, then $a_n \\\\to 0$ as $n \\\\to \\\\infty$.

\"", "name": "tr4", "description": ""}, "ch5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f1,if(f=2,f2,if(f=3,f3,f4)))", "name": "ch5", "description": ""}, "f16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ with $a_n \\\\neq 0$ for all $n$ has radius of convergence $R$ with $R<\\\\infty$, then $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.

\"", "name": "f16", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "f10", "f11", "f12", "f13", "f14", "f15", "f16", "tr1", "tr2", "tr3", "tr4", "tr5", "tr6", "tr7", "tr8", "tr9", "tr10", "tr11", "tr12", "tr13", "tr14", "tr15", "tr16", "t", "u", "v", "w", "f", "g", "h", "x", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "ch7", "ch8", "f20", "tr20"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "layout": {"expression": ""}, "choices": ["

{Ch1}

", "

{Ch2}

", "

{Ch3}

", "

{Ch4}

", "

{Ch5}

", "

{Ch6}

", "

{Ch7}

", "

{Ch8}

"], "matrix": [[1, -1], [1, -1], [1, -1], ["1", "-1"], [-1, 1], [-1, 1], ["-1", "1"], ["-1", "1"]], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "warningType": "none", "scripts": {}, "maxMarks": 0, "type": "m_n_x", "minMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "marks": 0, "answers": [true, false]}], "type": "gapfill", "prompt": "\n \n \n

[[0]]

\n \n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "

Answer the following question on series. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

", "tags": ["checked2015", "divergent series", "limits", "MAS1601", "MAS2224", "power series", "series"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

17/04/2015:

\n

(OK) new question based on a similar style question on sequences. Changed the statements to long text to enable better mathematical expressions. Encountered problems when editing (math expressions not recognised).

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

You should be able to work out the correct answers from your notes.

"}, {"name": "Find discontinuities in a piecewise-defined function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "w"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=3,1,0)", "description": "", "name": "v"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+random(1..3)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b+random(1..3)", "description": "", "name": "c"}, "er1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*0+(1-t)*random(1,2,3)", "description": "", "name": "er1"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=2,1,0)", "description": "", "name": "u"}, "er3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "v*0+(1-v)*random(1,2,3)", "description": "", "name": "er3"}, "er2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "u*0+(1-u)*random(1,2,3)", "description": "", "name": "er2"}, "lo1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "dis1", "description": "", "name": "lo1"}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "q1"}, "dis1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,b,a)", "description": "", "name": "dis1"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-random(1..3)", "description": "", "name": "q2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "name": "a"}, "dis2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=3,b,c)", "description": "", "name": "dis2"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "p"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,1,0)", "description": "", "name": "t"}}, "ungrouped_variables": ["q1", "a", "c", "q2", "er1", "er2", "er3", "dis1", "dis2", "p", "lo1", "u", "t", "w", "v", "b"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {"discont": {"type": "html", "language": "javascript", "definition": "var boxup=Math.max(3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)+q1*(b-a)+p+er1+er2+er3)+2;\nvar boxdown=Math.min(-3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)+q1*(b-a)+p+er1+er2+er3)-2;\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:true, boundingbox:[a-2,boxup,c+2,boxdown]});\nvar brd=div.board;\nvar l1=brd.create('functiongraph',[function(x){return p;},a-2,a],{strokeColor:'red'});\nvar l2=brd.create('functiongraph',[function(x){return q1*x+(p+er1-q1*a);},a,b],{strokeColor:'red'});\nvar l3=brd.create('functiongraph',[function(x){return q2*x+(p+er1+er2-q2*b+q1*(b-a));},b,c],{strokeColor:'red'});\nvar l4=brd.create('functiongraph',[function(x){return q2*(c-b)+q1*(b-a)+p+er1+er2+er3;},c,c+2],{strokeColor:'red'});\nreturn div;", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["p", "number"], ["q1", "number"], ["q2", "number"], ["er1", "number"], ["er2", "number"], ["er3", "number"]]}}, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "dis1", "minValue": "dis1", "correctAnswerFraction": false, "marks": 3}, {"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "dis2", "minValue": "dis2", "correctAnswerFraction": false, "marks": 3}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $f(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$ $\\var{p},$ $x \\leq \\var{a},$ $\\simplify{{q1}*x+{p+er1-q1*a}},$ $\\var{a} \\lt x \\leq \\var{b},$ $\\simplify{{q2}*x+{-q2*b+q1*(b-a)+p+er1+er2}},$ $\\var{b}\\lt x \\leq \\var{c},$ $\\var{q2*(c-b)+q1*(b-a)+p+er1+er2+er3},$ $x \\gt \\var{c}.$
\n

$f$ is discontinuous at $x=a$ where $a=\\;$[[0]].

\n

$f$ is discontinuous at $x=b$ where $b=\\;$[[1]] (remember that $b \\gt a$).

", "marks": 0}], "statement": "

Find the $2$ points $x=a$ and  $x=b$, where $a \\lt b$,  at which the following function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ is not continuous.

\n

", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

{discont(a,b,c,p,q1,q2,er1,er2,er3)}

\n

\n

The function is discontinuous at $x=\\var{dis1},\\;\\;x=\\var{dis2}$.

\n

At $x=\\var{dis1}$ we have:

\n

\$\\lim_{x \\nearrow\\; \\var{dis1}} f(x) \\neq \\lim_{x \\searrow\\; \\var{dis1}} f(x)\$

\n

\n

At $x=\\var{dis2}$ we have:

\n

\$\\lim_{x \\nearrow\\; \\var{dis2}} f(x) \\neq \\lim_{x \\searrow\\; \\var{dis2}} f(x)\$

\n

\n

See graph of $f$ above.

"}, {"name": "Find discontinuities of piecewise-defined function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "w"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=3,1,0)", "description": "", "name": "v"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+random(1..3)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b+random(1..3)", "description": "", "name": "c"}, "er1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*0+(1-t)*random(1,2,3)", "description": "", "name": "er1"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=2,1,0)", "description": "", "name": "u"}, "er3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "v*0+(1-v)*random(1,2,3)", "description": "", "name": "er3"}, "er2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "u*0+(1-u)*random(1,2,3)", "description": "", "name": "er2"}, "lo1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "dis1", "description": "", "name": "lo1"}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "q1"}, "dis1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,b,a)", "description": "", "name": "dis1"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-random(1..3)", "description": "", "name": "q2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "name": "a"}, "dis2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=3,b,c)", "description": "", "name": "dis2"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "p"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=1,1,0)", "description": "", "name": "t"}}, "ungrouped_variables": ["q1", "a", "c", "q2", "er1", "er2", "er3", "dis1", "dis2", "p", "lo1", "u", "t", "w", "v", "b"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {"discont": {"type": "html", "language": "javascript", "definition": "var boxup=Math.max(3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3)+2;\nvar boxdown=Math.min(-3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3)-2;\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:true, boundingbox:[Math.min(-2,a-2),boxup,Math.max(c+2,2),boxdown]});\nvar brd=div.board;\nvar l1=brd.create('functiongraph',[function(x){return p;},a-2,a],{strokeColor:'red'});\nvar l2=brd.create('functiongraph',[function(x){return q1*x+(p+er1-q1*a);},a,b],{strokeColor:'red'});\nvar l3=brd.create('functiongraph',[function(x){return q2*x*x-2*b*q2*x+(p+er1+er2+q2*b*b+q1*(b-a));},b,c],{strokeColor:'red'});\nvar l4=brd.create('functiongraph',[function(x){return q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3;},c,c+2],{strokeColor:'red'});\nreturn div;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["p", "number"], ["q1", "number"], ["q2", "number"], ["er1", "number"], ["er2", "number"], ["er3", "number"], ["dis1", "number"], ["dis2", "number"]]}}, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "dis1", "minValue": "dis1", "correctAnswerFraction": false, "marks": 3}, {"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "dis2", "minValue": "dis2", "correctAnswerFraction": false, "marks": 3}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $f(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$ $\\var{p},$ $x \\leq \\var{a},$ $\\simplify{{q1}*x+{p+er1-q1*a}},$ $\\var{a} \\lt x \\leq \\var{b},$ $\\simplify{{q2}*x^2+{-2*q2*b}*x+{q2*b^2+q1*(b-a)+p+er1+er2}},$ $\\var{b}\\lt x \\leq \\var{c},$ $\\var{q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3},$ $x \\gt \\var{c}.$
\n

$f$ is discontinuous at $x=a$ where $a=\\;$[[0]].

\n

$f$ is discontinuous at $x=b$ where $b=\\;$[[1]] (remember that $b \\gt a$).

", "marks": 0}], "statement": "

Find the $2$ points $x=a$ and $x=b$, where $a \\lt b$,  at which the following function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ is not continuous.

\n

", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

{discont(a,b,c,p,q1,q2,er1,er2,er3,dis1,dis2)}

\n

\n

The function is discontinuous at $x=\\var{dis1},\\;\\;x=\\var{dis2}$.

\n

At $x=\\var{dis1}$ we have:

\n

\$\\lim_{x \\nearrow\\; \\var{dis1}} f(x) \\neq \\lim_{x \\searrow\\; \\var{dis1}} f(x)\$

\n

\n

At $x=\\var{dis2}$ we have:

\n

\$\\lim_{x \\nearrow\\; \\var{dis2}} f(x) \\neq \\lim_{x \\searrow\\; \\var{dis2}} f(x)\$

\n

\n

See graph of $f$ above.

\n"}, {"name": "Find the discontinuity in a piecewise-defined function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "w", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=3,0,1)", "name": "v", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+random(1..3)", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b+random(1..3)", "name": "c", "description": ""}, "er1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "t*0+(1-t)*random(-2,-1,1,2,3)", "name": "er1", "description": ""}, "dis": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=1,a,if(w=2,b,c))", "name": "dis", "description": ""}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=2,0,1)", "name": "u", "description": ""}, "er3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "v*0+(1-v)*random(-2,-1,1,2,3)", "name": "er3", "description": ""}, "er2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "u*0+(1-u)*random(-2,-1,1,2,3)", "name": "er2", "description": ""}, "q1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "q1", "description": ""}, "q2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-random(1..3)", "name": "q2", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3)", "name": "a", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "name": "p", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(w=1,0,1)", "name": "t", "description": ""}}, "ungrouped_variables": ["q1", "a", "c", "q2", "er1", "er2", "er3", "p", "b", "u", "t", "w", "v", "dis"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"discont": {"type": "html", "language": "javascript", "definition": "var boxup=Math.max(3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3)+2;\nvar boxdown=Math.min(-3,p,p+er1,q1*(b-a)+p+er1,q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2,q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3)-2;\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:true, boundingbox:[Math.min(-2,a-2),boxup,Math.max(c+2,2),boxdown]});\nvar brd=div.board;\nvar l1=brd.create('functiongraph',[function(x){return p;},a-2,a],{strokeColor:'red'});\nvar l2=brd.create('functiongraph',[function(x){return q1*x+(p+er1-q1*a);},a,b],{strokeColor:'red'});\nvar l3=brd.create('functiongraph',[function(x){return q2*x*x-2*b*q2*x+(p+er1+er2+q2*b*b+q1*(b-a));},b,c],{strokeColor:'red'});\nvar l4=brd.create('functiongraph',[function(x){return q2*(c-b)*(c-b)+q1*(b-a)+p+er1+er2+er3;},c,c+2],{strokeColor:'red'});\nreturn div;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["p", "number"], ["q1", "number"], ["q2", "number"], ["er1", "number"], ["er2", "number"], ["er3", "number"], ["dis", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "dis", "correctAnswerFraction": false, "marks": 3, "maxValue": "dis"}], "type": "gapfill", "prompt": "\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $f(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$ $\\var{p},$ $x \\leq \\var{a},$ $\\simplify{{q1}*x+{p+er1-q1*a}},$ $\\var{a} \\lt x \\leq \\var{b},$ $\\simplify{{q2}*x^2+{-2*q2*b}*x+{q2*b^2+q1*(b-a)+p+er1+er2}},$ $\\var{b}\\lt x \\leq \\var{c},$ $\\var{q2*(c-b)^2+q1*(b-a)+p+er1+er2+er3},$ $x \\gt \\var{c}.$
\n

$f$ is discontinuous at $x=a$ where $a=\\;$[[0]].

\n

", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Find the point at which the following function $f:\\mathbb{R} \\rightarrow \\mathbb{R}$ is not continuous.

\n

", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "

{discont(a,b,c,p,q1,q2,er1,er2,er3,dis)}

\n

\n

The function is discontinuous at $x=\\var{dis}$.

\n

At $x=\\var{dis}$ we have:

\n

\$\\lim_{x \\nearrow\\; \\var{dis}} f(x) \\neq \\lim_{x \\searrow\\; \\var{dis}} f(x)\$

\n

\n

\n

See graph of $f$ above.

"}, {"name": "Find the limit of an algebraic fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "a"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9 except a)", "description": "", "name": "b"}}, "ungrouped_variables": ["a", "b"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "1/{(a-b)}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "

Input as a fraction or an integer.

", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$\\displaystyle \\simplify{Limit((x + { -a}) / (x ^ 2 + { -a -b} * x + {a * b}),x,{a}) }=\\;$[[0]] (input as a fraction or as an integer).

", "marks": 0}], "statement": "

Find the limit of the following function.

", "tags": ["checked2015", "MAS2224"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

Note that on putting $x=\\var{a}$ into $\\displaystyle \\simplify{(x + { -a}) / (x ^ 2 + { -a -b} * x + {a * b}) }$ we get a $0/0$ case and so we have to do more work.

\n

You can factorise $\\simplify{x ^ 2 + { -a -b} * x + {a * b}}$ and then see what happens.

"}, {"name": "Find the limit of an algebraic fraction as parameter tends to a given value", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a2*d2^2+2*a2*b2*d2+c2+a2*b2^2", "name": "w", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0..5)", "name": "d2", "description": ""}, "b3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "name": "b3", "description": ""}, "stat2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(2*a2*b3-sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "name": "stat2", "description": ""}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "name": "a3", "description": ""}, "stat1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(2*a2*b3+sqrt(4*a2^2*b3^2+4*a3*a2*(a3*c2-2*b3*b2*a2+a2*b2^2*a3)))/(-2*a3*a2)", "name": "stat1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-random(1..6)", "name": "b2", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a3*d2+b3", "name": "v", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "name": "a2", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "name": "c2", "description": ""}}, "ungrouped_variables": ["w", "stat1", "b3", "a3", "a2", "b2", "v", "c2", "d2", "stat2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"plotf": {"type": "html", "language": "javascript", "definition": "var f = function(t){ return (a3*t+b3)/(a2*t*t+2*a2*b2*t+c2+a2*b2*b2); };\nvar m1=Math.min(stat1,stat2);\nvar m2=Math.max(stat1,stat2);\nvar f1=f(stat1);\nvar f2=f(stat2);\nvar a=Math.abs(f1);\nvar b=Math.abs(f2);\nvar M=Math.max(a,b);\nvar div = Numbas.extensions.jsxgraph.makeBoard('300px','300px', {axis:true,showNavigation:false,boundingbox:[m1-10,M+2,m2+10,-M-2]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,m1-10,m2+10]);\n\n//brd.create('text',[c,-2,c]);\n//var i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;", "parameters": [["a2", "number"], ["b2", "number"], ["c2", "number"], ["a3", "number"], ["b3", "number"], ["stat1", "number"], ["stat2", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{v}/{w}", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "

Enter as a fraction or an integer and not as a decimal.

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all, fractionNumbers", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

Find the following limit:

\n

$\\displaystyle \\simplify{Limit(f(t),t,{d2}) }= \\;$[[0]].

\n

Enter your answer as a fraction or an integer and not as a decimal.

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "({a3}*a+{b3})/({a2}*a^2+{2*a2*b2}*a+{c2+a2*b2^2})", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"displayType": "radiogroup", "choices": ["

Because $f(t) \\neq 0, \\; \\forall t \\in \\mathbb{R}$.

", "

Because $f$ is continuous at all points in $\\mathbb{R}$.

", "

Because $f$ is a function defined in terms of polynomials.

", "

Because all ratios of polynomials are continuous.

", "

Because $f$ is differentiable at all points.

"], "matrix": [0, 1, 0, 0, 0], "distractors": ["Not true as $f(t)=0$ for a value of $t$.", "", "", "", ""], "shuffleChoices": true, "scripts": {}, "maxMarks": 0, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "marks": 0}], "type": "gapfill", "prompt": "

Also find:

\n

$\\displaystyle \\simplify{Limit(f(t),t,a) }= \\;$[[0]], where $a \\in \\mathbb{R}$ is any point.

\n

Why can we evaluate this limit? [[1]]

", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Let the function $f$ be given by $\\displaystyle f(t)=\\simplify{({a3} * t + {b3}) / ({a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2}) }$

", "tags": ["checked2015", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "

Graph of $f$.

\n

{plotf(a2,b2,c2,a3,b3,stat1,stat2)}

\n

\n

Note that $\\simplify{{a2} * t ^ 2 + {2 * b2 * a2} * t + {c2 + a2 * b2 ^ 2} ={a2}*(t+{b2})^2+{c2}} \\gt 0$.

\n

Hence the denominator of $f(t) \\neq 0,\\;\\forall t \\in \\mathbb{R}$ and so $f$ is continuous at all points in $\\mathbb{R}$.

\n

This means that in part a) we can take the limit by simply subsituting $t=\\var{d2}$ into the expression for $f(t)$ and we get:

\n

\$\\lim_{x \\to \\var{d2}}f(t)=\\simplify[all,!otherNumbers,fractionNumbers,!collectNumbers]{({a3} * {d2} + {b3}) / ({a2} * {d2}^ 2 + {2 * b2 * a2} * {d2}+ {c2 + a2 * b2 ^ 2})={v}/{w} }\$

\n

Similarly in part b) we have :

\n

\$\\lim_{x \\to a}f(t)=\\simplify[all,!collectNumbers,!otherNumbers,fractionNumbers]{({a3} * a + {b3}) / ({a2} * a^ 2 + {2 * b2 * a2} * a+ {c2 + a2 * b2 ^ 2})}\$

\n

\n

As noted above we can find this limit by simply putting $t=a$ into the formula for the function as $f$ is continuous at all points in $\\mathbb{R}$.

\n

"}, {"name": "Prove discontinuity of a function at a given point", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {"disc2": {"type": "html", "language": "javascript", "definition": "\nvar disc=function(x){\n if(x<=2){return x*x+1;}\n else {return x+4;}\n};\nvar div=Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n {axis:true,\nboundingbox:[-2,10,5,-2]});\nvar brd=div.board;\nvar f=brd.create('functiongraph',[disc,-2,5]);\n//var e=brd.create('slider', [[4,6],[4,4],[0,1.5,1.5]],{name:'ε'});\nvar d=brd.create('slider', [[1,-1],[3,-1],[0,1.5,1.5]],{name:'δ'});\nvar dis=brd.create('point',[2,disc(2)],{fixed:true,size:1,name:''});\nvar a=brd.create('point',[2,0],{size:2,face:'x',name:'2'});\nvar im=brd.create('point',[0,disc(2)],{size:2,face:'x',name:''});\nvar imPlus=brd.create('point',[0,function(){return disc(2)+0.5;}],{size:2,face:'x',name:'5.5'});\nvar imMinus=brd.create('point',[0,function(){return disc(2)-0.5;}],{size:2,face:'x',name:'4.5'});\nvar dPlus=brd.create('point',[function(){return 2+d.Value();},0],{size:2,face:'x',name:'2+δ'});\nvar fdPlus=brd.create('point',[function(){return 2+d.Value();},function(){return disc(2+d.Value());}],{size:2,face:'x',name:''});\nvar pt=brd.create('point',[function(){return 2+d.Value()/2;},0],{size:2,face:'x',name:'2+δ/2'});\nvar fpt=brd.create('point',[function(){return 2+d.Value()/2;},function(){return disc(2+d.Value()/2);}],{size:2,face:'x',name:''});\nvar impt=brd.create('point',[0,function(){return disc(2+d.Value()/2);}],{size:2,face:'x',name:'f(2+δ/2)'});\nvar dMinus=brd.create('point',[function(){return 2-d.Value();},0],{size:2,face:'x',name:'2-δ'});\nvar fdMinus=brd.create('point',[function(){return 2-d.Value();},function(){return disc(2-d.Value());}],{size:2,face:'x',name:''});\nvar seg1=brd.create('segment',[dMinus,fdMinus],{strokeColor:'black',dash:1});\nvar seg2=brd.create('segment',[dPlus,fdPlus],{strokeColor:'black',dash:1});\n\nvar ifdPlus=brd.create('point',[0,function(){return disc(2+d.Value());}],{size:2,face:'x',name:''});\nvar ifdMinus=brd.create('point',[0,function(){return disc(2-d.Value());}],{size:2,face:'x',name:''});\nvar seg3=brd.create('segment',[fdMinus,ifdMinus],{strokeColor:'black',dash:1});\nvar seg4=brd.create('segment',[fdPlus,ifdPlus],{strokeColor:'black',dash:1});\nvar seg5=brd.create('segment',[imMinus,imPlus],{strokeColor:'red'});\nvar seg6=brd.create('segment',[pt,fpt],{strokeColor:'red',dash:1});\nvar seg7=brd.create('segment',[fpt,impt],{strokeColor:'red',dash:1});\nreturn div;\n", "parameters": []}, "disc": {"type": "html", "language": "javascript", "definition": "\nvar disc=function(x){\n if(x<=2){return x*x+1;}\n else {return x+4;}\n};\nvar div=Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n {axis:true,\n boundingbox:[-2,10,5,-2]});\nvar brd=div.board;\nvar f=brd.create('functiongraph',[disc,-2,5]);\nvar e=brd.create('slider', [[4,6],[4,4],[0,1.5,1.5]],{name:'ε'});\nvar d=brd.create('slider', [[1,-1],[3,-1],[0,1.5,1.5]],{name:'δ'});\nvar dis=brd.create('point',[2,disc(2)],{fixed:true,size:1,name:''});\nvar a=brd.create('point',[2,0],{size:2,face:'x',name:'2'});\nvar im=brd.create('point',[0,disc(2)],{size:2,face:'x',name:'f(2)=5'});\nvar imPlus=brd.create('point',[0,function(){return disc(2)+e.Value();}],{size:2,face:'x',name:'5+ε'});\nvar imMinus=brd.create('point',[0,function(){return disc(2)-e.Value();}],{size:2,face:'x',name:'5-ε'});\nvar dPlus=brd.create('point',[function(){return 2+d.Value();},0],{size:2,face:'x',name:'2+δ'});\nvar fdPlus=brd.create('point',[function(){return 2+d.Value();},function(){return disc(2+d.Value());}],{size:2,face:'x',name:''});\nvar dMinus=brd.create('point',[function(){return 2-d.Value();},0],{size:2,face:'x',name:'2-δ'});\nvar fdMinus=brd.create('point',[function(){return 2-d.Value();},function(){return disc(2-d.Value());}],{size:2,face:'x',name:''});\nvar seg1=brd.create('segment',[dMinus,fdMinus],{strokeColor:'black',dash:1});\nvar seg2=brd.create('segment',[dPlus,fdPlus],{strokeColor:'black',dash:1});\n\nvar ifdPlus=brd.create('point',[0,function(){return disc(2+d.Value());}],{size:2,face:'x',name:'f(2+δ)'});\nvar ifdMinus=brd.create('point',[0,function(){return disc(2-d.Value());}],{size:2,face:'x',name:'f(2-δ)'});\nvar seg3=brd.create('segment',[fdMinus,ifdMinus],{strokeColor:'black',dash:1});\nvar seg4=brd.create('segment',[fdPlus,ifdPlus],{strokeColor:'black',dash:1});\nvar seg5=brd.create('segment',[imMinus,imPlus],{strokeColor:'red'});\nreturn div;\n", "parameters": []}}, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "1", "minValue": "1e-24", "correctAnswerFraction": false, "marks": 2}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Value of $\\epsilon=\\;$[[0]]  (enter as a decimal).

", "marks": 0}], "statement": "

{disc()}

\n

The graph above is of the function :

\n

\$\\begin{eqnarray} f(x)&=&x^2+1,\\;\\;\\;&x&\\leq 2\\\\ &=&x+4,\\;\\;\\;&x& \\gt 2 \\end{eqnarray}\$

\n

\n

The red dot indicates that the point $(2,5)$ lies on the graph.

\n

In order for the function $f$ to be continuous at $x=2$, you have to show that given any $\\epsilon \\gt 0$ you can find a $\\delta \\gt 0$ such that the interval $(2-\\delta,2+\\delta)$ is mapped entirely within the interval $(f(2)-\\epsilon,f(2)+\\epsilon)=(5-\\epsilon,5+\\epsilon)$.

\n

In this example, you have to show $f$ is not continuous at $x=2$. So you have to find an $\\epsilon \\gt 0$ such that there is no such $\\delta$.

\n

Using the diagram above, this boils down to finding an $\\epsilon\\gt 0$ so that the interval $(2-\\delta,2+\\delta)$ is not mapped inside the interval $(5-\\epsilon,5+\\epsilon)$ (in red), for any $\\delta \\gt 0$ .

\n

Experiment by changing the value of $\\epsilon$ using its slider to a value where no matter what positive value of $\\delta$ you choose by using its slider, the interval $(2-\\delta,2+\\delta)$ is not mapped by $f$ inside $(5-\\epsilon,5+\\epsilon)$. This gives a value of $\\epsilon$ you can use in a formal proof.

\n

", "tags": ["checked2015", "continuity", "continuous", "discontinuous", "functions", "Jsxgraph", "JSXgraph", "jsxgraph", "MAS2224"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

19/12/2013:

\n

Created.  Uses a fixed function. Next version will use functions created with random parameters.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

A graphical approach to aiding students in writing down a formal proof of discontinuity of a function at a given point.

\n

Uses JSXgraph to sketch the graphs and involves some interaction/experimentation by students in finding appropriate intervals.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

We see from the diagram and moving the intervals around that if $\\epsilon \\le1$ then we can never find a $\\delta \\gt 0$ such that $f$ maps $(2-\\delta,2+\\delta)$ inside $(f(2)-\\epsilon,f(2)+\\epsilon)$.

\n

{disc2()}

\n

We can now construct a formal proof using this:

\n

Take $\\epsilon = \\frac{1}{2}$, (you can choose any value of $\\epsilon, \\;\\;0 \\lt \\epsilon \\le 1$).

\n

Our task is now to show that for any $\\delta \\gt 0$ we choose, there is a point $x \\in (2-\\delta,2+\\delta)$ such that $f(x) \\notin (f(2)-\\epsilon,f(2)+\\epsilon)=(5-1/2,5+1/2)=(9/2,11/2)$.

\n

Looking at the diagram where we have chosen $\\epsilon = \\frac{1}{2}$ we see that the point $x=2+\\delta/2$ lies in $(2-\\delta,2+\\delta)$ and that $f(2+\\delta/2)=(2+\\delta/2)+4=6+\\delta/2 \\notin (9/2,11/2)$.

\n

Hence for this value of $\\epsilon$ we have shown that no matter what $\\delta \\gt 0$ we choose there is a point  $x \\in (2-\\delta,2+\\delta)$ such that $f(x) \\notin (f(2)-\\epsilon,f(2)+\\epsilon)=(9/2,11/2)$.

\n

So the function is discontinuous at $x=2$.

\n

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Choose the correct definitions of continuity at $x_0$ from the following:

\n

\n

[[0]]

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For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |x-x_0\\right | \\lt \\var{ep}$ whenever $|f(x)-f(x_0)| \\lt \\var{del}$ and $x \\in I$.

", "

There exists $\\var{ep} \\gt 0$ , such that for every $\\var{del} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{del}$ whenever $|x-x_0| \\lt \\var{ep}$ and $x \\in I$.

", "

There exists $\\var{del} \\gt 0$ , such that for every $\\var{ep} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

$\\lim_{x \\to x_0}f(x)=f(x_0)$.

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Let $\\mathbb{R}$ be the set of real numbers.

\n

Let $x_0$ be a point in the open interval $I \\subset \\mathbb{R}$ and let $f:I \\rightarrow \\mathbb{R}$ be a function.

\n

What does it mean to say that $f$ is continuous at $x_0$?

\n

", "tags": ["checked2015", "continuity", "continuity at a point", "continuous functions", "definition of continuity"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": ""}, {"name": "True/false statements about continuity of a function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["

For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-a\\right | \\lt \\var{ep}$ whenever $0 \\lt |x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-a\\right | \\lt \\var{ep}$ whenever $0 \\lt |x-x_0| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-a\\right | \\lt \\var{del}$ whenever $0 \\lt |x-x_0| \\lt \\var{ep}$ and $x \\in I$.

", "

For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $0 \\lt |x-a| \\lt \\var{del}$ and $x \\in I$.

", "

For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-a\\right | \\gt \\var{del}$ whenever $0 \\lt |x-x_0| \\lt \\var{ep}$ and $x \\in I$.

", "

For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-a\\right | \\lt \\var{ep}$ whenever $0 \\lt |x-x_0| \\gt \\var{del}$ and $x \\in I$.

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Choose one of the following as the correct definition:

\n

\n

[[0]]

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Let $x_0$ be a point in the open interval $I \\subset \\mathbb{R}$ and let $f:I \\setminus \\{x_0\\} \\rightarrow \\mathbb{R}$ be a function.

\n

What does it mean to say that $f(x) \\rightarrow a$ as $x \\rightarrow x_0$?

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Find the points $x=p,\\;x=q,\\;p \\leq q$ where the graph of $f$ cuts the $x$-axis.

\n\t\t\t

$p=\\;$[[0]]      $q=\\;$[[1]]

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "(b-a)/2", "minValue": "(b-a)/2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "(a+b)^2/4", "minValue": "(a+b)^2/4", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\t\t\t

Using elementary calculus find the area below the curve and above the interval $[p,q]$.

\n\t\t\t

Area= [[0]]  (enter to 3 decimal places).

\n\t\t\t

Also find the point $x=c$ at which the function attains its maximum value over the interval $[p,q]$.

\n\t\t\t

$c=\\;$[[1]]

\n\t\t\t

Maximum value $f(c)=\\;$[[2]]

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({(c * (c + 2) * (2 * c - 1))} / 12)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}, {"answer": "({(c * (c - 2) * (2 * c + 1))} / 12)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t

Let $p$ and $q$ be as above.

\n\t\t\t

Compute the upper and lower sums for the function $f$ over the interval $[p,q]$  using the partition:

\n\t\t\t

\$\\{p,\\;p+1,\\;\\ldots,\\;q\\}\$ with subintervals of length 1.

\n\t\t\t

Upper sum = [[0]]

\n\t\t\t

Lower sum = [[1]]

\n\t\t\t

\n\t\t\t

You will find useful formulae on clicking Show steps.

\n\t\t\t

\n\t\t\t", "steps": [{"type": "information", "prompt": "

\$\\begin{eqnarray} \\sum_{r=1}^n r&=&\\frac{n(n+1)}{2}\\\\ \\sum_{r=1}^n r^2&=&\\frac{n(n+1)(2n+1)}{6} \\end{eqnarray}\$

", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n\t\t\t

Let $p$ and $c$ be as above.

\n\t\t\t

Consider the partition of the interval $[p,c]$ into $\\var{n}$ subintervals each of length  $\\displaystyle \\frac{c-p}{\\var{n}}$:

\n\t\t\t

\$\\Delta=\\left\\{p,\\;p+\\frac{c-p}{\\var{n}},\\;p+\\frac{2(c-p)}{\\var{n}},\\;\\ldots,c\\right\\}\$

\n\t\t\t

Find the upper sum and lower sums for $f$  corresponding to this partition.

\n\t\t\t

Upper sum = [[0]] (enter to 3 decimal places)

\n\t\t\t

Lower sum = [[1]] (enter to 3 decimal places)

\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t

Let $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ be defined by $f(x)=\\simplify{{a*b}+{b-a}*x-x^2}$.

\n\t

{Riemann(a,b)}

Compute the upper and lower sums of $f$ for the partition of the interval $[\\var{c},\\var{d}]$ into subintervals each of length 1:

\n

\$\\Delta = \\{\\var{c},\\var{c+1},\\;\\ldots,\\var{d}\\}.\$

\n

Note that $f$ is {monotonic} function over this interval.

\n

Upper sum = ?[[0]] (Input to 3 decimal places.)

\n

Lower sum = ?[[1]] (Input to 3 decimal places.)

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "uppersum2", "minValue": "uppersum2", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lowersum2", "minValue": "lowersum2", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Suppose we refine the partition into $\\var{2*n}$ subintervals, each of length $\\simplify[all]{1/2}$.

\n

Find the new values of the upper and lower sums:

\n

Upper sum = ?[[0]]  (Input to 3 decimal places).

\n

Lowersum = ?[[1]]   (Input to 3 decimal places).

", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be the function $f(x)=\\simplify{{a}*x+{b}}$.

\n

\n

\n

\n", "tags": ["approximating integrals", "checked2015", "interactive", "jsxgraph", "JSXgraph", "Jsxgraph", "linear function", "lower sum", "MAS1601", "MAS2224", "numerics", "Riemann sums", "upper sum"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

09/12/2013:

\n\t\t

First version started.

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Approximating integral of a linear function by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

{plotf(a,b,c,d)}

\n

$f(x)$ is a linear function which is {monotonic} function. It follows that:

\n

1. The lower sum is given by taking the area over each interval $[x_j,x_{j+1}]$ to be $f(x_{\\simplify{j+{1-side}}})\\times(x_{j+1}-x_j)$ as $f(x_{\\simplify{j+{1-side}}})\\lt f(x_{\\simplify{j+{side}}})$

\n

2. The upper sum is given by taking the area over each interval $[x_j,x_{j+1}]$ to be  $f(x_{\\simplify{j+{side}}})\\times(x_{j+1}-x_j)$  as $f(x_{\\simplify{j+{side}}})\\gt f(x_{\\simplify{j+{1-side}}})$

\n

\n

a) In this case, $x_j=j,\\;\\;j=\\var{c},\\var{c+1},\\dots,\\var{d}$ and so the interval length  $x_{j+1}-x_{j}=1$ and so we have the for the lower sum:

\n

Lower sum.

\n

\$\\begin{eqnarray}\\mbox{LS}=\\sum_{j=\\var{c}}^{\\var{d-1}}f(x_{\\simplify{j+{1-side}}})=\\sum_{j=\\var{c}}^{\\var{d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{{a}*(j+{1-side})+{b}}\\right)&=&{\\var{a}}\\times\\sum_{j=\\var{c}}^{\\var{d-1}}(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{1-side}) })+\\var{n*b}\\\\&=&\\var{lowersum}\\end{eqnarray}\$

\n

We are using the arithmetic series sum: $\\displaystyle \\sum_{j=a}^b j=\\frac{1}{2}(a+b)(b-a+1)$ so that $\\sum_{j=\\var{c}}^{\\var{d-1}}(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{side}) })=\\var{(c+d-1+2*side)*(d-c)/2}$.

\n

Upper sum.

\n

\$\\begin{eqnarray}\\mbox{US}=\\sum_{j=\\var{c}}^{\\var{d-1}}f(x_{\\simplify{j+{side}}})=\\sum_{j=\\var{c}}^{\\var{d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{{a}*(j+{side})+{b}}\\right)&=&{\\var{a}}\\times\\sum_{j=\\var{c}}^{\\var{d-1}}(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{side}) })+\\var{n*b}\\\\&=&\\var{uppersum}\\end{eqnarray}\$

\n

We are using the arithmetic series sum: $\\displaystyle \\sum_{j=a}^b j=\\frac{1}{2}(a+b)(b-a+1)$ so that $\\sum_{j=\\var{c}}^{\\var{d-1}}(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{side}) })=\\var{(c+d-1+2*side)*(d-c)/2}$.

\n

{riemann(a,b,c,d)}

\n

You can use the slider above to see the effect of changing the number, $n$ of partitions between $\\var{c}$ and $\\var{d}$.

\n

b)

\n

If we double the number of  partitions them we have $x_j=\\frac{j}{2},\\;\\;j=\\var{2*c},\\var{2*c+1},\\ldots,\\var{2*d}$.

\n

Also we have to remember to multiply by the interval length $x_{j+1}-x_j=1/2$.

\n

We have:

\n

Lower sum.

\n

\$\\begin{eqnarray}\\mbox{LS}=\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}f(x_{\\simplify{j+{1-side}}})\\times \\frac{1}{2}=\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{{a}*((j+{1-side})/2)+{b}}\\right)\\times \\frac{1}{2}&=&\\frac{\\var{a}}{4}\\times\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{(1-side)}) }\\right)+\\var{n*b}\\\\&=&\\var{lowersum2}\\end{eqnarray}\$

\n

Upper sum.

\n

\$\\begin{eqnarray}\\mbox{US}=\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}f(x_{\\simplify{j+{side}}})\\times \\frac{1}{2}=\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{{a}*((j+{side})/2)+{b}}\\right)\\times \\frac{1}{2}&=&\\frac{\\var{a}}{4}\\times\\sum_{j=\\var{2*c}}^{\\var{2*d-1}}\\left(\\simplify[all,!collectNumbers,!noLeadingMinus]{(j+{side}) }\\right)+\\var{n*b}\\\\&=&\\var{uppersum2}\\end{eqnarray}\$

\n

True value of the integral is:

\n

\$\\int_{\\var{c}}^{\\var{d}}\\simplify{{a}*x+{b}}\\;dx=\\left[\\simplify{{a}/2*x^2+{b}*x}\\right]_{\\var{c}}^{\\var{d}}=\\var{theint}\$

\n"}, {"name": "Compute Riemann sums of a quadratic function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "lowersum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3*(2*n-1)*(4*n-1)/(24*n^2)+a*c^2*(2*n-1)/(4*n)+b*c,3)", "description": "", "name": "lowersum2"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "c"}, "uppersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tuppersum,3)", "description": "", "name": "uppersum"}, "uppersum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3*(2*n+1)*(4*n+1)/(24*n^2)+a*c^2*(2*n+1)/(4*n)+b*c,3)", "description": "", "name": "uppersum2"}, "theint": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3/3+a*c^2/2+b*c,3)", "description": "", "name": "theint"}, "lowersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tlowersum,3)", "description": "", "name": "lowersum"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..20#5)", "description": "", "name": "n"}, "tlowersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c^3*(n-1)*(2n-1)/(6*n^2)+a*c^2*(n-1)/(2*n)+b*c", "description": "", "name": "tlowersum"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "tuppersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c^3*(n+1)*(2n+1)/(6*n^2)+a*c^2*(n+1)/(2*n)+b*c", "description": "", "name": "tuppersum"}}, "ungrouped_variables": ["a", "tuppersum", "c", "b", "uppersum2", "theint", "n", "lowersum", "lowersum2", "tlowersum", "tol", "uppersum"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"plotf1": {"type": "html", "language": "javascript", "definition": "\nvar f = function(x){ return x*x+a*x+b; };\nvar div = Numbas.extensions.jsxgraph.makeBoard('200px','200px', {axis:true,showNavigation:false, boundingbox:[-a/2-5,f(-a/2+5),-a/2+5,Math.min(f(-a/2),-4)]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,-a/2-5,-a/2+5]);\n\n//brd.create('text',[c,-2,c]);\n//var i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"]]}, "riemann": {"type": "html", "language": "javascript", "definition": "\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:false,showNavigation:true, boundingbox:[-2-c-1,(c+1)*(c+1)+(c+1)*a+b+15,c+2,-15]});\nvar brd=div.board;\n//This slider varies the number of partitions. Start off with n\nvar s = brd.create('slider',[[-1,(c+1)*(c+1)+(c+1)*a+b+10],[1,(c+1)*(c+1)+(c+1)*a+b+10],[0,n,50]],{name:'n',snapWidth:1});\nn=s.Value();\n//The function for which we are estimating the integral. \n//This dispays to the right\nvar f = function(x){ return x*x+a*x+b; };\n//Same function but displaced to the left for the lower sum\nvar f1=function(x){return (x+2+c+1)*(x+2+c+1)+a*(x+2+c+1)+b;};\nvar plot = brd.create('functiongraph',[f,0,c+1]);\nvar plot1=brd.create('functiongraph',[f1,-2-c-1,-2]);\n//Two diagrams created using built in function.\nvar os = brd.create('riemannsum',[f,function(){return s.Value();},'upper',0, c ], \n{fillColor:'#ffff00', fillOpacity:0.3});\nvar os1 = brd.create('riemannsum',[f1,function(){return s.Value();},'lower',-2-c-1, -3 ], \n{fillColor:'#ffff00', fillOpacity:0.3});\n//Unfortunately the uppersum calculation in jsxgraph does not agree with mine!\n//So have written the function which gives the same result as the question.\n//Have checked this and is OK - still could be checked.\nvar usm=function(c,n,a,b){\n var s=Math.pow(c,3)*(n+1)*(2*n+1)/(6*Math.pow(n,2))+a*Math.pow(c,2)*(n+1)/(2*n)+b*c;\ns=Numbas.math.precround(Numbas.math.niceNumber(s),3);\nreturn s;}\n//This gives the lower sum using the inbuilt Jsx calculations.\n//Agrees with the question calculations.\nbrd.create('text',\n[-2-c-1,(c+1)*(c+1)+(c+1)*a+b+5,function(){ return 'Sum='+(JXG.Math.Numerics.riemannsum(f1,s.Value(),'lower',-2-c-1,-3)).toFixed(3); }]);\nbrd.create('text',\n[-2-c-1,-5,'Lower Sum']);\nbrd.create('text',[-2-c-1,-1,0]);\nbrd.create('text',[-3,-1,c]);\n//Custom calculation of the upper bound.\nbrd.create('text',\n[0,(c+1)*(c+1)+(c+1)*a+b+5,function(){ return 'Sum='+usm(c,s.Value(),a,b); }]);\nbrd.create('text',[0,-5,'Upper Sum']);\nbrd.create('text',[0,-1,0]);\nbrd.create('text',[c,-1,c]);\nreturn div;\n", "parameters": [["c", "number"], ["n", "number"], ["a", "number"], ["b", "number"]]}, "plotf": {"type": "html", "language": "javascript", "definition": "\nvar f = function(x){ return x*x+a*x+b; };\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:true,showNavigation:false, boundingbox:[-a/2-2,f(c+1),c+4,Math.min(f(-a/2)-2,-5)]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,-5,c+5]);\n//var p1=brd.create('point',[0,0],{visible:false,fixed:true});\n//var p2=brd.create('point',[c,-1],{visible:true,fixed:true,size:1,name:c});\n//var p3=brd.create('point',[c,f(c)],{visible:false,fixed:true,size:1,name:''});\n//var p4=brd.create('point',[0,f(0)],{visible:false,fixed:true,size:1,name:''});\n//var c1=brd.create('line',[p1,p2]);\n//var c2=brd.create('line',[p2,p3]);\n\nbrd.create('text',[c,-2,c]);\nvar i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "-a/2", "minValue": "-a/2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Find $a$ such that $f$ is an increasing function for $x \\ge a$ and a decreasing function for $x \\le a$.

\n

$a=\\;$[[0]] (input as a decimal).

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "uppersum+tol", "minValue": "uppersum-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lowersum+tol", "minValue": "lowersum-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n

Compute the upper and lower sums over the interval for the partition $[0,\\var{c}]$ into subintervals each of length $\\simplify[all]{{c}/{n}}$

\n

\$\\Delta = \\{0,\\;\\simplify[all]{{c}/{n}},\\;\\simplify[all]{{2*c}/{n}},\\ldots,\\;\\var{c}\\}\$

\n

\n

Upper sum = ?[[0]] (Input to 3 decimal places.)

\n

Lower sum = ?[[1]] (Input to 3 decimal places.)

\n", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "uppersum2+tol", "minValue": "uppersum2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lowersum2+tol", "minValue": "lowersum2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Suppose we refine the partition into $\\var{2*n}$ subintervals, each of length $\\simplify[all]{{c}/{2*n}}$.

\n

Find the new values of the upper and lower sums:

\n

Upper sum = ?[[0]]  (Input to 3 decimal places).

\n

Lowersum = ?[[1]]   (Input to 3 decimal places).

", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be the function $f(x)=\\simplify{x^2+{a}*x+{b}}$.

\n

{plotf1(a,b,c)}

\n

\n

\n

\n", "tags": ["approximating integrals", "checked2015", "interactive", "Jsxgraph", "JSXgraph", "jsxgraph", "lower sum", "MAS2224", "numerics", "quadratic", "Riemann sums", "upper sum"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

06/12/2013:

\n\t\t

Started - to be finished with a fully interactive jsxgraph in Advice (or Steps).

\n\t\t

09/12/2013:

\n\t\t

First version finished.

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n

$f(x)$ is a quadratic and its stationary point is given by $\\frac{df}{dx}=0$ which after solving gives the minimum point  $x=-\\simplify[all]{{a}/2}$.

\n

{plotf(a,b,c)}

\n

We see that the function is increasing in the range $[0,\\var{c}]$ and it follows that the lower sums are given by taking the value of $f$ on the left hand side of each interval, multiplying by the interval length and summing up over all intervals. The upper sums by taking the right hand side of each interval etc.

\n

The following diagrams illustrate this. Move the slider to see the effect on the upper and lower sums by varying the number $n$ of subintervals.

\n

{riemann(c,n,a,b)}

\n

Finding the Upper and Lower sums Directly.

\n

First you have to know the following sums of sequences:

\n

\$\\begin{eqnarray*}\\sum_{r=1}^m r &=& \\frac{m(m+1)}{2}\\\\ \\sum_{r=1}^{m} r^2&=&\\frac{m(m+1)(2m+1)}{6}\\end{eqnarray*}\$

\n

Also the interval points between $0$ and $\\var{c}$ are given by $x_j=\\simplify[all,fractionNumbers]{({c}/{n})j},\\;j=0,\\dots,\\var{n}$

\n

For the lower sums, each rectangle with base the interval $[x_r,x_{r+1}]$ contributes an area $f(x_r)(x_{r+1}-x_r)=\\simplify[all,fractionNumbers]{f({c}/{n}*r)*({c}/{n})},\\;r=0,\\ldots,\\var{n-1}$.

\n

We sum from $r=0$ to $r=\\var{n-1}$ as we have to take the left value of each interval.

\n

So the lower sum is:

\n

\$\\begin{eqnarray}\\mbox{LS}=\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers]{f(({c}/{n})*r)*({c}/{n})}&=&\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{((({c}/{n})*r)^2+{a}*(({c}/{n})*r)+{b})*({c}/{n})}\\\\ &=&\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^3*r^2}+\\var{a}\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^2*r}+\\var{b}\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})}\\\\ &=&\\simplify[all,fractionNumbers]{({c}/{n})^3}\\sum_{r=0}^{\\var{n-1}}r^2+\\var{a}\\times\\simplify[all,fractionNumbers]{({c}/{n})^2}\\sum_{r=0}^{\\var{n-1}}r+\\var{b}\\times\\var{c}\\end{eqnarray}\$

\n

Using \$\\sum_{r=0}^{\\var{n-1}}r=\\frac{\\var{n-1}\\times \\var{n}}{2}=\\var{(n-1)*n/2},\\;\\;\\;\\sum_{r=0}^{\\var{n-1}}r^2=\\frac{\\var{n-1}\\times \\var{n}\\times \\var{2*n-1}}{6}=\\var{(n-1)*n*(2*n-1)/6}\$

\n

and substituting into the above equation we obtain:

\n

LS = $\\var{tlowersum}=\\var{lowersum}$ to 3 decimal places.

\n

As for the upper sum we have $\\mbox{US}=\\mbox{LS}+(f(\\var{c})-f(0))\\times\\simplify[all,fractionNumbers]{{c}/{n}}=\\mbox{LS}+(\\var{c^2+a*c+b}-\\var{b})\\times \\simplify[all,fractionNumbers]{{c}/{n}}=\\var{tlowersum}+\\var{tuppersum-tlowersum}=\\var{tuppersum}=\\var{uppersum}$ to 3 decimal places .

\n

Or if we take the more long-winded approach using the same method as for LS we have:

\n

\$\\begin{eqnarray}\\mbox{US}=\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers]{f(({c}/{n})*r)*({c}/{n})}&=&\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{((({c}/{n})*r)^2+{a}*(({c}/{n})*r)+{b})*({c}/{n})}\\\\&=&\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^3*r^2}+\\var{a}\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^2*r}+\\var{b}\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})}\\\\&=&\\simplify[all,fractionNumbers]{({c}/{n})^3}\\sum_{r=1}^{\\var{n}}r^2+\\var{a}\\times\\simplify[all,fractionNumbers]{({c}/{n})^2}\\sum_{r=1}^{\\var{n}}r+\\var{b}\\times\\var{c}\\\\&=&\\var{tuppersum}=\\var{uppersum}\\end{eqnarray}\$

\n

to 3 decimal places.

\n

The true value is given by the integral

\n

\$\\int_{0}^{\\var{c}} \\simplify[all]{x^2+{a}*x+{b}}\\;dx=\\left[\\simplify[all]{x^3/3+{a}/2*x^2+{b}*x}\\right]_{0}^{\\var{c}}=\\var{theint}\$ to 3 decimal places.

\n

\n

For $\\var{2*n}$ intervals we obtain:

\n

$\\mbox{LS}=\\var{lowersum2}$.

\n

$\\mbox{US}=\\var{uppersum2}$.

\n

Both to 3 decimal places.

\n

\n

\n

\n

\n

\n"}, {"name": "Compute Riemann sums on an interval", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "lowersum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3*(2*n-1)*(4*n-1)/(24*n^2)+a*c^2*(2*n-1)/(4*n)+b*c,3)", "description": "", "name": "lowersum2"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "c"}, "uppersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tuppersum,3)", "description": "", "name": "uppersum"}, "uppersum2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3*(2*n+1)*(4*n+1)/(24*n^2)+a*c^2*(2*n+1)/(4*n)+b*c,3)", "description": "", "name": "uppersum2"}, "theint": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(c^3/3+a*c^2/2+b*c,3)", "description": "", "name": "theint"}, "lowersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tlowersum,3)", "description": "", "name": "lowersum"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..20#5)", "description": "", "name": "n"}, "tlowersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c^3*(n-1)*(2n-1)/(6*n^2)+a*c^2*(n-1)/(2*n)+b*c", "description": "", "name": "tlowersum"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "tuppersum": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c^3*(n+1)*(2n+1)/(6*n^2)+a*c^2*(n+1)/(2*n)+b*c", "description": "", "name": "tuppersum"}}, "ungrouped_variables": ["a", "tuppersum", "c", "b", "uppersum2", "theint", "n", "lowersum", "lowersum2", "tlowersum", "tol", "uppersum"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"riemann": {"type": "html", "language": "javascript", "definition": "\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:false,showNavigation:true, boundingbox:[-2-c-1,(c+1)*(c+1)+(c+1)*a+b+15,c+2,-15]});\nvar brd=div.board;\n//This slider varies the number of partitions. Start off with n\nvar s = brd.create('slider',[[-1,(c+1)*(c+1)+(c+1)*a+b+10],[1,(c+1)*(c+1)+(c+1)*a+b+10],[0,n,50]],{name:'n',snapWidth:1});\nn=s.Value();\n//The function for which we are estimating the integral. \n//This dispays to the right\nvar f = function(x){ return x*x+a*x+b; };\n//Same function but displaced to the left for the lower sum\nvar f1=function(x){return (x+2+c+1)*(x+2+c+1)+a*(x+2+c+1)+b;};\nvar plot = brd.create('functiongraph',[f,0,c+1]);\nvar plot1=brd.create('functiongraph',[f1,-2-c-1,-2]);\n//Two diagrams created using built in function.\nvar os = brd.create('riemannsum',[f,function(){return s.Value();},'upper',0, c ], \n{fillColor:'#ffff00', fillOpacity:0.3});\nvar os1 = brd.create('riemannsum',[f1,function(){return s.Value();},'lower',-2-c-1, -3 ], \n{fillColor:'#ffff00', fillOpacity:0.3});\n//Unfortunately the uppersum calculation in jsxgraph does not agree with mine!\n//So have written the function which gives the same result as the question.\n//Have checked this and is OK - still could be checked.\nvar usm=function(c,n,a,b){\n var s=Math.pow(c,3)*(n+1)*(2*n+1)/(6*Math.pow(n,2))+a*Math.pow(c,2)*(n+1)/(2*n)+b*c;\ns=Numbas.math.precround(Numbas.math.niceNumber(s),3);\nreturn s;}\n//This gives the lower sum using the inbuilt Jsx calculations.\n//Agrees with the question calculations.\nbrd.create('text',\n[-2-c-1,(c+1)*(c+1)+(c+1)*a+b+5,function(){ return 'Sum='+(JXG.Math.Numerics.riemannsum(f1,s.Value(),'lower',-2-c-1,-3)).toFixed(3); }]);\nbrd.create('text',\n[-2-c-1,-5,'Lower Sum']);\nbrd.create('text',[-2-c-1,-1,0]);\nbrd.create('text',[-3,-1,c]);\n//Custom calculation of the upper bound.\nbrd.create('text',\n[0,(c+1)*(c+1)+(c+1)*a+b+5,function(){ return 'Sum='+usm(c,s.Value(),a,b); }]);\nbrd.create('text',[0,-5,'Upper Sum']);\nbrd.create('text',[0,-1,0]);\nbrd.create('text',[c,-1,c]);\nreturn div;\n", "parameters": [["c", "number"], ["n", "number"], ["a", "number"], ["b", "number"]]}, "plotf": {"type": "html", "language": "javascript", "definition": "\nvar f = function(x){ return x*x+a*x+b; };\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px', {axis:true,showNavigation:false, boundingbox:[-a/2-2,f(c+1),c+4,Math.min(f(-a/2)-2,-5)]});\n\nvar brd=div.board;\n\nvar plot = brd.create('functiongraph',[f,-5,c+5]);\n//var p1=brd.create('point',[0,0],{visible:false,fixed:true});\n//var p2=brd.create('point',[c,-1],{visible:true,fixed:true,size:1,name:c});\n//var p3=brd.create('point',[c,f(c)],{visible:false,fixed:true,size:1,name:''});\n//var p4=brd.create('point',[0,f(0)],{visible:false,fixed:true,size:1,name:''});\n//var c1=brd.create('line',[p1,p2]);\n//var c2=brd.create('line',[p2,p3]);\n\nbrd.create('text',[c,-2,c]);\nvar i1 = brd.create('integral', [[0, c], plot]);\n\nreturn div;\n", "parameters": [["a", "number"], ["b", "number"], ["c", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "-a/2", "minValue": "-a/2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Find $a$ such that $f$ is an increasing function for $x \\ge a$ and a decreasing function for $x \\le a$.

\n

$a=\\;$[[0]]  (input as a decimal).

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "uppersum+tol", "minValue": "uppersum-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lowersum+tol", "minValue": "lowersum-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Compute the upper and lower sums of $f$ for the partition of the interval $[0,\\var{c}]$ into subintervals each of length $\\simplify[all]{{c}/{n}}$:

\n

\$\\Delta = \\{0,\\;\\simplify[all]{{c}/{n}},\\;\\simplify[all]{{2*c}/{n}},\\ldots,\\;\\var{c}\\}.\$

\n

\n

Upper sum = ?[[0]] (Input to 3 decimal places.)

\n

Lower sum = ?[[1]] (Input to 3 decimal places.)

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "uppersum2+tol", "minValue": "uppersum2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lowersum2+tol", "minValue": "lowersum2-tol", "correctAnswerFraction": false, "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "

Suppose we refine the partition into $\\var{2*n}$ subintervals, each of length $\\simplify[all]{{c}/{2*n}}$.

\n

Find the new values of the upper and lower sums:

\n

Upper sum = ?[[0]]  (Input to 3 decimal places).

\n

Lowersum = ?[[1]]   (Input to 3 decimal places).

", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be the function $f(x)=\\simplify{x^2+{a}*x+{b}}$.

\n

\n

\n

\n", "tags": ["approximating integrals", "checked2015", "Jsxgraph", "JSXgraph", "jsxgraph", "lower sum", "MAS2224", "numerics", "Riemann sums", "upper sum"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

06/12/2013:

\n\t\t

Started - to be finished with a fully interactive jsxgraph in Advice (or Steps).

\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Approximating integral of a quadratic by Riemann sums . Will include an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

$f(x)$ is a quadratic and its stationary point is given by $\\frac{df}{dx}=0$ which after solving gives the minimum point  $x=-\\simplify[all]{{a}/2}$.

\n

{plotf(a,b,c)}

\n

We see that the function is increasing in the range $[0,\\var{c}]$ and it follows that the lower sums are given by taking the value of $f$ on the left hand side of each interval, multiplying by the interval length and summing up over all intervals. The upper sums by taking the right hand side of each interval etc.

\n

The following diagrams illustrate this. Move the slider to see the effect on the upper and lower sums by varying the number $n$ of subintervals.

\n

{riemann(c,n,a,b)}

\n

Finding the Upper and Lower sums Directly.

\n

First you have to know the following sums of sequences:

\n

\$\\begin{eqnarray*}\\sum_{r=1}^m r &=& \\frac{m(m+1)}{2}\\\\ \\sum_{r=1}^{m} r^2&=&\\frac{m(m+1)(2m+1)}{6}\\end{eqnarray*}\$

\n

Also the interval points between $0$ and $\\var{c}$ are given by $x_j=\\simplify[all,fractionNumbers]{({c}/{n})j},\\;j=0,\\dots,\\var{n}$

\n

For the lower sums, each rectangle with base the interval $[x_r,x_{r+1}]$ contributes an area $f(x_r)(x_{r+1}-x_r)=\\simplify[all,fractionNumbers]{f({c}/{n}*r)*({c}/{n})},\\;r=0,\\ldots,\\var{n-1}$.

\n

We sum from $r=0$ to $r=\\var{n-1}$ as we have to take the left value of each interval.

\n

So the lower sum is:

\n

\$\\begin{eqnarray}\\mbox{LS}=\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers]{f(({c}/{n})*r)*({c}/{n})}&=&\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{((({c}/{n})*r)^2+{a}*(({c}/{n})*r)+{b})*({c}/{n})}\\\\ &=&\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^3*r^2}+\\var{a}\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^2*r}+\\var{b}\\sum_{r=0}^{\\var{n-1}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})}\\\\ &=&\\simplify[all,fractionNumbers]{({c}/{n})^3}\\sum_{r=0}^{\\var{n-1}}r^2+\\var{a}\\times\\simplify[all,fractionNumbers]{({c}/{n})^2}\\sum_{r=0}^{\\var{n-1}}r+\\var{b}\\times\\var{c}\\end{eqnarray}\$

\n

Using \$\\sum_{r=0}^{\\var{n-1}}r=\\frac{\\var{n-1}\\times \\var{n}}{2}=\\var{(n-1)*n/2},\\;\\;\\;\\sum_{r=0}^{\\var{n-1}}r^2=\\frac{\\var{n-1}\\times \\var{n}\\times \\var{2*n-1}}{6}=\\var{(n-1)*n*(2*n-1)/6}\$

\n

and substituting into the above equation we obtain:

\n

LS = $\\var{tlowersum}=\\var{lowersum}$ to 3 decimal places.

\n

As for the upper sum we have $\\mbox{US}=\\mbox{LS}+(f(\\var{c})-f(0))\\times\\simplify[all,fractionNumbers]{{c}/{n}}=\\mbox{LS}+(\\var{c^2+a*c+b}-\\var{b})\\times \\simplify[all,fractionNumbers]{{c}/{n}}=\\var{tlowersum}+\\var{tuppersum-tlowersum}=\\var{uppersum}$ to 3 decimal places .

\n

Or if we take the more long-winded approach using the same method as for LS we have:

\n

\$\\begin{eqnarray}\\mbox{US}=\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers]{f(({c}/{n})*r)*({c}/{n})}&=&\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{((({c}/{n})*r)^2+{a}*(({c}/{n})*r)+{b})*({c}/{n})}\\\\&=&\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^3*r^2}+\\var{a}\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})^2*r}+\\var{b}\\sum_{r=1}^{\\var{n}}\\simplify[all,fractionNumbers,!collectNumbers]{({c}/{n})}\\\\&=&\\simplify[all,fractionNumbers]{({c}/{n})^3}\\sum_{r=1}^{\\var{n}}r^2+\\var{a}\\times\\simplify[all,fractionNumbers]{({c}/{n})^2}\\sum_{r=1}^{\\var{n}}r+\\var{b}\\times\\var{c}\\\\&=&\\var{tuppersum}=\\var{uppersum}\\end{eqnarray}\$

\n

to 3 decimal places.

\n

The true value is given by the integral

\n

\$\\int_{0}^{\\var{c}} \\simplify[all]{x^2+{a}*x+{b}}\\;dx=\\left[\\simplify[all]{x^3/3+{a}/2*x^2+{b}*x}\\right]_{0}^{\\var{c}}=\\var{theint}\$ to 3 decimal places.

\n

\n

For $\\var{2*n}$ intervals we obtain:

\n

$\\mbox{LS}=\\var{lowersum2}$.

\n

$\\mbox{US}=\\var{uppersum2}$.

\n

Both to 3 decimal places.

\n

\n

\n

\n

\n

"}, {"name": "True/false statements about continuity and differentiability, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "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/"}], "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\n \n \n

[[0]]

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{Ch2}

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{Ch3}

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{Ch4}

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If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is continuous at $c \\\\in \\\\mathbb{R}$, then it is differentiable at $c$.

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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 $c \\\\in (a,b)$.

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If a function $f: \\\\mathbb{R} \\\\to \\\\mathbb{R}$ is differentiable at $c \\\\in \\\\mathbb{R}$, then it is continuous at $c$.

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If a function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then $f\\'(c)=0$ for some $c \\\\in (a,b)$.

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Given any function defined on $[a,b]$ with $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.

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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 $c \\\\in (a,b)$.

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If a function $f$ is differentiable on $(a,b)$,  then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.

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If a function $f$ is continuous on $[a,b]$ and $f(a) < \\\\gamma < f(b)$, then $f(c)=\\\\gamma$ for some $c \\\\in (a,b)$.

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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 $c \\\\in (a,b)$.

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If a function $f$ is continuous and differentiable on $(a,b)$,  then $f\\'(c)=\\\\dfrac{f(b)-f(a)}{b-a}$ for some $c \\\\in (a,b)$.

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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 $f(b)>f(a)$.

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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 $c \\\\in (a,b)$.

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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", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (2 correct out of 4) covering properties of continuity and differentiability. Selection of questions from a pool.

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Can 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.

You should be able to work out the correct answers from your notes.

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If $a_n \\\\neq 0$  for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell$ with $|\\\\ell | <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

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If $a_n \\\\geq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

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If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for all $x$ with $|x|<R$.

\"", "name": "tr13", "description": ""}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ diverges.

\"", "name": "tr3", "description": ""}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If the series $\\\\Sigma a_n$ diverges, then $a_n \\\\not\\\\to 0$ as $n \\\\to \\\\infty$.

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If  $a_n \\\\to 0$ as $n \\\\to \\\\infty$, then the series $\\\\Sigma a_n$ converges.

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If $a_n \\\\geq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr2", "description": ""}, "f15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0<R<\\\\infty$, then $a_n \\\\neq 0$ and $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.

\"", "name": "f15", "description": ""}, "f9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is convergent then it is absolutely convergent.

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If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ does not converge to $0$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr11", "description": ""}, "f10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is not divergent then it is absolutely convergent.

\"", "name": "f10", "description": ""}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell >1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr6", "description": ""}, "tr16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ diverges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\leq |X|$

\"", "name": "tr16", "description": ""}, "ch8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(x=1,f13,if(x=2,f14,if(f=x,f15,f16)))", "name": "ch8", "description": ""}, "f13": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series converges for $x=R$.

\"", "name": "f13", "description": ""}, "f14": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for $x=R$

\"", "name": "f14", "description": ""}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n^2}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

\"", "name": "tr1", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "g", "description": ""}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "u", "description": ""}, "w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "w", "description": ""}, "f11": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is not decreasing, then $\\\\Sigma a_n$ diverges.

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If a power series $\\\\Sigma a_n x^n$ has radius of convergence $R$ with $0 < R <\\\\infty$, then the power series diverges for all $x$ with $|x|>R$.

\"", "name": "tr14", "description": ""}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,if(t=3,tr3,tr4)))", "name": "ch1", "description": ""}, "tr10": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.

\"", "name": "tr10", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "t", "description": ""}, "tr12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n >0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr12", "description": ""}, "tr20": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is divergent then it is not absolutely convergent.

\"", "name": "tr20", "description": ""}, "f7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$  $\\\\ell >0$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f7", "description": ""}, "tr7": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\infty$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

\"", "name": "tr7", "description": ""}, "tr9": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $\\\\Sigma a_n$ is absolutely convergent then it is convergent.

\"", "name": "tr9", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\geq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $a_n \\\\leq \\\\dfrac{1}{n}$ for all $n \\\\in \\\\mathbb{N}$, then $\\\\Sigma a_n$ converges.

\"", "name": "f2", "description": ""}, "tr15": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ converges for some value $x=X \\\\neq 0$, then the radius of convergence $R$ satisfies $R \\\\geq |X|$.

\"", "name": "tr15", "description": ""}, "ch6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(g=1,f5,if(g=6,f2,if(g=3,f7,f8)))", "name": "ch6", "description": ""}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell <1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "tr5", "description": ""}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n > 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n} \\\\to \\\\ell =1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f5", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "f", "description": ""}, "ch7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(h=1,f9,if(h=2,f10,if(h=3,f11,f12)))", "name": "ch7", "description": ""}, "f8": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n \\\\neq 0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell<1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ converges.

\"", "name": "f8", "description": ""}, "f12": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n = (-1)^{n-1} u_n$ with $u_n \\\\geq 0$ and if $\\{u_n\\}$ is increasing, then $\\\\Sigma a_n$ diverges.

\"", "name": "f12", "description": ""}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $a_n>0$ for all $n \\\\in \\\\mathbb{N}$ and if $\\\\dfrac{a_{n+1}}{a_n}$ $\\\\to$ $\\\\ell=1$ as $n \\\\to \\\\infty$, then $\\\\Sigma a_n$ diverges.

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If the series $\\\\Sigma a_n$ converges, then $a_n \\\\to 0$ as $n \\\\to \\\\infty$.

\"", "name": "tr4", "description": ""}, "ch5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f1,if(f=2,f2,if(f=3,f3,f4)))", "name": "ch5", "description": ""}, "f16": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a power series $\\\\Sigma a_n x^n$ with $a_n \\\\neq 0$ for all $n$ has radius of convergence $R$ with $R<\\\\infty$, then $\\\\dfrac{a_{n+1}}{a_n}$ tends to a limit as $n \\\\to \\\\infty$.

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{Ch1}

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{Ch2}

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{Ch3}

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{Ch4}

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{Ch5}

", "

{Ch6}

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{Ch7}

", "

{Ch8}

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[[0]]

\n \n \n \n", "showCorrectAnswer": true, "marks": 0}], "statement": "

Answer the following question on series. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

", "tags": ["checked2015", "divergent series", "limits", "MAS1601", "MAS2224", "power series", "series"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

17/04/2015:

\n

(OK) new question based on a similar style question on sequences. Changed the statements to long text to enable better mathematical expressions. Encountered problems when editing (math expressions not recognised).

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (4 correct out of 8) covering properties of convergent and divergent series and including questions on power series. Selection of questions from a pool.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

You should be able to work out the correct answers from your notes.

"}, {"name": "True/false statements about limits of sequences", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "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/"}], "variable_groups": [], "variables": {"tr8": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A divergent sequence can have a convergent subsequence.

\"", "name": "tr8", "description": ""}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "x", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A bounded sequence is convergent.

\"", "name": "f1", "description": ""}, "ch2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "name": "ch2", "description": ""}, "f20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for an unbounded sequence to have a bounded subsequence.'", "name": "f20", "description": ""}, "f4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

If {$\\{x_n\\}$} converges, then {$\\{x_{n+i}\\}$} could diverge for some natural number $i$.

\"", "name": "f4", "description": ""}, "f3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A convergent sequence is either increasing or decreasing.

\"", "name": "f3", "description": ""}, "tr2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

If {$\\{x_{n+i}\\}$} diverges for some natural number $i$, then {$\\{x_n\\}$} diverges.

\"", "name": "tr2", "description": ""}, "f9": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'There exists a sequence that is not bounded but which converges.'", "name": "f9", "description": "

There exists a sequence that is not bounded but which converges.

"}, "tr11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A finite set is bounded.', 'A set that is not bounded has an infinite number of elements.')", "name": "tr11", "description": ""}, "tr3": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

If a sequence is not bounded, then it does not converge.

\"", "name": "tr3", "description": ""}, "tr6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A sequence with only a finite number of non zero terms converges to 0.'", "name": "tr6", "description": ""}, "ch8": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(x=1,f11,if(x=2,f12,if(x=3,f13,f14)))", "name": "ch8", "description": ""}, "f13": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded set never has a maximum element.', 'A bounded set never has a minimum element.')", "name": "f13", "description": ""}, "f14": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'A set with an infinite number of elements cannot be bounded.'", "name": "f14", "description": ""}, "tr1": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

A convergent sequence is bounded.

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All convergent sequences of positive terms converge to a value $> 0$.

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There exists a sequence with all terms greater than zero and with limit 0.

\"", "name": "tr9", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..4)", "name": "h", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

Every divergent sequence is unbounded.

\"", "name": "f2", "description": ""}, "tr20": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'If a sequence has the subsequence given by the even terms converges to the same limit as the subsequence of odd terms, then the sequence also converges to that limit.'", "name": "tr20", "description": ""}, "f10": {"group": "Ungrouped variables", "templateType": "anything", "definition": "'It is not possible for a sequence to be both increasing and decreasing.'", "name": "f10", "description": ""}, "tr5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

It is possible for a sequence to be both increasing and decreasing.

\"", "name": "tr5", "description": ""}, "f5": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

There exists a convergent sequence {$\\{x_n\\}$} with $x_n >0$ for all $n \\\\in \\\\mathbb{N}$ and limit $\\\\ell <0$.

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If {$\\{x_n\\}$} diverges, then {$\\{x_{n+i}\\}$} could converge for some natural number $i$.

\"", "name": "f8", "description": ""}, "tr7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('A bounded increasing sequence converges.','A bounded decreasing sequence converges.')", "name": "tr7", "description": ""}, "f6": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

There exists a bounded increasing sequences that does not converge.

\"", "name": "f6", "description": ""}, "ch3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(v=1,tr7,if(v=2,tr8,tr9))", "name": "ch3", "description": ""}, "ch4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(f=1,f1,if(f=2,f2,f3))", "name": "ch4", "description": ""}, "tr4": {"group": "Ungrouped variables", "templateType": "long string", "definition": "\"

In a convergent sequence, all subsequences converge to the same limit.

\"", "name": "tr4", "description": ""}, "ch5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(g=1,f4,if(g=2,f5,f6))", "name": "ch5", "description": ""}}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "t", "tr9", "tr8", "tr1", "u", "tr3", "tr2", "tr5", "tr4", "tr7", "tr6", "tr20", "g", "f", "h", "ch1", "ch2", "ch3", "ch4", "ch5", "ch6", "v", "f10", "f20", "w", "x", "f11", "f12", "f13", "f14", "tr10", "tr11", "tr12", "ch7", "ch8"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (4 correct out of 8) covering properties of convergent and divergent sequences and boundedness of sets. Selection of questions from a pool.

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{Ch1}

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{Ch2}

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{Ch3}

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{Ch4}

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{Ch5}

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{Ch6}

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{Ch7}

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{Ch8}

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[[0]]

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Answer the following question on sequences and sets. Note that a sequence is said to be unbounded if it is not bounded.

\n

Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

", "tags": ["bounded sequences", "bounded sets", "checked2015", "convergence", "convergent sequences", "decreasing sequence", "divergence", "divergent sequences", "increasing sequence", "limits", "monotone sequence", "monotonic sequence", "sequence", "sequences", "subsequence", "tested1", "unbounded sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

You should be able to work out the correct answers from your notes.

"}, {"name": "True/false statements about properties of continuity and limits, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "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/"}], "variable_groups": [], "variables": {"v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "v"}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If for some sequence {$u_n$} converging to $c$, the sequence {$f(u_n)$} converges to $l$, then $f(x) \\\\to l$ as $x \\\\to c$.

\"", "description": "", "name": "f1"}, "ch2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,tr4,if(u=2,tr5,tr6))", "description": "", "name": "ch2"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "u"}, "f4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $f(x) \\\\not\\\\to l$ as $x$ tends to $c$, then $f(x_n) \\\\not\\\\to l$ as $n \\\\to \\\\infty$ for every sequence {$x_n$} converging to $c$.

\"", "description": "", "name": "f4"}, "f5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If the limit of $f(x)$ as $x \\\\to c$ exists then the limit is $f(c)$.

\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If for some sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.

\"", "description": "", "name": "f2"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $g(x) \\\\to m$ as $x \\\\to c$, then $\\\\dfrac{f(x)}{g(x)} \\\\to \\\\dfrac{\\\\ell}{m}$ as $x \\\\to c$.

\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

There exists a function $f$ such that the limit of $f(x)$ as $x \\\\to c$ exists and $f(c)$ exists, but $f$ is not continuous at $c$.

\"", "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 the limit of $f(x)$ as $x \\\\to c$ exists and the limit is $f(c)$, then $f$ is continuous at $c$.

\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a function $f$ is continuous at $c$, then for any sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$.

\"", "description": "", "name": "tr2"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "f"}, "tr3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If for every sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.

\"", "description": "", "name": "tr3"}, "f6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If the limit of $f(x)$ as $x \\\\to c$ exists and if $f(c)$ exists, then $f$ is continuous at $c$.

\"", "description": "", "name": "f6"}, "ch3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,f1,if(v=2,f2,f3))", "description": "", "name": "ch3"}, "ch4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(f=1,f4,if(f=2,f5,f6))", "description": "", "name": "ch4"}, "tr4": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $f(x) \\\\not\\\\to \\\\ell$ as $x \\\\to c$, then $f(x_n) \\\\not\\\\to \\\\ell$ as $n \\\\to \\\\infty$ for some sequence {$x_n$} converging to $c$.

\"", "description": "", "name": "tr4"}, "tr1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $x_n \\\\to c$ as $n \\\\to \\\\infty$ (with each $x_n \\\\neq c$), then $f(x_n) \\\\to \\\\ell$ as $n \\\\to \\\\infty$.

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[[0]]

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{Ch2}

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{Ch3}

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{Ch4}

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Answer the following question on continuity and limits of functions. You may assume that the functions $f$ are $:\\mathbb{R} \\to \\mathbb{R}$. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.

You should be able to work out the correct answers from your notes.

"}, {"name": "True/false statements about Riemann integration, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"tr8": {"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 $U(P_n) \\\\to \\\\ell$ and $L(P_n) \\\\to \\\\ell$ as $n \\\\to \\\\infty$, then $f$ is Riemann integrable on $[a,b]$ and $\\\\int_a^b f(x) dx = \\\\ell$.

\"", "description": "", "name": "tr8"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "v"}, "f1": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"

If a function $f$ is bounded on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

\"", "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 $L(P) \\\\geq U(Q)$.

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If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $L(P) \\\\leq L(Q)$.

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If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and increasing on $[a,b]$.

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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 $L(P) \\\\leq L(Q)$.

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If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $U(P \\\\cup Q) \\\\leq U(P)$.

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If a function $f$ is bounded and decreasing on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

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If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is bounded and decreasing on $[a,b]$.

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If a function $f$ is continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

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If a function $f$ is Riemann integrable on $[a,b]$, then $f$ is continuous on $[a,b]$.

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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 $n \\\\to \\\\infty$.

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If $f$ is a bounded function on $[a,b]$, and if $P$ and $Q$ are partitions of $[a,b]$, then $L(P) \\\\leq U(Q)$.

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If a function $f$ is bounded and increasing on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

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{Ch4}

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Answer the following question on continuity and differentiability. Note that every correct answer is worth 1 mark, but every wrong answer loses a mark.

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17/04/15:

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(OK) new question adapting the format of an older question

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Multiple response question (2 correct out of 4) covering properties of Riemann integration. Selection of questions from a pool.