A set of Numbas exercises for students transitioning from school to University. Designed to help students gain familiarity with using Numbas to enter mathematics, and as revision for algebra, geometry and calculus.
", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "duration": 0, "percentPass": "0", "showQuestionGroupNames": true, "showstudentname": true, "question_groups": [{"name": "Getting started", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "questions": [{"name": "How to enter numbers - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"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": "0.01", "description": "", "name": "tol"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,4,5,6,8,9,10,12)", "description": "", "name": "a1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "c"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7,11,13)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(a1/b1,2)", "description": "", "name": "ans1"}}, "ungrouped_variables": ["a", "c", "b", "ans1", "a1", "b1", "tol"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Find the result of this calculation: (This is an example of a randomised question - the next time you use this example you will probably be given a different calculation to do):
\n$\\var{a}\\times\\var{b}+\\var{c}=\\;$[[0]]
\nYou have to input a whole number - it could be in decimal form. If the answer was $2$ then you could input 2 or 2.0 - try both forms.
Decimals
\nMany calculations will result in numbers which need to be entered in decimal notation, and the question will ask for a certain number of decimal places.
\nOften there is a small tolerance built in so that if you get the result wrong by 1 in the last decimal place then it will be marked as correct. But accuracy is important, so make sure that you get the calculations correct.
\nFor example:
\nInput $\\displaystyle \\frac{\\var{a1}}{\\var{b1}}$ as a decimal correct to 2 decimal places here: [[0]]
\nTry entering the correct value and submitting. Then vary the last decimal place by 1 either way and submitting, and then the last place by 2 either way and submitting.
\nTry putting in the fraction as it is (i.e. $\\var{a1}/\\var{b1}$ ) and see what happens.
\nThe system gives an error message as what you have put in is not a direct representation of a number. But you can always re-enter.
\nSo be careful - always check after submitting your answer that the input field contains the answer that you thought you entered.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{ans1-tol}", "maxValue": "{ans1+tol}", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Fractions
\nYou will find that some questions may ask you to input fractions and not decimals.
\nFor example, find the following sum as a fraction:
\n$\\displaystyle \\frac{1}{\\var{a1}}+\\frac{1}{\\var{b1}}=\\;$[[0]]
\n(input as a fraction and not a decimal)
\nHint: the answer is {a1+b1}/{a1*b1}
\nTry inputting the decimal version of this to as many places as you like (for example given by the calculator on the PC - you can copy this from the calculator and paste into the input field) and see what happens.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{a1+b1}/{a1*b1}", "vsetRangePoints": 5, "notallowed": {"message": "Simplify into a single fraction. Do not enter as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["+", "."]}, "checkingType": "absdiff", "vsetRange": [0, 1], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": [], "variableReplacements": [], "failureRate": 1, "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showPreview": true, "checkVariableNames": false, "unitTests": [], "scripts": {}, "answerSimplification": "all, fractionNumbers", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "prompt": "As this question is in practice mode, if you click on the Reveal answers button all of the question fields are filled with the correct answers. Also, if available, there will be a full solution given under the heading Advice. Just scroll down to see this. However, there is no advice available for this question as it is not needed.
\nFinally as you are in practice mode, if you click on the Try another question like this one button at the bottom you will get this question again but with different numbers (usually!), and you can try it again. This is true for all practice mode questions which are randomised.
", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "unitTests": []}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example we show how to enter numbers, either as
\nDetails on inputting numbers into Numbas.
"}, "advice": "No advice available.
"}, {"name": "How to enter algebraic expressions - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "d", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..9)", "name": "c", "description": ""}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s", "description": ""}}, "ungrouped_variables": ["a", "s", "b", "c", "d"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "{a}*x^{b}+{c}x+{d}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "all", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Inputting polynomials such as $3x^2+5x-2$ is easy : just input 3*x^2+5*x-2.
Try this:
\nInput this polynomial: $\\simplify[all]{{a}*x^{b}+{c}*x+{d}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^2+{a+c}x*y+{a*c}y^2", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": true, "expectedvariablenames": ["x", "y"], "notallowed": {"showStrings": false, "message": "Do not include brackets in your answer.
", "strings": ["("], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Now consider this problem.
\nExpand the brackets and input the resulting expression:
\n$\\simplify[all]{(x+{a}y)(x+{c}y)}=\\;$[[0]]
\nMake sure that you input an expression in your answer such as $xy$ as x*y.
(Do not include brackets in your answer.)
", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example, we look at how you enter algebraic expressions - those involving symbols.
\nThe box next to your input shows what you've written in mathematical notation and is very important as you can check it against the expression you had in mind.
", "tags": ["algebraic expressions", "checked2015", "input", "introduction", "notation", "Numbas", "numbas", "polynomials", "symbols"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Inputting algebraic expressions into Numbas.
"}, "advice": ""}, {"name": "How to enter powers - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"scripts": {}, "gaps": [{"answer": "e^({a+b}*x)", "vsetrange": [0, 0.1], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "To test your input of powers try the following examples:
\nInput as a single power of $x$:
\n$\\simplify[all]{e^({a}*x)e^({b}*x)}=\\;$[[0]]
\n(The answer is $\\simplify[all]{e^({a+b}x)}$ but you have to enter it properly.)
\nYour input is shown in mathematical notation in a box next to your input so that you can check that you have entered it correctly.
\nClick on Submit part to check on your answer.
\nClick on the input field and edit your answer by inputting without brackets around the powers to see what happens.
\n\n
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^({c+d})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "
Input $x^{\\var{c}}x^{\\var{d}}$ as a single power of $x$.
\nFor example, you would input $x^{-6}x^{-5}$ as x^(-11).
$x^{\\var{c}}x^{\\var{d}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "x^{f}*y^{f}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input in the form $x^a*y^b$ for suitable values of $a$ and $b$.
", "showStrings": false, "partialCredit": 0, "strings": ["xy", "x*y"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "all", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Input $(x \\cdot y)^{\\var{f}}$ in the form $x^a \\times y^b$ for suitable values of $a$ and $b$.
\n$(x \\cdot y)^{\\var{f}}=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12 except a)*s", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(6..12)*s", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..-6)", "description": "", "name": "c"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..-6)", "description": "", "name": "d"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5,-4,-3,-2,-1)", "description": "", "name": "f"}}, "ungrouped_variables": ["a", "c", "b", "d", "f", "s"], "variable_groups": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "In this example we show you how to input powers. It is important that you get this right as many questions ask for such inputs.
\nThe standard way of inputting powers is as follows:
\n$a^b$ is input as a^b - and this is the only way to input powers.
But you have to be careful with inputting expressions such as $e^{2x}$ and $(xy)^2$. In these cases brackets should be used, as we now show:
\n| Power | Correct Input | Incorrect Input |
|---|---|---|
| $e^{2x}$ | \ne^(2*x) | \ne^2*x (system thinks this is $e^2 \\times x$) | \n
| $(xy)^2$ | \n(x*y)^2 | \nx*y^2 (system thinks this is $x \\times y^2$) | \n
So make sure that you use brackets to properly define your powers. This is a major source of input inaccuracies.
", "tags": ["brackets", "checked2015", "input", "introduction", "Numbas", "numbas", "powers"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Information on inputting powers
"}, "advice": ""}, {"name": "How to enter functions - Getting Started", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "d", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..9)", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "c", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d"], "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "Input:
\nInput:
\nInput:
\nFUNCTIONS
\nsin(x) not sinx, ln(a) not lna.abs(a).Here are some examples for you to try:
\n(If you want help, press Reveal Answers to see correct inputs in the Advice section.)
", "tags": ["arctan", "brackets", "checked2015", "functions", "input", "introduction", "Numbas", "numbas", "standard functions"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Dealing with functions in Numbas.
"}, "advice": "Correct inputs for these questions are as follows, although there may be other correct ways of inputting these:
\nsin(cos({a}x)+{b})cos(sin({a}x + {b}))abs((x + {c}) / (x + {d}))ln(abs((x + {a}) / (x + {d}))){a}t^({-b})*e^({-c}t)*sin({b}t) + (t + {d}t ^ 3)*e ^ ({c}t)arctan(({c}y ^ 2 + {d}) / ((y + {a})*(y + {b})))Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "\nRearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now solve this linear equation.
\\[\\simplify{{s} / ({a} * x + {b}) = {t} / ({c} * x + {d})}\\]
\n$x=\\;$ [[0]]
\nIf you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "stepsPenalty": 1}], "statement": "\nSolve the following equation for $x$.
\nInput your answer as a fraction or an integer as appropriate and not as a decimal.
\n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "subject of an equation"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{s}{ax+b} = \\frac{t}{cx+d}$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Rearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now this is a linear equation which is solved in the following steps: \\[\\simplify{{s*c-t*a}*x={t*b-s*d}}\\] and then \\[\\simplify{x={t*b-s*d}/{s*c-t*a}}.\\]
Factorise three quadratic equations of the form $x^2+bx+c$.
\nThe first has two negative roots, the second has one negative and one positive, and the third is the difference of two squares.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Factorise the following quadratic equations.
\n", "variables": {"v1": {"name": "v1", "group": "Part A ", "definition": "random(1..10)", "templateType": "anything", "description": ""}, "v2": {"name": "v2", "group": "Part A ", "definition": "random(2..6 except v1)", "templateType": "anything", "description": ""}, "v4": {"name": "v4", "group": "Part A ", "definition": "random(1..10 except -v3)", "templateType": "anything", "description": ""}, "v5": {"name": "v5", "group": "Part A ", "definition": "random(2..10)", "templateType": "anything", "description": ""}, "v3": {"name": "v3", "group": "Part A ", "definition": "random(-8..-1)", "templateType": "anything", "description": ""}, "v6": {"name": "v6", "group": "Part A ", "definition": "-v5", "templateType": "anything", "description": ""}}, "tags": ["Factorisation", "factorisation", "factorising quadratic equations", "Factorising quadratic equations", "taxonomy"], "ungrouped_variables": [], "functions": {}, "preamble": {"js": "question.is_factorised = function(part,penalty) {\n penalty = penalty || 0;\n if(part.credit>0) {\n // Parse the student's answer as a syntax tree\n var studentTree = Numbas.jme.compile(part.studentAnswer,Numbas.jme.builtinScope);\n\n // Create the pattern to match against \n // we just want two sets of brackets, each containing two terms\n // or one of the brackets might not have a constant term\n // or for repeated roots, you might write (x+a)^2\n var rule = Numbas.jme.compile('m_all(m_any(x,x+m_pm(m_number),x^m_number,(x+m_pm(m_number))^m_number))*m_nothing');\n\n // Check the student's answer matches the pattern. \n var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n // If not, take away marks\n if(!m) {\n part.multCredit(penalty,'Your answer is not fully factorised.');\n }\n }\n}", "css": ""}, "advice": "Quadratic equations of the form
\n\\[x^2+bx+c=0\\]
\ncan be factorised to create an equation of the form
\n\\[(x+m)(x+n)=0\\text{.}\\]
\nWhen we expand a factorised quadratic expression we obtain
\n\\[(x+m)(x+n)=x^2+(m+n)x+(m \\times n)\\text{.}\\]
\nTo factorise an equation of the form $x^2+bx+c$, we need to find two numbers which add together to make $b$, and multiply together to make $c$.
\n\\[\\simplify{x^2+{v1+v2}x+{v1*v2}=0}\\]
\nWe need to find two values that add together to make $\\var{v1+v2}$ and multiply together to make $\\var{v1*v2}$.
\n\\[\\begin{align}
\\var{v1} \\times \\var{v2}&=\\var{v1*v2}\\\\
\\var{v1}+\\var{v2}&=\\var{v1+v2}\\\\
\\end{align} \\]
So the factorised form of the equation is
\n\\[\\simplify{(x+{v1})(x+{v2})}=0\\text{.}\\]
\n\nWe can begin factorising by finding factors of $\\var{v3*v4}$ that add together to give $\\var{v3+v4}$.
\n\\[\\begin{align}
\\var{v3} \\times \\var{v4}&=\\var{v3*v4}\\\\
\\var{v3}+\\var{v4}&=\\var{v3+v4}\\\\
\\end{align} \\]
So the factorised form of the equation is
\n\\[\\simplify{(x+{v3})(x+{v4})}=0\\text{.}\\]
\nWhen factorising the quadratic expression
\n\\[\\simplify{x^2+{v5*v6}=0}\\]
\nwe need to find two values that add together to make $0$ and multiply together to make $\\var{v5*v6}$.
\n\\begin{align}
\\var{v5} \\times \\var{v6}& = \\var{v5*v6}\\\\
\\simplify[]{ {v5} + {v6}} &= 0 \\\\
\\end{align}
So the factorised form of the equation is
\n\\[\\simplify{(x+{v5})(x+{v6})}=0\\text{.}\\]
", "type": "question", "variable_groups": [{"variables": ["v1", "v2", "v3", "v4", "v5", "v6"], "name": "Part A "}], "parts": [{"scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"mustmatchpattern": {"pattern": "(`+-x^$n`? + `+- $n)`* * $z", "message": "Your answer is not fully factorised.", "partialCredit": 0, "nameToCompare": ""}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "checkingAccuracy": 0.001, "scripts": {}, "failureRate": 1, "checkVariableNames": false, "marks": 1, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "useCustomName": false, "showPreview": true, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "answer": "(x+{v1})(x+{v2})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "prompt": "$\\simplify{x^2+{v1+v2}x+{v1*v2}=0}$
\n[[0]] $=0$
\n", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "showCorrectAnswer": true, "gaps": [{"mustmatchpattern": {"pattern": "(`+-x^$n`? + `+- $n)`* * $z", "message": "Your answer is not fully factorised.", "partialCredit": 0, "nameToCompare": ""}, "variableReplacementStrategy": "originalfirst", "unitTests": [], "checkingAccuracy": 0.001, "scripts": {}, "failureRate": 1, "checkVariableNames": false, "marks": 1, "valuegenerators": [{"name": "x", "value": ""}], "variableReplacements": [], "useCustomName": false, "showPreview": true, "customName": "", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "answer": "(x+{v3})(x+{v4})", "customMarkingAlgorithm": "", "showFeedbackIcon": true, "vsetRangePoints": 5}], "type": "gapfill", "marks": 0, "customMarkingAlgorithm": "", "prompt": "
$\\simplify{x^2+{v3+v4}x+{v3*v4}}=0$
\n[[0]] $=0$
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$\\simplify{x^2+{v5*v6}}=0$
\n[[0]] $=0$
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\n\\[\\log_a(x^y)=y\\log_a(x)\\text{.}\\]
\nThis can also be useful for removing integers from the front of logarithms.
", "advice": "i)
\nWe need to use the rule
\n\\[k\\log_a(x)=\\log_a(x^k)\\text{.}\\]
\nSubsituting in our values for $x$ and $k$ gives
\n\\[\\var{x1[3]}\\log_a(\\var{z1[0]})=\\log_a(\\var{z1[0]^x1[3]})\\text{.}\\]
\nii)
\nWe need to use the rule
\n\\[k\\log_a(x)=\\log_a(x^k)\\text{.}\\]
\nSubsituting in our values for $x$ and $k$ gives
\n\\[\\var{x1[1]}\\log_a(\\var{z1[1]})=\\log_a(\\var{z1[1]^x1[1]})\\text{.}\\]
\ni)
\nThe rule for indices in logarithms also works the other way around,
\n\\[\\log_a(x^k)=k\\log_a(x)\\text{.}\\]
\nWe can use this to rearrange our expression by substituting in values for $x$ and $k$.
\n\\[\\begin{align}
\\log_a(\\var{x1[3]^z1[5]})&=k\\log_a(\\var{x1[3]})\\\\
\\var{x1[3]^z1[5]}&=\\var{x1[3]}^k\\\\
\\var{x1[3]^z1[5]}&=\\var{x1[3]}^\\var{z1[5]}\\\\
k&=\\var{z1[5]}\\\\
\\log_a(\\var{x1[3]^z1[5]})&=\\var{z1[5]}\\log_a(\\var{x1[3]})
\\end{align}\\]
ii)
\nAs with i) we can use the rule
\n\\[\\log_a(x^k)=k\\log_a(x)\\text{.}\\]
\nWe can use this to rearrange our expression by substituting in values for $x$ and $k$.
\n\\[\\begin{align}
\\log_a(\\var{x1[5]^z1[6]})&=k\\log_a(\\var{x1[5]})\\\\
\\var{x1[5]^z1[6]}&=\\var{x1[5]}^k\\\\
\\var{x1[5]^z1[6]}&=\\var{x1[5]}^\\var{z1[6]}\\\\
k&=\\var{z1[6]}\\\\
\\log_a(\\var{x1[5]^z1[6]})&=\\var{z1[6]}\\log_a(\\var{x1[5]})
\\end{align}\\]
i)
\nFrom the structure of this question we can tell that the answer can be written in the form $k\\log_a(\\var{x1[3]})$, meaning all of the values in the expression
\n\\[\\log_a(\\var{x1[3]^z1[2]})+\\log_a(\\var{x1[3]})\\]
\ncan be written in the form $k\\log_a(\\var{x1[3]})$.
\nIf we look at each log individually we can make sure they all take this form.
\n\\[\\begin{align}
\\log_a(\\var{x1[3]^z1[2]})&=k\\log_a(\\var{x1[3]})\\\\
\\var{x1[3]^z1[2]}&=\\var{x1[3]}^k\\\\
\\var{x1[3]^z1[2]}&=\\var{x1[3]}^\\var{z1[2]}\\\\
k&=\\var{z1[2]}\\\\
\\log_a(\\var{x1[3]^z1[2]})&=\\var{z1[2]}\\log_a(\\var{x1[3]})
\\end{align}\\]
We can now write our expression as
\n\\[\\begin{align}
\\log_a(\\var{x1[3]^z1[2]})+\\log_a(\\var{x1[3]})&=\\var{z1[2]}\\log_a(\\var{x1[3]})+\\log_a(\\var{x1[3]})\\\\
&=\\var{z1[2]+1}\\log_a(\\var{x1[3]})\\text{.}
\\end{align}\\]
ii)
\nFrom this question we know our answer is written in the form $k\\log_a(\\var{x1[4]})$, meaning all of the values in the expression
\n\\[\\log_a(\\var{x1[4]^z1[1]})+\\log_a(\\var{x1[4]^z1[0]})\\]
\ncan be written in the form $k\\log_a(\\var{x1[4]})$.
\nIf we look at each log individually we can make sure they all take this form.
\n\\[\\begin{align}
\\log_a(\\var{x1[4]^z1[1]})&=k\\log_a(\\var{x1[4]})\\\\
\\var{x1[4]^z1[1]}&=\\var{x1[4]}^k\\\\
\\var{x1[4]^z1[1]}&=\\var{x1[4]}^\\var{z1[1]}\\\\
k&=\\var{z1[1]}\\\\
\\log_a(\\var{x1[4]^z1[1]})&=\\var{z1[1]}\\log_a(\\var{x1[4]})
\\end{align}\\]
\\[\\begin{align}
\\log_a(\\var{x1[4]^z1[0]})&=k\\log_a(\\var{x1[4]})\\\\
\\var{x1[4]^z1[0]}&=\\var{x1[4]}^k\\\\
\\var{x1[4]^z1[0]}&=\\var{x1[4]}^\\var{z1[0]}\\\\
k&=\\var{z1[0]}\\\\
\\log_a(\\var{x1[4]^z1[0]})&=\\var{z1[0]}\\log_a(\\var{x1[4]})
\\end{align}\\]
We can now write our expression as
\n\\[\\begin{align}
\\log_a(\\var{x1[4]^z1[1]})+\\log_a(\\var{x1[4]^z1[0]})&=\\var{z1[1]}\\log_a(\\var{x1[4]})+\\var{z1[0]}\\log_a(\\var{x1[4]})\\\\
&=\\var{z1[1]+z1[0]}\\log_a(\\var{x1[4]})\\text{.}
\\end{align}\\]
iii)
\nFrom this question we know our answer is written in the form $k\\log_a(\\var{x1[5]})$, meaning all of the values in the expression
\n\\[\\log_a(\\var{x1[5]^z1[1]})+\\log_a(\\var{x1[5]^z1[2]})-\\log_a(\\var{x1[5]^z1[4]})\\]
\ncan be written in the form $k\\log_a(\\var{x1[5]})$.
\nIf we look at each log individually we can make sure they all take this form.
\n\\[\\begin{align}
\\log_a(\\var{x1[5]^z1[1]})&=k\\log_a(\\var{x1[5]})\\\\
\\var{x1[5]^z1[1]}&=\\var{x1[5]}^k\\\\
\\var{x1[5]^z1[1]}&=\\var{x1[5]}^\\var{z1[1]}\\\\
k&=\\var{z1[1]}\\\\
\\log_a(\\var{x1[5]^z1[1]})&=\\var{z1[1]}\\log_a(\\var{x1[5]})
\\end{align}\\]
\\[\\begin{align}
\\log_a(\\var{x1[5]^z1[2]})&=k\\log_a(\\var{x1[5]})\\\\
\\var{x1[5]^z1[2]}&=\\var{x1[5]}^k\\\\
\\var{x1[5]^z1[2]}&=\\var{x1[5]}^\\var{z1[2]}\\\\
k&=\\var{z1[2]}\\\\
\\log_a(\\var{x1[5]^z1[2]})&=\\var{z1[2]}\\log_a(\\var{x1[5]})
\\end{align}\\]
\\[\\begin{align}
\\log_a(\\var{x1[5]^z1[4]})&=k\\log_a(\\var{x1[5]})\\\\
\\var{x1[5]^z1[4]}&=\\var{x1[5]}^k\\\\
\\var{x1[5]^z1[4]}&=\\var{x1[5]}^\\var{z1[4]}\\\\
k&=\\var{z1[4]}\\\\
\\log_a(\\var{x1[5]^z1[4]})&=\\var{z1[4]}\\log_a(\\var{x1[5]})
\\end{align}\\]
We can now write our expression as
\n\\[\\begin{align}
\\log_a(\\var{x1[5]^z1[1]})+\\log_a(\\var{x1[5]^z1[2]})-\\log_a(\\var{x1[5]^z1[4]})&=\\var{z1[1]}\\log_a(\\var{x1[5]})+\\var{z1[0]}\\log_a(\\var{x1[5]})-\\var{z1[4]}\\log_a(\\var{x1[5]})\\\\
&=\\var{z1[1]+z1[2]-z1[4]}\\log_a(\\var{x1[5]})\\text{.}
\\end{align}\\]
Simplify the following expressions.
\ni)
\n$\\var{z1[0]}\\log_a(\\var{x1[3]})=\\log_a($ [[0]]$)$
\nii)
\n$\\var{z1[1]}\\log_a(\\var{x1[1]})=\\log_a($ [[1]]$)$
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\ni)
\n$\\log_a(\\var{x1[3]^z1[5]})=$ [[0]] $\\log_a(\\var{x1[3]})$
\nii)
\n$\\log_a(\\var{x1[5]^z1[6]})=$ [[1]] $\\log_a(\\var{x1[5]})$
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\n$\\log_a(\\var{x1[3]^z1[2]})+\\log_a(\\var{x1[3]})=$ [[0]]$\\log_a(\\var{x1[3]})$
\nii)
\n$\\log_a(\\var{x1[4]^z1[1]})+\\log_a(\\var{x1[4]^z1[0]})=$ [[1]]$\\log_a(\\var{x1[4]})$
\niii)
\n$\\log_a(\\var{x1[5]^z1[1]})+\\log_a(\\var{x1[5]^z1[2]})-\\log_a(\\var{x1[5]^z1[4]})=$ [[2]]$\\log_a(\\var{x1[5]})$
", "marks": 0}], "ungrouped_variables": ["x1", "y1", "z1", "b1", "c", "b4", "b", "b2"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Use the rule $\\log_a(n^b) = b\\log_a(n)$ to rearrange some expressions.
"}, "preamble": {"css": "", "js": ""}, "functions": {}}, {"name": "Using the Logarithm Equivalence $\\log_ba=c \\Longleftrightarrow a=b^c$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}], "type": "question", "statement": "Changing the subject of an equation involving logarithms often requires the use of the equivalence
\n\\[\\log_ba=c \\Longleftrightarrow a=b^c\\text{.}\\]
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\n$\\log_\\var{f}(x)=\\var{f1}$
\n$x=$ [[0]]
", "type": "gapfill"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"checkingtype": "absdiff", "scripts": {}, "showpreview": true, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "expectedvariablenames": [], "marks": 1, "variableReplacements": [], "answer": "{g1}^(y+{g2})", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "Make $x$ the subject of the following equation.
\n$\\log_\\var{g1}(x)=y+\\var{g2}$
\n$x=$ [[0]]
Make $x$ the subject of the equation, leaving your answer in the form $a^{\\frac{1}{b}}$.
\n$\\log_x(y+\\var{h1})=\\var{h2}$
\n$x=$ [[0]]
", "type": "gapfill"}, {"maxAnswers": 0, "minMarks": 0, "distractors": ["", "", "", "", "", ""], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "choices": ["$\\log_a(a^x)$
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", "$e^{\\ln(x)}$
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", "$\\log_e(x)$
", "$\\ln(e^x)$
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"}, "preamble": {"css": "", "js": ""}, "advice": "i)
\nWe can rearrange logarithms using indices.
\n\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]
\nUsing this equivalence we can rewrite $\\log_\\var{f}x=\\var{f1}$.
\n\\[\\begin{align}
x&= \\var{f}^\\var{f1} \\\\
&=\\var{f^f1}
\\end{align}\\]
\n
i)
\nWe can use the equivalence to rewrite our equation.
\n\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]
\nWe can write out our values to makes it easier.
\n\\[\\begin{align}
a&=x \\\\
b&=\\var{g1}\\\\
c&=y+\\var{g2}
\\end{align}\\]
Then we can write out our equation in the required form.
\n\\[x=\\var{g1}^{y+\\var{g2}}\\]
\n\n
We can use the same equivalence as in part b).
\n\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]
\nWe have
\n\\begin{align}
a&=y+\\var{h1} \\\\
b&=x\\\\
c&=\\var{h2}\\text{.} \\\\ \\\\
\\log_{x}(y+\\var{h1}) &= \\var{h2} \\\\
\\implies y+\\var{h1} &= x^{\\var{h2}} \\\\
x &= (y+\\var{h1})^{\\frac{1}{\\var{h2}}}
\\end{align}
The two in this list that don't equal $x$ are $\\log_e(x)$ and $\\log_{10}(x)$.
\n\\[\\begin{align}
\\log_e(x)&=\\ln(x)\\\\
\\log_{10}(x)&=\\log(x)\\text{.}
\\end{align}\\]
This question tests the student's ability to solve simple linear equations by elimination. Part a) involves only having to manipulate one equation in order to solve, and part b) involves having to manipulate both equations in order to solve.
", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "type": "question", "ungrouped_variables": [], "advice": "\\begin{align}
\\var{h}x+\\var{k}y&=\\var{m}\\text{,}\\\\
\\var{j}x-\\var{l}y&=\\var{n}\\text{.}\\\\
\\end{align}
To find the solution to these equations, we need to cancel one of the unknowns.
\nNotice that $\\var{h}x$ in the first equation can be multiplied by $\\var{j/h}$ to match $\\var{j}x$ in the second equation. This means that we will only have to manipulate the first equation and can leave the second equation as it is.
\nWe have to multiply the entire first equation by $\\var{j/h}$, not just the $x$ term to ensure the equation still holds.
\n$\\var{h}x+\\var{k}y=\\var{m}$ multiplied by $\\var{j/h}$ gives $\\var{j}x+\\var{k*(j/h)}y=\\var{m*(j/h)}.$
\nWe now have a common $x$ term and we can cancel this by subtracting one equation from the other to find the $y$ term.
\n\\begin{align}
&&\\var{j}x+\\var{k*{j/h}}y&=\\var{m*(j/h)}\\\\
-&&\\var{j}x-\\var{l}y&=\\var{n}\\\\
&&\\overline{\\qquad} & \\overline{\\qquad}\\\\
&&0x+\\var{k*(j/h)+l}y&=\\var{m*(j/h)-n}\\\\[1em]
&&y&=\\frac{\\var{m*j/h-n}}{\\var{k*j/h+l}}\\\\
&&y&=\\var{y1}
\\end{align}
We can find the corresponding value of $x$ by substituting this value for $y$ back into either of the original equations.
\n\\begin{align}
\\var{h}x+(\\var{k}\\times\\var{y1})&=\\var{m}\\text{,}\\\\
\\var{h}x+\\var{k*y1}&=\\var{m}\\text{,}\\\\
\\var{h}x&=\\var{m-(k*y1)}\\text{,}\\\\
x&=\\var{x1}\\text{.}\\\\
\\end{align}
Therefore, $x=\\var{x1}$ and $y=\\var{y1}$.
\n\\begin{align}
\\var{a}x+\\var{b}y&=\\var{c}\\text{,}\\\\
\\var{d}x+\\var{f}y&=\\var{g}\\text{.}\\\\
\\end{align}
To be able to solve the equations, we need to cancel one of the unknowns by manipulating the two equations so that the variable we wish to cancel is of the same value in each equation.
\nAlthough we can choose to cancel either variable, $x$ or $y$, a good rule of thumb is to look at the lowest common multiples of the coefficients for each variable and cancel the variable with the lowest LCM.
\nThe LCM of the coefficients of the $x$ terms is $\\var{lcm(a,d)}$.
\nThe LCM of the coefficients of the $y$ terms is $\\var{lcm(b,f)}$.
\nTherefore, we will choose to cancel the $x$ terms.
\nWe need to multiply the equations individually to achieve the lowest common multiple identified.
\n\\begin{align}
\\simplify{ {a}x + {b}y } &= \\var{c} &\\text{multiply by } \\var{lcm(a,d)/a} \\text { to obtain } && \\simplify{ {lcm(a,d)}x + {b*lcm(a,d)/a}y} &= \\var{c*lcm(a,d)/a} \\\\
\\simplify{ {d}x + {f}y } &= \\var{g} &\\text{multiply by } \\var{lcm(a,d)/d} \\text { to obtain } && \\simplify{ {lcm(a,d)}x + {b*lcm(a,d)/d}y} &= \\var{c*lcm(a,d)/d}
\\end{align}
We now have a common $x$ term, and can cancel this by subtracting one equation from the other.
\n\\begin{align}
&& \\simplify{ {lcm(a,d)}x+{b*lcm(a,d)/a}y } = \\var{c*lcm(a,d)/a} \\\\
- && \\simplify{ {lcm(a,d)}x + {f*lcm(a,d)/d}y } = \\var{g*lcm(a,d)/d} \\\\
&& \\overline{\\simplify[]{ 0x+{b*lcm(a,d)/a-f*lcm(a,d)/d}y} = \\var{c*lcm(a,d)/a-g*lcm(a,d)/d}}
\\end{align}
\\begin{align}
\\var{(b*lcm(a,d)/a)-(f*lcm(a,d)/d)}y &= \\var{(c*lcm(a,d)/a)-(g*lcm(a,d)/d)}\\text{,}\\\\
y &= \\var{y2}\\text{.}
\\end{align}
We can find the corresponding value of $x$ by substituting thsi value of $y$ value back into either of the original equations.
\n\\begin{align}
\\simplify[]{ {a}x + {b}{y2}} &= \\var{c} \\\\
\\simplify[]{ {a}x + {b*y2}} &= \\var{c} \\\\
\\var{a}x&=\\var{c-b*y2} \\\\
x &= \\var{x2} \\text{.}
\\end{align}
Therefore, $x=\\var{x2}$ and $y=\\var{y2}$.
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\n\\begin{align}
\\simplify{{h}x+{k}y} &= \\var{m} \\text{,} \\\\
\\simplify{{j}x+{l}y} &= \\var{n} \\text{.}
\\end{align}
$x =$ [[0]]
\n$y =$ [[1]]
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\n\\begin{align}
\\simplify{{a}x + {b}y} &= \\var{c} \\text{,} \\\\
\\simplify{{d}x + {f}y} &= \\var{g} \\text{.}
\\end{align}
$x =$ [[0]]
\n$y =$ [[1]]
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", "group": "Part b", "definition": "random(a+1..9 except map(j*a,j,0..10/a))", "name": "d", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": "lcm(a,d)The quadratic formula is
\n\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]
\nFrom the equation, we can read off values for $a$, $b$ and $c$:
\n\\[\\begin{align}
a&=1\\text{,}\\\\
b&=\\var{a+m}\\text{,}\\\\
c&=\\var{a*m} \\text{.}
\\end{align}\\]
Substituting these values into the quadratic formula,
\n\\[x = \\frac {-\\var{a+m}\\pm\\sqrt{\\var{a+m}^2-4\\times \\var{a*m}}}{2}\\text{.}\\]
\nNote the $\\pm$ symbol in the formula. This means there are two solutions: one using $+$, the other using $-$.
\nThe two solutions are
\n\\[\\begin{align}
x_1&=\\var{m}\\text{,}\\\\
x_2&=\\var{a}\\text{.}
\\end{align}\\]
Note that the right-hand side of the given equation is not zero. We need to rewrite it in the form $ax^2+bx+c=0$:
\n\\[\\begin{align}
\\simplify{{a1}x^2+{a2}x+{a3}}&=\\var{a4}\\\\
\\simplify{{a1}x^2+{a2}x+{a3-a4}}&=0\\text{.}
\\end{align}\\]
Then we can read off values for $a$, $b$ and $c$:
\n\\[\\begin{align}
a&=\\var{a1}\\\\
b&=\\var{a2}\\\\
c&=\\var{a3-a4} \\text{.}
\\end{align}\\]
We can now substitute these values into the quadratic formula:
\n\\[x = {\\frac {-\\var{a2}\\pm\\sqrt{\\var{a2}^2-4\\times \\var{a1}\\times \\var{a3-a4}}}{2\\times\\var{a1}}}\\text{.}\\]
\nSo the two solutions are
\n\\[\\begin{align}
x_1&=\\var{dpformat(x1,2)}\\\\
x_2&=\\var{dpformat(x2,2)}\\text{.}
\\end{align}\\]
We first rearrange our equation into the form $ax^2+bx+c=0$:
\n\\[\\begin{align}
\\simplify{{b1}x^2+{b2}x+{b3}}&=0=\\var{b4}x\\\\
\\simplify{{b1}x^2+{b2-b4}x+{b3}}&=0\\text{.}
\\end{align}\\]
We can then read off the values for $a, b$ and $c$, which are
\n\\[\\begin{align}
a&=\\var{b1}\\text{,}\\\\
b&=\\var{b2-b4}\\text{,}\\\\
c&=\\var{b3}\\text{.}
\\end{align}\\]
Substituting these values into the quadratic formula,
\n\\[x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\\]
\nwe obtain solutions
\n\\[\\begin{align}
x_1&=\\var{dpformat(p1,2)}\\text{,}\\\\
x_2&=\\var{dpformat(p2,2)}\\text{.}
\\end{align}\\]
When quadratic equations can't be factorised, or if equations are difficult to factorise (perhaps if the coefficients are large), we need to use the quadratic formula to solve the equations.
\nUse the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1<x_2$.
", "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "prompt": "$\\simplify{x^2+{a+m}x+{a*m}=0}$
\n$x_1=$ [[0]]
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An equation of the form
\n\\[ax^2+bx+c=0\\text{,}\\]
\n\ncan be solved using the quadratic formula
\n\\[x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\\]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Answer the following questions by substituting the correct values into the given equations.
", "advice": "When inserting numbers into your calculator, make sure that you place brackets correctly.
\nWe can see from the diagram that the radius of the frisbee is $\\var{mccall[2]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the area of a circle with $\\var{mccall[2]}$ gives,
\\begin{align}
\\mathrm{Area} &= \\pi r^2 \\\\
&= \\pi\\times(\\var{mccall[2]})^2 \\\\
&= \\var{dpformat((mccall[2])^2, 2)}\\pi\\, \\mathrm{cm}^2 \\\\
&= \\var{dpformat(pi (mccall[2])^2, 2)}\\, \\mathrm{cm}^2 \\quad \\text{to 2 d.p.}
\\end{align}
We can see from the diagram that the radius of the cone is $\\var{r}$ $\\mathrm{cm}$ and the height is $\\var{h}$ $\\mathrm{cm}$.
Replacing the letters $r$ and $h$ in the formula for the volume of a cone with $\\var{r}$ $\\mathrm{cm}$ and $\\var{h}$ $\\mathrm{cm}$ respectively gives,
\\begin{align}
\\mathrm{Volume} &= \\frac{h}{3} \\pi r^2 \\\\
&= \\frac{(\\var{h})}{3} \\times \\pi \\times (\\var{r})^2 \\\\
&= \\var{dpformat((pi)*(h/3)*(r)^2 , 5)}\\, \\mathrm{cm}^3 \\\\
&=\\var{dpformat(h/3 * pi * (r)^2, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\n
We can see from the diagram that the radius of the tennis ball is $\\var{mccall[1]}$ $\\mathrm{cm}$.
Replacing the letter $r$ in the formula for the volume of a sphere with $\\var{mccall[1]}$ gives,
\\begin{align}
\\mathrm{Volume} &= \\frac{4}{3} \\pi r^3 \\\\
&= \\frac{(4)}{(3)} \\times \\pi \\times (\\var{mccall[1]})^3 \\\\
&= \\var{dpformat((4/3)*pi*mccall[1]^3, 5)}\\, \\mathrm{cm}^3 \\\\
&= \\var{precround(((4/3)* pi) *(mccall[1])^3, 1)}\\, \\mathrm{cm}^3 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
We can see from the diagram that the trapezium has two parallel sides with length $\\var{trap_length_a}$ $\\mathrm{cm}$, $\\var{trap_length_b}$ $\\mathrm{cm}$ and height $\\var{trap_h}$ $\\mathrm{cm}$.
Replacing the letters $a$, $b$ and $h$ in the formula for the area of a trapezium with $\\var{trap_length_a}$, $\\var{trap_length_b}$ and $\\var{trap_h}$ respectively gives,
\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
\\begin{align}
\\mathrm{Area} &= \\frac{1}{2} (a + b) h \\\\
&= \\frac{1}{2} \\times (\\var{trap_length_a} + \\var{trap_length_b}) \\times \\var{trap_h} \\\\
&= \\var{precround((0.5)(trap_length_a +trap_length_b) trap_h, 2)}\\, \\mathrm{cm}^2 \\\\
&= \\var{precround((0.5) (trap_length_a + trap_length_b) trap_h, 1)}\\, \\mathrm{cm}^2 \\quad \\text{to 1 d.p.} \\\\
\\end{align}
Rounded value for the length of c.
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", "templateType": "anything"}, "b": {"name": "b", "group": "Triangle variables", "definition": "vector(-3,0)", "description": "Position of the point A in Geogebra. This point is fixed so the triangle doesn't hang in one corner or the whole page.
", "templateType": "anything"}, "n": {"name": "n", "group": "Name variables", "definition": "random(0..4#1)", "description": "n is a random number between 0 and 4 that picks a name from {name} and then picks the next in the list for the other name such that there is always a male and a female in the question.
", "templateType": "anything"}, "name2": {"name": "name2", "group": "Name variables", "definition": "[\"Andrew\",\"Susan\",\"Tom\",\"Geraldine\",\"Joshua\",\"Chantel\"]", "description": "List of names to randomise. Can change to any name inserted
", "templateType": "anything"}, "trap_rand": {"name": "trap_rand", "group": "Trapezium variables", "definition": "random(1..3#1)", "description": "A random number to define the height of the trapezium.
", "templateType": "anything"}, "c_theta": {"name": "c_theta", "group": "Triangle variables", "definition": "(180/pi)*arccos(((length_b)^2+(length_c)^2-(length_a)^2)/(2(length_b)(length_c))) ", "description": "Theta is randomised by the lengths
", "templateType": "anything"}, "trap_areadp1": {"name": "trap_areadp1", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h,1)", "description": "Calculates the area of the trapezium
", "templateType": "anything"}, "defs": {"name": "defs", "group": "Triangle variables", "definition": "[\n ['A',a],\n ['B',b],\n ['C',c]\n ]", "description": "Creates the points in Geogebra is not used directly in the question but to create the image in Geogebra.
", "templateType": "anything"}, "length_a": {"name": "length_a", "group": "Triangle variables", "definition": "sqrt((a[0]-b[0])^2+(a[1]-b[1])^2)", "description": "For triangle - The length of the vector AB
", "templateType": "anything"}, "trap_d": {"name": "trap_d", "group": "Trapezium variables", "definition": "vector(random(5..7#0.1), -4)", "description": "Creates the point D on the trapezium
", "templateType": "anything"}, "c": {"name": "c", "group": "Triangle variables", "definition": "vector(\n random(2..5#0.1),\n random(2..5#0.1)\n )", "description": "Triangle - A variable point which ultimately decides how the triangle looks.
", "templateType": "anything"}, "trap_a": {"name": "trap_a", "group": "Trapezium variables", "definition": "vector(1,-4)", "description": "Creates the point A on the trapezium
", "templateType": "anything"}, "trap_c": {"name": "trap_c", "group": "Trapezium variables", "definition": "vector(random(4..5.5#0.1), trap_rand)", "description": "Creates the point C on the trapezium
", "templateType": "anything"}, "mccall": {"name": "mccall", "group": "RNG", "definition": "[0,random(3.1..3.7#0.1),random(5..20#0.1)]\n", "description": "Matrix of random variables used to create length in the questions.
", "templateType": "anything"}, "x2": {"name": "x2", "group": "Quadratic variables", "definition": "random(1..10#1)", "description": "The x^2 coefficient
", "templateType": "anything"}, "trap_areadp2": {"name": "trap_areadp2", "group": "Trapezium variables", "definition": "precround(0.5*(trap_length_a + trap_length_b)*trap_h, 2)", "description": "", "templateType": "anything"}, "trap_b": {"name": "trap_b", "group": "Trapezium variables", "definition": "vector(random(1.5..2.5#0.1), trap_rand)", "description": "Creates the point B on the trapezium
", "templateType": "anything"}, "x1": {"name": "x1", "group": "Quadratic variables", "definition": "random(1..50)", "description": "The x coefficient
", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Trapezium variables", "variables": ["trap_a", "trap_b", "trap_c", "trap_d", "trap_h", "trap_rand", "trap_defs", "trap_length_a", "trap_length_b", "trap_e", "trap_areadp1", "trap_areadp2"]}, {"name": "Triangle variables", "variables": ["Triangle_area", "c", "b", "a", "c_theta", "length_a", "length_b", "length_c", "defs", "length_bdp2", "length_cdp2", "c_thetadp2"]}, {"name": "Name variables", "variables": ["name", "name2", "pronoun", "n"]}, {"name": "Quadratic variables", "variables": ["x2", "x1", "const"]}, {"name": "Cone variables", "variables": ["r", "h"]}, {"name": "RNG", "variables": ["mccall"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate the area of a frisbee, assuming that the frisbee can be modelled as a circle, given the formula for the area of a circle is
\n\\[\\mathrm{Area} = \\pi r^2.\\]
\n\n$\\mathrm{Area}$ = [[0]] $\\mathrm{cm}^2$ Round your answer to 2 decimal places.
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "pi*{mccall[2]}^2 - 0.05", "maxValue": "pi*{mccall[2]}^2 + 0.05", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate the volume of a cone given the formula for the volume of a cone is
\n\\[\\mathrm{Volume} = \\frac{h}{3} \\pi r^2.\\]
\n\n$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.
\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{(h/3)}*{r}^2*pi -0.5", "maxValue": "{(h/3)}*{r}^2*pi +0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "{name[n]} has a tennis ball and {pronoun} wants to find the volume of the ball. Using the diagram and the formula for the volume of a sphere, calculate the volume of the ball.
\n\\[\\mathrm{Volume}= \\frac{4}{3} \\pi r^3.\\]
\n\n$\\mathrm{Volume}$ = [[0]] $\\mathrm{cm}^3$ Round your answer to 1 decimal place.
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "4/3 * pi * {mccall[1]}^3 - 0.5", "maxValue": "4/3 * pi * {mccall[1]}^3 + 0.5", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Find the area of the trapezium given the formula for the area of a trapezium is
\n\\[\\mathrm{Area} = \\frac{1}{2}(a+b) h .\\]
\n{geogebra_applet('https://www.geogebra.org/m/Gtjzajb6',trap_defs)}
\nAll lengths are given in metres.
\n$\\mathrm{Area}$ = [[0]] $\\mathrm{m}^2$ Round your answer to 1 decimal place.
\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} - 0.1", "maxValue": "0.5({trap_length_a}+{trap_length_b})*{trap_h} + 0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Find the equation of a line through two points - positive gradient", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}], "tags": [], "metadata": {"description": "Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.
\nThe answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.
\nThis particular example has a positive gradient.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "In this question we will identify the equation of the straight line passing through points $A=(\\var{xa},\\var{ya})$ and $B=(\\var{xb},\\var{yb})$ in the form $y = mx + c$.
\n{plotPoints()}
", "advice": "We find the equation of a straight line passing through two points by finding the gradient and the $y$-intercept of the line.
\nWe can find the gradient ($m$) using the points $A = (x_1,y_1)=(\\var{xa},\\var{ya})$ and $B = (x_2,y_2)=(\\var{xb},\\var{yb})$.
\nAs the definition of gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).
The equation for gradient is,
\\begin{align}
m &= \\frac{y_2-y_1}{x_2-x_1} \\\\[0.5em]
&= \\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}} \\\\[0.5em]
&= \\frac{\\simplify[]{{yb}-{ya}}}{\\simplify{{xb}-{xa}}} \\\\[0.5em]
&= \\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}
\\end{align}
Rearranging the equation $y=mx+c$ and substituting either of the points gives
\n\\[c = y_1-mx_1 \\quad \\mathrm{or} \\quad c = y_2-mx_2 \\,\\text{.} \\]
\nWe can then also use this equation with the other point's coordinates to check our answer.
\nLet's use point $A$ first:
\n\\[
\\begin{align}
c &= y_1-mx_1 \\\\
&= \\var{ya}-\\var[fractionnumbers]{m}\\times\\var{xa} \\\\
& = \\simplify[fractionnumbers]{{ya-m*xa}}\\text{.}
\\end{align}
\\]
We then check this against point $B$:
\n\\[
\\begin{align}
y_2 &= mx_2 + c \\\\[0.5em]
&= \\simplify[fractionNumbers]{{m}{xb}+{c}} \\\\[0.5em]
&= \\var[fractionnumbers]{m*xb+c}\\text{.}
\\end{align}
\\]
We can now substitute these values for $m$ and $c$ into $y=mx+c$ to get:
\n\\[y=\\simplify[!noLeadingMinus,fractionNumbers,unitFactor]{{m} x+ {c}}\\text{.}\\]
\nThe green line drawn on the graph represents the above line equation.
\n{correctPoints()}
", "rulesets": {}, "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "(ya-yb)/(xa-xb)", "description": "", "templateType": "anything"}, "yb": {"name": "yb", "group": "Ungrouped variables", "definition": "ya+random([2,4])", "description": "", "templateType": "anything"}, "xa": {"name": "xa", "group": "Ungrouped variables", "definition": "random(-4..-1)", "description": "", "templateType": "anything"}, "ya": {"name": "ya", "group": "Ungrouped variables", "definition": "random(-4..2)", "description": "", "templateType": "anything"}, "xb": {"name": "xb", "group": "Ungrouped variables", "definition": "xa+random([2,4] except -xa)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "ya-m*xa", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "\n", "maxRuns": 100}, "ungrouped_variables": ["xa", "xb", "ya", "yb", "m", "c"], "variable_groups": [], "functions": {"correctPoints": {"parameters": [], "type": "html", "language": "javascript", "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\n\n\nquestion.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\n//define ans as this \ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}, "plotPoints": {"parameters": [], "type": "html", "language": "javascript", "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n var ans;\n var nscope = new Numbas.jme.Scope([scope,{variables:{x:new Numbas.jme.types.TNum(0)}}]);\n//this is the beating heart of whatever plots the function,\n//I've changed this from being curve to functiongraph\n var line = board.create('functiongraph',[function(x){\nif(ans) {\n try {\nnscope.variables.x.value = x;\n var val = Numbas.jme.evaluate(ans,nscope).value;\n return val;\n }\n catch(e) {\nreturn 13;\n }\n}\nelse\n return 13;\n },-12,12]\n , {strokeColor:\"blue\",strokeWidth: 4} );\n \nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: false, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\nquestion.lines = {\n l:line, c:correct_line\n}\n\n question.signals.on('HTMLAttached',function() {\nko.computed(function(){\nvar expr = question.parts[2].gaps[0].display.studentAnswer();\n\n//define ans as this \ntry {\n ans = Numbas.jme.compile(expr,scope);\n}\ncatch(e) {\n ans = null;\n}\nline.updateCurve();\ncorrect_line.updateCurve();\nboard.update();\n});\n });\n\n\nreturn div;"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate the gradient, $m$, of the straight line between these two points.
\n$m=$ [[0]]
\n", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "m", "maxValue": "m", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Use this gradient and the coordinates of the points to calculate the $y$-intercept, $c$.
\n$c=$ [[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "c", "maxValue": "c", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Give the equation of the straight line through these points in the form $y=mx+c$.
\n$\\displaystyle y=$ [[0]]
\nUse the graph to plot your answer and check that it goes through these points.
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "m", "part": "p0", "must_go_first": true}, {"variable": "c", "part": "p1", "must_go_first": true}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{m}*x+{c}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "notallowed": {"strings": ["c", "m"], "showStrings": false, "partialCredit": 0, "message": "You must input your answer in the form y = mx +c where m and c are numbers.
"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Calculus", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", ""], "questions": [{"name": "Chain rule - binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["Calculus", "calculus", "chain rule", "checked2015", "derivative of a function of a function", "differentiation", "Differentiation", "function of a function", "Steps", "steps"], "metadata": {"description": "Differentiate $\\displaystyle (ax^m+b)^{n}$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Differentiate the following function $f(x)$ using the chain rule.
", "advice": "$\\simplify[std]{f(x) = ({a} * x^{m}+{b})^{n}}$
\nThe chain rule says that if $f(x)=g(h(x))$ then
\n\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]
\nOne way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.
\nThen we use the chain rule in the form:
\n\\[\\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\frac{\\mathrm{d}u}{\\mathrm{d}x}\\frac{\\mathrm{d}f(u)}{\\mathrm{d}u}\\]
\nOnce you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.
\nFor this example, we let $u=\\simplify[std]{{a} * x^{m}+{b}}$ and we have $f(u)=\\simplify[std]{u^{n}}$.
\nThis gives
\n\\begin{align}
\\frac{\\mathrm{d}u}{\\mathrm{d}x} &= \\simplify[std]{{m*a}x ^ {m -1}} \\\\[1em]
\\frac{\\mathrm{d}f(u)}{\\mathrm{d}u} &= \\simplify[std]{{n}u^{n-1}}
\\end{align}
Hence on substituting into the chain rule above we get:
\n\\begin{align}
\\frac{\\mathrm{d}f}{\\mathrm{d}x} &= \\simplify[std]{{m*a}x ^ {m-1} * ({n}*u^{n-1})} \\\\
&= \\simplify[std]{{m*a*n}x^{m-1}u^{n-1}} \\\\
&= \\simplify[std]{{m*a*n}x^{m-1}({a}*x^{m}+{b})^{n-1}}
\\end{align}
on replacing $u$ by $\\simplify[std]{{a}x^{m}+{b}}$.
", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(5..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\simplify[std]{f(x) = ({a} * x^{m}+{b})^{n}}\\]
\n$\\displaystyle \\frac{\\mathrm{d}f}{\\mathrm{d}x}=$ [[0]]
\nClick on Show steps for more information. You will not lose any marks by doing so.
", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The chain rule says that if $f(x)=g(h(x))$ then
\n\\[\\simplify[std]{f'(x) = h'(x)g'(h(x))}\\]
\nOne way to find $f'(x)$ is to let $u=h(x)$ then we have $f(u)=g(u)$ as a function of $u$.
\nThen we use the chain rule in the form:
\n\\[\\frac{\\mathrm{d}f}{\\mathrm{d}x} = \\frac{\\mathrm{d}u}{\\mathrm{d}x}\\frac{\\mathrm{d}f}{\\mathrm{d}u}\\]
\nOnce you have worked this out, you replace $u$ by $h(x)$ and your answer is now in terms of $x$.
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*m*n}x ^ {m-1} * ({a} * x^{m}+{b})^{n-1}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Chain rule - product of two functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}], "tags": [], "metadata": {"description": "The derivative of $\\displaystyle x ^ {m}(ax^2+b)^{n}$ is of the form $\\displaystyle x^{m-1}(ax^2+b)^{n-1}g(x)$. Find $g(x)$.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Differentiate the following function $f(x)$.
", "advice": "\n\t \n\t \n\tThe product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = ({a} * x^2+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {2*n*a}*x * ({a} * x^2+{b})^{n-1}}\\]
\n\t \n\t \n\t \n\tFor this last differentiation we used the chain rule.
\n\t \n\t \n\t \n\tHence on substituting into the product rule above we get:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*}\\frac{df}{dx} &=& \\simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+x^{m} *{2*n*a}*x* ({a} * x^2+{b})^{n-1}}\\\\\n\t \n\t &=& \\simplify[std]{{m}x ^ {m-1} * ({a} * x^2+{b})^{n}+{2*n*a}*x^{m+1}* ({a} * x^2+{b})^{n-1}}\\\\\n\t \n\t &=& \\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m}*({a}*x^2+{b})+{2*n*a}x^{2})} \\\\\n\t \n\t &=&\\simplify[std]{x ^ {m-1} * ({a} * x^2+{b})^{n-1}*({m*a+2*a*n}*x^2+{m*b})}\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t \n\tHence $\\simplify[std]{g(x)={m*a+2*a*n}*x^2+{m*b}}$
\n\t \n\t \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t$\\simplify[std]{f(x) = x ^ {m} * ({a} * x^2+{b})^{n}}$
The answer is in the form
\\[\\frac{df}{dx}=\\simplify[std]{x^{m-1}({a}x^2+{b})^{n-1}*g(x)}\\] for a polynomial $g(x)$.
You have to find $g(x)$.
\n\t\t\t$g(x)=\\;$[[0]]
\n\t\t\tClick on Show steps for more information. You will not lose any marks by doing so.
\n\t\t\t", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "You should use the the product rule and the chain rule for this example.
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{m*a+2*a*n}*x^2+{m*b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Differentiate product of trig function and binomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}], "tags": [], "metadata": {"description": "Differentiate $ (a+bx) ^ {m} \\sin(nx)$
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Differentiate the following function $f(x)$ using the product rule.
", "advice": "\n\t \n\t \n\tThe product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = ({a} + {b} * x) ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = sin({n} * x)} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n} * cos({n} * x)}\\]
\n\t \n\t \n\t \n\tHence on substituting into the product rule above we get:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{Diff(f,x,1) = {m*b}({a} + {b} * x) ^ {m-1} * sin({n} * x)+{n}*({a} + {b} * x) ^ {m} * cos({n} * x)}\\]
\n\t \n\t \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..5)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t$\\simplify[std]{f(x) = ({a} + {b} * x) ^ {m} * sin({n} * x)}$
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\n\t\t\tClicking on Show steps gives you more information, you will not lose any marks by doing so.
\n\t\t\t", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
Differentiate $x^m\\cos(ax+b)$
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Differentiate the following function $f(x)$ using the product rule.
", "advice": "\n\t \n\t \n\tThe product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
For this example:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]
\n\t \n\t \n\t \n\t\\[\\simplify[std]{v = cos({a} * x+{b})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = -{a} * sin({a} * x+{b})}\\]
\n\t \n\t \n\t \n\tHence on substituting into the product rule above we get:
\n\t \n\t \n\t \n\t\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * cos({a} * x+{b})-{a}x^{m} * sin({a} * x+{b})}\\]
\n\t \n\t \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t$\\simplify[std]{f(x) = x ^ {m} * cos({a} * x+{b})}$
\n\t\t\t$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\n\t\t\tClicking on Show steps gives you more information, you will not lose any marks by doing so.
\n\t\t\t", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
Students must find $\\int \\frac{1}{x-a} \\, dx$.
", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "Integrate the following,
", "advice": "Using the method of substitution with $u = \\simplify{x-{a}}$, we see that $du = dx$ and so \\[\\int \\simplify{1/(x-{a})} \\, dx = \\int \\frac{1}{u} \\, du = \\ln{u} + c = \\ln{(\\simplify{x-{a}})} + c.\\] The answer is therefore $\\ln{(\\simplify{x-{a}})}$
", "rulesets": {}, "variables": {"limLo": {"name": "limLo", "group": "Ungrouped variables", "definition": "a+random(1..5)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything"}, "limHi": {"name": "limHi", "group": "Ungrouped variables", "definition": "limLo+random(1..5)", "description": "", "templateType": "anything"}, "lims": {"name": "lims", "group": "Ungrouped variables", "definition": "[limLo,limHi]", "description": "", "templateType": "anything"}, "intLims": {"name": "intLims", "group": "Ungrouped variables", "definition": "map(\n ln(x-a),\n x,\n lims\n )", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "lims", "intLims", "limLo", "limHi"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\int \\simplify{1/(x-{a})} dx \\,\\] = [[0]] $+C$
", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln(x-{a})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}], "tags": [], "metadata": {"description": "Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx $ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\tFind the following indefinite integrals.
\n\tInput all numbers as fractions or integers and not decimals.
\n\tInput the constant of integration as $C$ where needed.
\n\t", "advice": "\n\ta)
\n\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \\int \\left( \\simplify[std]{(-{a}/{b})*cos({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \\end{eqnarray} \\]
b)
\n\tFor this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \\int \\left( \\simplify[std]{({a}/{b})*sin({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \\end{eqnarray} \\]
c)
\n\tUsing the results from Parts a and b, we have \\[\\begin{eqnarray*}I &=& \\int \\simplify[std]{{a1}x*sin({b}x+{c})} dx + \\int \\simplify[std]{{a2}x*cos({b}x+{c})} dx\\\\ &=& \\simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\\\ &=&\\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\\\ &=&\\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \\end{eqnarray*}\\]
Hence
$\\displaystyle \\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$
$\\displaystyle \\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$
\n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n\t\t\t", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Do not input numbers as decimals, only as integers without the decimal point, or fractions
"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\t", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
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$\\displaystyle I=\\int \\simplify[std]{{a1}x*sin({b}x+{c})+{a2}x*cos({b}x+{c})} dx $
You are given that \\[I=\\simplify[std]{f(x)*sin({b}x+{c})+g(x)*cos({b}x+{c})+C}\\]
where $f(x)$ and $g(x)$ are polynomials of degree 1. You have to find $f(x)$ and $g(x)$.
$f(x)=\\;$[[0]] $\\;\\;\\;\\;\\;g(x)=\\;$[[1]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
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"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}], "tags": [], "metadata": {"description": "Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\nFind the following integral.
\nYou must input the constant of integration as $C$.
\nInput all numbers as integers or fractions.
\nYou can click on Show steps to get a hint. You will lose 1 mark if you do so.
\nNote that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+C$ and you must include the absolute value in the argument of $\\ln$. You input $|x|$ as abs(x).
\n ", "advice": "\nThis exercise is best solved by using substitution.
\nNote that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\nSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\nHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\nHence the integral becomes:
\n\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
\nThe using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]
\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n$I=\\;$[[0]]
\nInput all numbers as integers or fractions.
\nDo not forget to include the constant of integration $C$.
\n ", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\nTry the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
\nNote that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$
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\nThis is an example of online, formative assessment. You will be presented with a set of mathematical exercises to work through. If you are unsure how to complete any you can click for feedback or look at the answer. Then refresh the question and have another go. We will not record any details of your attempts - this is purely for you to practice ahead of starting Year 1.
\nYou can try the whole test or any question as many times as you like, in any order.
\nThe first few questions introduce you to the way these online tests work - how to enter your answers - followed by some questions on algebra, geometry and then calculus. You might find some rather easy, but hopefully there are some that require serious thought and some working out with pen and paper. And you might need more than one attempt for some. That's ok - it's all good practice!
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