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Try these questions as a little refresher on what you did in first year. These are the type of thing you should know going into second year. If you find any questions tricky then Maths Cafe is a great place to go and get a little support.
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", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variable_groups": [], "ungrouped_variables": ["ba21", "a21", "a22", "ba22", "cb21", "b22", "b21", "cb22", "ac22", "ac21", "ab22", "ab21", "b12", "b11", "c12", "c11", "c22", "a11", "cb11", "cb12", "a12", "c21", "ba11", "ba12", "ab12", "ab11", "ac12", "ac11", "a13", "a23", "b31", "b32", "ba13", "ba23", "ba31", "ba32", "ba33"], "statement": "Do the following matrix problems :
Let
\\[A=\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix},\\;\\; B=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix},\\;\\; \\]
Calculate the following products of these matrices:
$AB = \\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\var{b31}&\\var{b32}\\end{pmatrix} = $ [[0]]
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\n\\begin{align}
\\simplify{{a}x+{b}y}&=\\var{c}\\text{,}\\\\
\\simplify{{a1}x+{b1}y}&=\\var{c1}\\text{.}
\\end{align}
Solve to find the values of $x$ and $y$.
\n$x=$ [[0]]
\n$y=$ [[1]]
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\n\\begin{align}
\\simplify{{a*b1}x+{b*b1}y} &= \\var{c*b1} \\\\
\\simplify{{a1*b}x+{b1*b}y} &= \\var{c1*b}
\\end{align}
Next, subtract the second equation from the first to get
\n\\[ \\simplify[std]{{a*b1-a1*b}x} = \\var{c*b1-c1*b} \\]
\nSo $x = \\displaystyle \\simplify[std]{{(c*b1-c1*b)/(a*b1-a1*b)}}$.
\nSubstitute this value of $x$ into the first equation and rearrange to obtain $y$:
\n\\begin{align}
\\simplify[std]{{a}*{(c*b1-c1*b)/(a*b1-a1*b)} + {b}y} &= \\var{c} \\\\
\\simplify[std]{{b}y} &= \\simplify[std]{{c}-{a*(c*b1-c1*b)/(a*b1-a1*b)}} \\\\
y &= \\simplify[std]{{(c-a*(c*b1-c1*b)/(a*b1-a1*b))/b}}
\\end{align}
Solve a system of three simultaneous linear equations
", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "statement": "Solve the following system of three simultaneous linear equations:
\n\\(\\var{a1}x+2y+4z=\\var{r1}\\)
\nand
\n\\(2x+\\var{b1}y+3z=\\var{r2}\\)
\nand
\n\\(5x+6y+\\var{c1}z=\\var{r3}\\)
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\n(ii) \\(2x+\\var{b1}y+3z=\\var{r2}\\)
\n(iii) \\(5x+6y+\\var{c1}z=\\var{r3}\\)
\nFirst reduce the three equations in three unknowns to a two equations in two unknowns problem by eliminating one of the variables.
\nWe can eliminate \\(x\\) using equations (i) and (ii)
\n2*(i) \\(\\simplify{2*{a1}}x+4y+8z=\\simplify{2*{r1}}\\)
\n\\(\\var{a1}\\)*(ii) \\(\\simplify{2*{a1}}x+\\simplify{{a1}*{b1}}y+\\simplify{3*{a1}}z=\\simplify{{a1}*{r2}}\\)
\nSubtracting gives us a new equation
\n(iv) \\(\\simplify{(4-{a1}{b1})y+(8-3*{a1})z}=\\simplify{2*{r1}-{a1}*{r2}}\\)
\nWe can also eliminate \\(x\\) using equations (ii) and (iii)
\n5*(ii) \\(10x +\\simplify{5*{b1}}y+15z=\\simplify{5*{r2}}\\)
\n2*(iii) \\(10x+12y+\\simplify{2*{c1}}z=\\simplify{2*{r3}}\\)
\nSubtracting gives us another new equation
\n(v) \\(\\simplify{(5*{b1}-12)y+(15-2*{c1})z}=\\simplify{5*{r2}-2*{r3}}\\)
\nWe could then eliminate the \\(y\\) from these two new equations
\n\\(\\simplify{5*{b1}-12}\\)*(iv) \\(\\simplify{(5*{b1}-12)*(4-{a1}{b1})y+(5*{b1}-12)*(8-3*{a1})z}=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})}\\)
\n\\(\\simplify{4-{a1}{b1}}\\)*(v) \\(\\simplify{(4-{a1}{b1})*(5*{b1}-12)y+(4-{a1}{b1})*(15-2*{c1})z}=\\simplify{(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)
\nSubtracting gives us
\n\\(\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}z=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)
\nThus
\n\\(z=\\frac{\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}}{\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}}=\\simplify{decimal{((5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}*{b1})*(5*{r2}-2*{r3}))/( (5*{b1}-12)*(8-3*{a1})-(4-{a1}*{b1})*(15-2*{c1}))}}\\)
\nWe can now back substitute this value for \\(z\\) into equation (iv) to find the correct value for \\(y\\) and then back substitute both these values into equation (i) to calculate \\(x\\).
\n", "tags": [], "variables": {"b1": {"templateType": "randrange", "description": "", "definition": "random(2..10#1)", "group": "Ungrouped variables", "name": "b1"}, "r1": {"templateType": "randrange", "description": "", "definition": "random(20..42#1)", "group": "Ungrouped variables", "name": "r1"}, "a1": {"templateType": "randrange", "description": "", "definition": "random(2..8#1)", "group": "Ungrouped variables", "name": "a1"}, "c1": {"templateType": "randrange", "description": "", "definition": "random(3..12#1)", "group": "Ungrouped variables", "name": "c1"}, "r3": {"templateType": "randrange", "description": "", "definition": "random(30..60#1)", "group": "Ungrouped variables", "name": "r3"}, "r2": {"templateType": "randrange", "description": "", "definition": "random(18..50#1)", "group": "Ungrouped variables", "name": "r2"}}, "type": "question", "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "prompt": "Input the value of \\(x\\) that satisfies the three equations.
\n\\(x = \\) [[0]]
\nInput the value of \\(y\\) that satisfies the three equations.
\n\\(y = \\) [[1]]
\nInput the value of \\(z\\) that satisfies the three equations.
\n\\(z = \\) [[2]]
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\n\\(A =\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\)
", "variables": {"k": {"templateType": "randrange", "definition": "random(1 .. 6#1)", "description": "", "name": "k", "group": "Ungrouped variables"}, "a21": {"templateType": "randrange", "definition": "random(1 .. 5#1)", "description": "", "name": "a21", "group": "Ungrouped variables"}, "a11": {"templateType": "randrange", "definition": "random(1 .. 10#1)", "description": "", "name": "a11", "group": "Ungrouped variables"}, "a12": {"templateType": "anything", "definition": "k*(a11-c1)", "description": "", "name": "a12", "group": "Ungrouped variables"}, "c1": {"templateType": "randrange", "definition": "random(14 .. 20#1)", "description": "", "name": "c1", "group": "Ungrouped variables"}, "a22": {"templateType": "anything", "definition": "{k}*{a21}+{c1}", "description": "", "name": "a22", "group": "Ungrouped variables"}, "lambda2": {"templateType": "anything", "definition": "max({c1},{a11}+{a22}-{c1})", "description": "", "name": "lambda2", "group": "Ungrouped variables"}, "lambda1": {"templateType": "anything", "definition": "min({c1},{a11}+{a22}-{c1})", "description": "", "name": "lambda1", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "advice": "The eigenvalues of a matrix are the values of \\(\\lambda\\) that satisfy the relation
\n\\(|A-\\lambda I| = 0\\)
\n\\(\\begin{vmatrix} \\var{a11}-\\lambda&\\var{a12}\\\\ \\var{a21}&\\var{a22}-\\lambda\\\\ \\end{vmatrix}=0\\)
\nThis gives:
\n\\((\\var{a11}-\\lambda)*(\\var{a22}-\\lambda)-(\\var{a12})*(\\var{a21})=0\\)
\n\\(\\lambda^2-\\simplify{{a11}+{a22}}\\lambda+\\simplify{{a11}*{a22}-{a21}*{a12}}=0\\)
\nThis can be solved using factorisation or by the quadratic formula to give:
\n\\(\\lambda_1 =\\var{lambda1}\\) and \\(\\lambda_2 =\\var{lambda2}\\)
\nAn eigenvector \\({\\bf v}=\\begin{pmatrix} x\\\\ y\\\\ \\end{pmatrix}\\) corresponding to an eigenvalue \\(\\lambda\\) must satisfy the relation: \\((A-\\lambda I){\\bf v} = {\\bf 0}\\)
\nso for \\(\\lambda_1=\\var{lambda1}\\)
\n\\(\\begin{pmatrix} \\simplify{{a11}-{lambda1}}&\\var{a12}\\\\ \\var{a21}&\\simplify{{a22}-{lambda1}}\\\\ \\end{pmatrix}\\begin{pmatrix} x\\\\ \\var{a21}\\\\ \\end{pmatrix}={\\bf 0}\\)
\nthus
\n\\(\\var{a21}x+\\simplify{{a22}-{lambda1}}*\\var{a21}=0\\)
\n\\(\\Rightarrow ~ \\var{a21}x=-\\simplify{({a22}-{lambda1})*{a21}}\\)
\n\\(\\Rightarrow ~ x=-\\simplify{({a22}-{lambda1})}\\)
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\n\\(\\lambda_1\\) is the lesser of the two eigenvalues and \\(\\lambda_2\\) is the greater of the two eigenvalues;
\n\\(\\lambda_1\\) = [[0]]
\n\\(\\lambda_2\\) = [[1]]
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\nEnter the value for \\(x=\\) [[0]]
\n", "useCustomName": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true}]}, {"name": "True or false - algebra", "extensions": [], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.
", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates totrue
or false
.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Maria Pickett", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3492/"}], "tags": [], "advice": "For help go to the maths cafe
", "metadata": {"description": "", "licence": "None specified"}, "functions": {}, "variable_groups": [], "statement": "Which of these statements are true in general:
", "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "parts": [{"marks": 1, "variableReplacementStrategy": "originalfirst", "prompt": "\\[(x+y)^2 = x^2+y^2\\]
", "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "customName": "", "type": "yes-no", "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "settings": {"correct_answer_expr": "false"}, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "prompt": "\\[\\sqrt{x^2+y^2} = x+y\\]
", "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "customName": "", "type": "yes-no", "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "settings": {"correct_answer_expr": "false"}, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "prompt": "\\[\\frac{a^2+2a}{4a+b} = \\frac{a+2}{4+b}\\]
", "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "customName": "", "type": "yes-no", "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "settings": {"correct_answer_expr": "false"}, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "prompt": "\\[s(s^2+1)-3(s^2+2) +3 = (s-3)(s^2+1)\\]
", "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "customName": "", "type": "yes-no", "adaptiveMarkingPenalty": 0, "showCorrectAnswer": true, "useCustomName": false, "showFeedbackIcon": true, "settings": {"correct_answer_expr": "true"}, "scripts": {}, "extendBaseMarkingAlgorithm": true}], "variables": {}, "ungrouped_variables": []}, {"name": "Factorise a quadratic", "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": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}], "advice": "", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "The student is asked to factorise a quadratic $x^2 + ax + b$. A custom marking script uses pattern matching to ensure that the student's answer is of the form $(x+a)(x+b)$, $(x+a)^2$, or $x(x+a)$.
\nTo find the script, look in the Scripts tab of part a.
"}, "rulesets": {}, "statement": "", "tags": [], "functions": {}, "variables": {"b": {"description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables", "name": "b"}, "a": {"description": "", "templateType": "anything", "definition": "random(-5..5)", "group": "Ungrouped variables", "name": "a"}}, "variable_groups": [], "parts": [{"extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "useCustomName": false, "scripts": {"mark": {"order": "after", "script": "// Parse the student's answer as a syntax tree\nvar studentTree = Numbas.jme.compile(this.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\nvar rule = Numbas.jme.compile('m_any( (x+??)(x+??), (x+??)^2, x*(x+??) )');\n\n// Check the student's answer matches the pattern. \nvar m = Numbas.jme.display.matchTree(rule,studentTree,true);\n// If not, take away marks\nif(!m) {\n this.multCredit(0.5,'Your answer is not in the form $(\\\\dots)(\\\\dots)$.');\n}\n"}}, "customName": "", "marks": "1", "showCorrectAnswer": true, "unitTests": [], "type": "jme", "answer": "(x+{a})(x+{b})", "customMarkingAlgorithm": "", "failureRate": 1, "variableReplacements": [], "checkingType": "absdiff", "adaptiveMarkingPenalty": 0, "vsetRange": [0, 1], "prompt": "Factorise $\\simplify{x^2+{a+b}x+{a*b}}$
", "valuegenerators": [{"value": "", "name": "x"}], "checkVariableNames": false, "showFeedbackIcon": true, "showPreview": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst"}], "ungrouped_variables": ["a", "b"]}, {"name": "Solve Equations of the Form $ax^2 +bx+c=0$", "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/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}], "tags": [], "rulesets": {}, "statement": "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 $x_1<x_2$.
", "preamble": {"js": "", "css": ""}, "parts": [{"sortAnswers": false, "type": "gapfill", "marks": 0, "scripts": {}, "showFeedbackIcon": true, "variableReplacements": [], "stepsPenalty": 0, "extendBaseMarkingAlgorithm": true, "steps": [{"extendBaseMarkingAlgorithm": true, "type": "information", "showCorrectAnswer": true, "prompt": "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{.}\\]
\n", "marks": 0, "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "prompt": "$\\simplify{x^2+{a+m}x+{a*m}=0}$
\n$x_1=$ [[0]]
\n$x_2=$ [[1]]
$\\simplify{{a1}x^2+{a2}x+{a3}={a4}}$
\n$x_1=$ [[0]]
\n$x_2=$ [[1]]
\n", "unitTests": [], "gaps": [{"minValue": "x1", "mustBeReduced": false, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "mustBeReducedPC": 0, "variableReplacements": [], "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "precisionType": "dp", "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "maxValue": "x1", "scripts": {}, "precision": "2", "extendBaseMarkingAlgorithm": true, "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst"}, {"minValue": "x2", "mustBeReduced": false, "showFeedbackIcon": true, "type": "numberentry", "showCorrectAnswer": true, "mustBeReducedPC": 0, "variableReplacements": [], "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "precisionType": "dp", "allowFractions": false, "precisionPartialCredit": 0, "marks": 1, "maxValue": "x2", "scripts": {}, "precision": "2", "extendBaseMarkingAlgorithm": true, "precisionMessage": "You have not given your answer to the correct precision.", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}, {"sortAnswers": false, "marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "showCorrectAnswer": true, "prompt": "
$\\simplify{{b1}x^2+{b2}x+{b3}={b4}x}$
\n$x_1=$ [[0]]
\n$x_2=$ [[1]]
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"}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "advice": "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}}&=\\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}\\]
$f'(x)=\\;$ [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*{p}*x^({p}-1)+{b}*{q}*x^({q}-1)+{c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "Find $f'(x),$ the derivate of $f(x),$ if $\\simplify{f(x)={a}x^{p}+{b}x^{q}+{c}x+{d}}.$
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-100..100 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-100..100 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "q": {"definition": "random(-10..10 except 0 except 1 except p)", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "random(-10..10 except 0 except 1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}}, "metadata": {"notes": "", "description": "Finding the derivative of a simple polynomial function.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Differentiation: Product rule", "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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "s1", "b", "m"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "{m}x ^ {m-1} * cos({a} * x+{b})-{a}x^{m} * sin({a} * x+{b})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "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", "steps": [{"type": "information", "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 the following function $f(x)$ using the product rule.
", "tags": ["calculus", "Calculus", "checked2015", "derivative of a product", "differentiating a product", "differentiating trigonometric functions", "differentiation", "MAS1601", "mas1601", "product rule", "Steps", "steps", "trigonometric functions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t31/07/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tSteps problem to be addressed. Now resolved.
\n\t\tChecked calculation.OK.
\n\t\tImproved prompt display.
\n\t\tClicking on Show steps does not lose any marks.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Differentiate $x^m\\cos(ax+b)$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"}, {"name": "Differentiation: product and chain rule, (a+bx)^m e^(nx), factorise answer", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"description": "Differentiate the function $f(x)=(a + b x)^m e ^ {n x}$ using the product and chain rule. Find $g(x)$ such that $f^{\\prime}(x)= (a + b x)^{m-1} e ^ {n x}g(x)$. Non-calculator. Advice is given.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Differentiate the following function $f(x)$.
", "variables": {"b": {"templateType": "anything", "description": "", "definition": "s1*random(1..5)", "group": "Ungrouped variables", "name": "b"}, "m": {"templateType": "anything", "description": "", "definition": "random(2..8)", "group": "Ungrouped variables", "name": "m"}, "a": {"templateType": "anything", "description": "", "definition": "random(1..4)", "group": "Ungrouped variables", "name": "a"}, "s1": {"templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s1"}, "n": {"templateType": "anything", "description": "", "definition": "random(2..6)", "group": "Ungrouped variables", "name": "n"}}, "tags": [], "parts": [{"customName": "", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "prompt": "$\\simplify{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$
\nYou are told that $\\simplify{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}$, for a polynomial $g(x)$.
\n\nYou have to find $g(x)$.
\n$g(x)=\\;$[[0]]
", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "unitTests": [], "useCustomName": false, "gaps": [{"customName": "", "customMarkingAlgorithm": "", "failureRate": 1, "checkingType": "absdiff", "showCorrectAnswer": true, "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "scripts": {}, "answer": "({((m * b) + (n * a))} + ({(n * b)} * x))", "valuegenerators": [{"value": "", "name": "x"}], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "type": "jme", "answerSimplification": "all", "marks": "4", "unitTests": [], "showPreview": true, "useCustomName": false, "variableReplacements": []}], "variableReplacements": []}], "rulesets": {}, "advice": "\n$f(x)$ is the product of the two functions $\\simplify{({a} + {b}*x)^{m}}$ and $\\simplify{e ^ ({n} * x)}$, so we need to use the product rule.
\n\nDifferentiating the first part, keeping the second half the same, gives the term: $\\simplify{{m} *{ b} * ({a} + {b} * x) ^ {m -1}} \\times \\simplify{e ^ ({n} * x)}$.
\nNote that that we needed the chain rule to do this differentiation.
\n\n\nDifferentiating the second part, keeping the first half the same, gives the term: $\\simplify{{n} * e ^ ({n} * x)} \\times \\simplify{({a} + {b}x)^{m}}$.
\nAgain, we needed the chain rule to do this differentiation.
\n\nHence, $\\simplify{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$.
\n$= \\simplify{({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}$, (by doing some factorising)
\n\nHence, $\\simplify{g(x) = {m * b + n * a} + {n * b} * x}$.
"}, {"name": "Basic Integration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}], "metadata": {"description": "Integration techniques for monomials and simple polynomials.
", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "variable_groups": [], "statement": "Find the following definite integrals.
", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "b", "n"], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "advice": "$\\int_a^bx^n=\\frac{b^{n+1}-a^{n+1}}{n+1}$
", "tags": [], "variables": {"b": {"templateType": "randrange", "description": "upper bound of integration
", "definition": "random(4..8#1)", "group": "Ungrouped variables", "name": "b"}, "n": {"templateType": "randrange", "description": "the power on x
", "definition": "random(2..10#1)", "group": "Ungrouped variables", "name": "n"}, "a": {"templateType": "randrange", "description": "lower bound of integration
", "definition": "random(0..3#1)", "group": "Ungrouped variables", "name": "a"}}, "type": "question", "parts": [{"marks": 1, "correctAnswerFraction": false, "scripts": {}, "minValue": "1/(n+1)", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "$\\int_{0}^{1}{x^{\\var{n}}}$
", "allowFractions": true, "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry", "maxValue": "1/(n+1)"}, {"marks": 1, "correctAnswerFraction": false, "scripts": {}, "minValue": "(b^(n+1)-a^(n+1))/(n+1)", "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "$\\int_{\\var{a}}^{\\var{b}}{x^{\\var{n}}}$
", "allowFractions": true, "mustBeReducedPC": 0, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true, "type": "numberentry", "maxValue": "(b^(n+1)-a^(n+1))/(n+1)"}]}, {"name": "Indefinite Integration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}], "tags": [], "parts": [{"checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "failureRate": 1, "marks": 1, "unitTests": [], "prompt": "$\\int{(\\frac{1}{4}\\sqrt{x}-3\\sqrt{x^5})}\\mathrm{dx}$
", "answer": "1/6x^(3/2)-6/7x^(7/2)+c", "showCorrectAnswer": true, "checkingType": "absdiff", "expectedVariableNames": [], "vsetRangePoints": 5, "checkingAccuracy": 0.001, "vsetRange": [0, 1], "showFeedbackIcon": true, "customMarkingAlgorithm": "malrules:\n [\n [\"1/6x^(3/2)-6/7x^(7/2)\",\"Don't forget the constant of integration!\",0.9],\n [\"1/6x^(3/2)-2(x^5)^(3/2)+C\", \"Check the second term again. Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/6x^(3/2)-2(x^5)^(3/2)\", \"Check the second term again. Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/4x^(1/2)-2(x^5)^(3/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/4x^(1/2)-2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/4x^(1/2)-9/2(x^5)^(3/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/4x^(1/2)-9/2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/4x^(1/2)-6/7x^(7/2)\", \"You need to look at the first term again. $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated.\",0],\n [\"1/4x^(1/2)-6/7x^(7/2)+C\", \"You need to look at the first term again. $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated.\",0],\n [\"1/4x^(1/2)-21/2x^(7/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"1/4x^(1/2)-21/2x^(7/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"1/2x^(1/2)-2(x^5)^(3/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/2x^(1/2)-2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/2x^(1/2)-9/2(x^5)^(3/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/2x^(1/2)-9/2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"1/2x^(1/2)-6/7x^(7/2)\", \"You need to look at the first term again. $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated.\",0],\n [\"1/2x^(1/2)-6/7x^(7/2)+C\", \"You need to look at the first term again. $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated.\",0],\n [\"1/2x^(1/2)-21/2x^(7/2)\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"1/2x^(1/2)-21/2x^(7/2)+C\", \"You need to look at both terms again. First term: $\\\\sqrt{x}=x^{\\\\frac{1}{2}}$ but this has not actually been integrated. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"3/8x^(3/2)-2(x^5)^(3/2)\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"3/8x^(3/2)-2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"3/8x^(3/2)-9/2(x^5)^(3/2)\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"3/8x^(3/2)-9/2(x^5)^(3/2)+C\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: Try to write the power on the $x$ as a single power rather than $(x^5)^{\\\\frac{1}{2}}$ before integrating. Remember, if the powers are side by side, multiply them: $\\\\sqrt{x^5}=(x^5)^{\\\\frac{1}{2}}=x^{5 \\\\times \\\\frac{1}{2}}=x^{\\\\frac{5}{2}}$.\",0],\n [\"3/8x^(3/2)-6/7x^(7/2)\", \"You need to look at the first term again. It looks like you have multiplied by the new power of $\\\\frac{3}{2}$.\",0],\n [\"3/8x^(3/2)-6/7x^(7/2)+C\", \"You need to look at the first term again. It looks like you have multiplied by the new power of $\\\\frac{3}{2}$.\",0],\n [\"3/8x^(3/2)-21/2x^(7/2)\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"3/8x^(3/2)-21/2x^(7/2)+C\", \"You need to look at both terms again. First term: It looks like you have multiplied by the new power of $\\\\frac{3}{2}$. Second term: It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"1/6x^(3/2)-21/2x^(7/2)\", \"You need to look at the second term again. It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0],\n [\"1/6x^(3/2)-21/2x^(7/2)+C\", \"You need to look at the second term again. It looks like you have multiplied by the new power of $\\\\frac{7}{2}$...\",0]\n ]\n\n\nparsed_malrules: \n map(\n [\"expr\":parse(x[0]),\"feedback\":x[1],\"credit\":x[2]],\n x,\n malrules\n )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n map(\n len(filter(not x ,x,map(\n try(\n resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n message,\n false\n ),\n vars,\n vset\n )))Solve the following indefinite integrals, using $C$ to represent an unknown constant.
", "functions": {}, "preamble": {"css": "", "js": ""}, "variables": {"f": {"group": "Ungrouped variables", "description": "", "definition": "random(1..8 except d)", "name": "f", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "random(2..9)", "name": "a", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "random(2..9 except a)", "name": "b", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "random(1..9 except a except b)", "name": "c", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "random(1..8)", "name": "d", "templateType": "anything"}}, "type": "question", "rulesets": {}, "advice": "Indefinite Integrals
", "variablesTest": {"condition": "", "maxRuns": 100}}, {"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/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..5)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a}/{c})*x+{c*b-a}/{c^2}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*e^({c}x)} dx $
You are given that the answer is of the form \\[I=g(x)e^{\\var{c}x}+C\\] for a polynomial $g(x)$. You have to find $g(x)$.
$g(x)=\\;$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n\t\t\t", "steps": [{"type": "information", "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
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "all", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\tUse the result from the first part to find:
\n\t\t\t$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*e^({c}x)} dx $
\n\t\t\tYou are given that the answer is of the form \\[I=h(x)e^{\\var{c}x}+C\\] for a polynomial $h(x)$. You have to find $h(x)$.
\n\t\t\t$h(x)=\\;$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tFind the following indefinite integrals.
\n\tInput all numbers as fractions or integers and not decimals.
\n\t", "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "exponential function", "integration", "integration by parts", "integration of exponential function"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t3/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tGot rid of redundant instructions about inputting constant of integration.
\n\t\tPenalised use of steps in first part, 1 mark. Added message to that effect in first part.
\n\t\tAdded message about not inputting decimals in appropriate places.
\n\t\tChanged marks reflecting the use of steps and degree of difficulty in second part.
\n\t\tImproved Advice display.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Given $\\displaystyle \\int (ax+b)e^{cx}\\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\\displaystyle \\int (ax+b)^2e^{cx}\\;dx =h(x)e^{cx}+C$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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 $u = \\simplify[std]{{a}x+{b}}$ and $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\n\tSo $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})*e^({c}x) - (1/{c})*Int(({a})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})*e^({c}x) -({a}/{c^2})*e^({c}x) + C}\\\\ &=& \\simplify[std]{(({a}x+{b})/{c}-{a}/{c^2})*e^({c}*x) + C}\\\\ &=& \\simplify[std]{(({a}/{c})x+{b*c-a}/{c^2})*e^({c}*x) + C} \\end{eqnarray} \\]
Hence $\\displaystyle \\simplify[std]{g(x)=({a}/{c})*x+{c*b-a}/{c^2}}$
\n\tb)
\n\tFor this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\n\tHence,
\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})^2*e^({c}x) - (1/{c})*Int({2*a}*({a}x+{b})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*Int(({a}x+{b})*e^({c}x),x)}\\dots (*) \\end{eqnarray*}\\]
But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*e^({c}*x),x)=(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}$
\n\tSo on substituting this in equation (*) we find:
\\[ \\begin{eqnarray*}I&=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}\\\\ &=& \\simplify[std]{({a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3})*e^({c}x) +C} \\end{eqnarray*}\\]
Hence $\\displaystyle \\simplify[std]{h(x)={a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}}$
\n\t"}, {"name": "Indefinite integration using standard integrals", "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 Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}], "statement": "\nIntegrate the following function $f(x)$.
\n
You must input the constant of integration as $C$.
Integrate $f(x) = ae ^ {bx} + c\\sin(dx) + px^q$. Must input $C$ as the constant of integration.
", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": [], "advice": "\nSplitting the integral into three parts and using the information in Steps we have:
\n\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&=&\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &=&\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]
\n ", "parts": [{"stepsPenalty": 0, "adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "marks": 0, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "\n$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nEnter all numbers as integers or fractions and not as decimals.
\n ", "sortAnswers": false, "showFeedbackIcon": true, "gaps": [{"adaptiveMarkingPenalty": 0, "variableReplacements": [], "marks": 3, "useCustomName": false, "answerSimplification": "std", "notallowed": {"partialCredit": 0, "message": "Enter all numbers as integers or fractions and not as decimals.
", "showStrings": false, "strings": ["."]}, "customMarkingAlgorithm": "", "unitTests": [], "vsetRange": [0, 1], "showCorrectAnswer": true, "vsetRangePoints": 5, "failureRate": 1, "variableReplacementStrategy": "originalfirst", "type": "jme", "scripts": {}, "checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "answer": "({b}/{a}) * e ^({a}*x) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "showPreview": true, "showFeedbackIcon": true, "valuegenerators": [{"value": "", "name": "c"}, {"value": "", "name": "x"}], "customName": ""}], "showCorrectAnswer": true, "customName": "", "type": "gapfill", "steps": [{"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "marks": 0, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "unitTests": [], "prompt": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
", "showFeedbackIcon": true, "showCorrectAnswer": true, "customName": "", "type": "information", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}], "functions": {}, "variables": {"b": {"definition": "s2*random(2..9)", "group": "Ungrouped variables", "name": "b", "templateType": "anything", "description": ""}, "c3": {"definition": "s5*random(2..8)", "group": "Ungrouped variables", "name": "c3", "templateType": "anything", "description": ""}, "s3": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s3", "templateType": "anything", "description": ""}, "a1": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "a1", "templateType": "anything", "description": ""}, "a": {"definition": "s1*random(2..5)", "group": "Ungrouped variables", "name": "a", "templateType": "anything", "description": ""}, "b1": {"definition": "s3*random(2..9)", "group": "Ungrouped variables", "name": "b1", "templateType": "anything", "description": ""}, "s1": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s1", "templateType": "anything", "description": ""}, "s5": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s5", "templateType": "anything", "description": ""}, "s4": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s4", "templateType": "anything", "description": ""}, "a2": {"definition": "s4*random(3..9)", "group": "Ungrouped variables", "name": "a2", "templateType": "anything", "description": ""}, "s2": {"definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s2", "templateType": "anything", "description": ""}}}, {"name": "Number of roots and stationary points of a graph", "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Nick Walker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2416/"}, {"name": "Thomas Waters", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3649/"}], "tags": [], "advice": "(i) Definition: A 'root' of a function $f(x)$ is a value of $x$ which makes $f(x)=0$. Visually a root can be found be seeing when the $y$-coordinate of the graph is $0$, i.e., when the graph crosses the $x$-axis. Therefore, to count the roots, you need to count how many times the graph crosses the $x$-axis. In this question, the graph crosses the $x$-axis $\\var{num_roots}$ time(s), so there are $\\var{num_roots}$ roots.
\n(ii) Definition: A 'stationary point' of a function is a point on the graph where $f'(x)=0$. Remember that $f'$ tells us the gradient of $f$, so visually a stationary point is where the gradient of the curve is 0. In this question, there is/are $\\var{num_stat}$ place(s) where the gradient of the graph is $0$, so the answer is $\\var{num_stat}$.
", "metadata": {"description": "A graph (of a cubic) is given. The question is to determine the number of roots and number of stationary points the graph has. Non-calculator. Advice is given.
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\nAbove is the graph of some function $f$.
\nHow many roots does $f$ have? [[0]]
\nHow many stationary points does $f$ have? [[1]]
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