// Numbas version: exam_results_page_options {"name": "Implicit 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"c": {"definition": "random(2..9 except -a+1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c"}, "d": {"definition": "random(-3..3 except 0)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d"}, "b": {"definition": "c-1", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "a": {"definition": "-random(1..9)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}}, "parts": [{"scripts": {}, "unitTests": [], "gaps": [{"scripts": {}, "unitTests": [], "showFeedbackIcon": true, "answerSimplification": "all,!collectNumbers", "vsetRange": [0, 1], "checkingType": "absdiff", "marks": 2, "answer": "(({( - a)} + ( - (2 * x))-{d}y) / ({b} + (2 * y)+{d}x))", "customMarkingAlgorithm": "", "showPreview": true, "variableReplacements": [], "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "expectedVariableNames": [], "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "type": "jme", "notallowed": {"showStrings": false, "message": "

Input all numbers as integers or as fractions, not as decimals.

", "strings": ["."], "partialCredit": 0}, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "prompt": "

$\\displaystyle \\frac{dy}{dx}=$ []

", "marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst"}], "ungrouped_variables": ["a", "c", "b", "d"], "rulesets": {"std": ["all", "fractionNumbers"]}, "extensions": [], "preamble": {"js": "", "css": ""}, "statement": "

Given the following relation between $x$ and $y$
\$\\simplify[all,!collectNumbers]{x^2+y^2+{d}x y+{a}x+{b}y}=\\var{c}\$

\n

Using implicit differentiation find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.

", "variable_groups": [], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\t

Implicit differentiation.

\n \t\t

Given $x^2+y^2+dxy +ax+by=c$ find $\\displaystyle \\frac{dy}{dx}$ in terms of $x$ and $y$.

\n \t\t

Also find two points on the curve where $x=0$ and find the equation of the tangent at those points.

\n \t\t

\n \t\t"}, "name": "Implicit 2", "functions": {}, "advice": "

#### a)

\n

On differentiating both sides of the equation implicitly we get
\$2x + \\simplify[all,!collectNumbers]{2y*Diff(y,x,1) +{d}(y+x*Diff(y,x,1))+ {a} + {b} *Diff(y,x,1)} = 0\$
Collecting terms in $\\displaystyle\\frac{dy}{dx}$ and rearranging the equation we get
\$( \\simplify[all,!collectNumbers]{({b} + 2y+{d}x)} )\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{{ -a} -2x-{d}y}\$ and hence on further rearranging:

\n

\$\\frac{dy}{dx} = \\simplify[all,!collectNumbers]{({ - a} - 2 * x-{d}y) / ({b} + (2 * y)+{d}x)}\$

\n

#### b)

\n

On putting $x=0$ in the relation we get:

\n

\$\\simplify{y^2+{b}y={c}} \\Rightarrow \\simplify{y^2+{b}y-{c}=0 }\\Rightarrow (y+\\var{c})(y-1)=0\$

\n

Hence $a=-\\var{c}$ and $b=1$.

\n

#### c)

\n

First we find the tangent at the point $(0,-\\var{c})$.

\n

We find using the formula we found for $\\frac{dy}{dx}$ in part a) that the gradient at  $(0,-\\var{c})$ is:

\n

\$\\frac{dy}{dx}=\\frac{\\simplify[all,!collectnumbers]{{-a}+{d*c}}}{\\var{b}-\\var{2*c}}=\\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}\$

\n

As the tangent goes through the point $(0,\\var{-c})$ i.e. at $x=0,\\;\\;y=-\\var{c}$ we see that the equation of the tangent is:

\n

\$y=\\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}x-\\var{c}\$

\n

Next we find that the gradient at  $(0,1)$ is:

\n

\$\\frac{dy}{dx}=\\frac{\\simplify[all,!collectnumbers]{{-a}-{d}}}{\\var{b}+2}=\\simplify[all,fractionNumbers]{-{a+d}/{c+1}}\$

\n

As the tangent goes through the point $(0,1)$ i.e. at $x=0,\\;\\;y=1$ we see that the equation of the tangent is:

\n

\$y=\\simplify[all,fractionNumbers]{-{a+d}/{c+1}}x+1\$

", "type": "question", "contributors": [{"name": "Leticija Dubickaite", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2461/"}]}]}], "contributors": [{"name": "Leticija Dubickaite", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2461/"}]}