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Manipulation of an exponential function
", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "extensions": [], "ungrouped_variables": ["k", "c", "m", "d"], "variables": {"c": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "m": {"description": "", "definition": "random(1.6..5#0.2)", "templateType": "randrange", "group": "Ungrouped variables", "name": "m"}, "k": {"description": "", "definition": "random(2..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "k"}, "d": {"description": "", "definition": "random(1..14#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "name": "Manipulation of formula 1", "parts": [{"type": "gapfill", "prompt": "\\(x =\\) [[0]]
", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"type": "jme", "answer": "(ln((1-y/{k})/{c})-{d})/{m}", "scripts": {}, "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showFeedbackIcon": true, "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": "2"}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)
\nWorking from the outside in, we divide across by \\(\\var{k}\\)
\n\\(\\frac{y}{\\var{k}}=1-\\var{c}e^{\\var{m}x+\\var{d}}\\)
\nWe can bring the \\(x\\) variable to the left hand side and move the \\(y\\) variable to the right hand side
\n\\(\\var{c}e^{\\var{m}x+\\var{d}}=1-\\frac{y}{\\var{k}}\\)
\nAgain working from the outside in we divide across by \\(\\var{c}\\)
\n\\(e^{\\var{m}x+\\var{d}}=\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\)
\nTaking the natural log of both sides eliminates the \\(e\\) from the left hand side.
\n\\(\\var{m}x+\\var{d}=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)\\)
\nSubtract \\(\\var{d}\\) from both sides
\n\\(\\var{m}x=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}\\)
\nand finally divide by \\(\\var{m}\\) to get
\n\\(x=\\frac{ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}}{\\var{m}}\\)
", "statement": "Rearrange the following expression to make \\(x\\) the subject:
\n\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)
", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}