Calculate the marginal and average cost for a given cost function. Find the corresponding startup/shutdown price.
Maximize the profit function at a given price.
a) en b) From the expression \\[ C(q) = \\simplify[]{{a}*q^3-{b}*q^2+{c}*q+{d}}\\] we get the marginal cost
\n\\[ MC(q) = \\frac {dC}{dq} (q) =\\simplify[]{{3*a}*q^2-{2*b}*q+{c}}\\] and the average cost
\n\\[ AC(q) = \\frac {C(q)}{q} = \\simplify[]{{a}*q^2-{b}*q+{c}+{d}/q} .\\]
\n\nc) The profit equals the difference of the revenues and the costs:
\n\\[ W(q) = R(q) - C(q) = \\var{prijs} \\cdot q - \\left( \\simplify[]{{a}*q^3-{b}*q^2+{c}*q+{d}} \\right) = \\simplify[]{{-a}*q^3+{b}*q^2+{prijs-c}*q-{d}}\\]
\nd) In order to maximize the profit, the derivative of \\( W \\) with respect to \\( q \\) must be equal to 0:
\n\\[ \\frac {dW}{dq} (q) = \\simplify[]{{-3*a}*q^2+{2*b}*q+{prijs-c}} . \\]
\nThis first order derivative equals 0 at the values \\( q_1 = \\var{q1} \\) and \\( q_2 = \\var{q2} \\).
\nSince
\n\\[ \\frac {d^2 W}{dq^2} (q) = \\simplify[]{{-6*a}*q+{2*b}} \\]
\nit follows
\n\\[ \\frac {d^2 W}{dq^2} (\\var{q1}) = \\var{-6*a*q1+2*b} > 0 \\]
\nand
\n\\[ \\frac {d^2 W}{dq^2} (\\var{q2}) = \\var{-6*a*q2+2*b} < 0 . \\]
\nThis implies that the pofit is maximal at \\( q = \\var{q2} \\).
\n\ne) \\( W( \\var{q2}) = \\var{-a*q2^3+b*q2^2+(prijs-c)*q2-d}\\)
\nGraph of the profit function:
\n{ShowWinst()}
\n\n\nf) The startup/shutdown price is the minimal value of the average variable cost function:
\n\\[ p_{startup/shutdown} = min \\, AVC ,\\]
\nwhere \\( AVC(q) \\) equals the total variable cost divided by \\( q \\):
\n\\[ AVC(q) = \\frac {TVC(q)} {q} .\\]
\nIn this problem
\n\\[ AVC(q) = \\simplify[]{{a}*q^2-{b}*q+{c}} \\]
\nhas as derivative
\n\\[ \\frac {dAVC}{dq} (q) = \\simplify[]{{2*a}*q-{b}} .\\]
\nThis derivative is equal to 0 at
\n\\[ q = \\simplify[all,FractionNumbers]{{b}/{2*a}} .\\]
\nThis value indeed leads to a minimum of \\( AVC \\), with
\n\\[ AVC \\left( \\simplify[all,FractionNumbers]{{b}/{2*a}} \\right) = \\simplify[all,FractionNumbers]{{a*(b/(2*a))^2-b*(b/(2*a))+c}} ,\\]
\nsuch that rounded up to 2 decimals \\( p_{startup/shutdown} \\) is given by
\n\\[ \\simplify[]{{a*(b/(2*a))^2-b*(b/(2*a))+c}} .\\]
\nGraphs of MC , AC and AVC :
\n{ShowKosten()}
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\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('600px','600px',\n{boundingBox:[-2,grensy,grensq+2,-30],grid: false,axis:true,keepaspectratio:false,\ndefaultAxes:{x:{ticks:{label:{visible:false}}},y:{ticks:{label:{visible:false}}}}});\n// boundingbox:[left, top, right, bottom]\nvar board = div.board;\n\n\txaxis = board.create('axis', [[0, 0], [1,0]], \n\t {name:'q', \n\t\t\twithLabel: true, \n\t\t\tlabel: {position: 'rt', // possible values are 'lft', 'rt', 'top', 'bot'\n\t\t\t\t\t offset: [-15, 10] // (in pixels)\n\t\t\t\t\t }\n\t\t\t});\n yaxis = board.create('axis', [[0, 0], [0, 1]], \n\t\t {name:'\u20ac', \n\t\t\twithLabel: true, \n\t\t\tlabel: {\n\t\t\t position: 'rt', // possible values are 'lft', 'rt', 'top', 'bot'\n\t\t\t offset: [-20, 0] // (in pixels)\n\t\t\t\t}\n\t\t\t});\n\n \nvar grafMC = board.create('functiongraph',\n [function(x){ return 3*a*x*x-2*b*x+c;},0,b/(2*a)], \n{strokeColor:\"red\",setLabelText:'MC',visible: true, strokeWidth: 2, dash:2, highlightStrokeColor: 'red'} \n ); \n\nvar grafMC2 = board.create('functiongraph',\n [function(x){ return 3*a*x*x-2*b*x+c;},b/(2*a),grensq+2], \n{strokeColor:\"red\",setLabelText:'MC',visible: true, strokeWidth: 4, highlightStrokeColor: 'red'} \n ); \n\nvar grafAC = board.create('functiongraph',\n [function(x){ return a*x*x-b*x+c+d/x;},0,grensq+2], \n{strokeColor:\"green\",setLabelText:'AC',visible: true, strokeWidth: 2, dash:2, highlightStrokeColor: 'green'} \n ); \nvar grafAVC = board.create('functiongraph',\n [function(x){ return a*x*x-b*x+c;},0,grensq+2], \n{strokeColor:\"blue\",setLabelText:'AVC',visible: true, strokeWidth: 4, highlightStrokeColor: 'blue'} \n ); \n\nreturn div;"}, "ShowWinst": {"parameters": [], "type": "html", "language": "javascript", "definition": "//variables\nvar a = Numbas.jme.unwrapValue(scope.variables.a);\nvar b = Numbas.jme.unwrapValue(scope.variables.b);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\nvar d = Numbas.jme.unwrapValue(scope.variables.d);\nvar s = Numbas.jme.unwrapValue(scope.variables.s);\nvar v = Numbas.jme.unwrapValue(scope.variables.v);\nvar prijs = Numbas.jme.unwrapValue(scope.variables.prijs);\nvar bovenaan = Numbas.jme.unwrapValue(scope.variables.bovenaan);\nvar onderaan = Numbas.jme.unwrapValue(scope.variables.onderaan);\nif(bovenaan>10) {bovengrens = bovenaan} else {bovengrens = 50};\n\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox:[-3,bovengrens,s+v+5,onderaan],grid: false,axis:true,keepaspectratio:false,\ndefaultAxes:{x:{ticks:{label:{visible:false}}},y:{ticks:{label:{visible:false}}}}});\n// boundingbox:[left, top, right, bottom]\nvar board = div.board;\n\n\txaxis = board.create('axis', [[0, 0], [1,0]], \n\t {name:'q', \n\t\t\twithLabel: true, \n\t\t\tlabel: {position: 'rt', // possible values are 'lft', 'rt', 'top', 'bot'\n\t\t\t\t\t offset: [-15, 10] // (in pixels)\n\t\t\t\t\t }\n\t\t\t});\n yaxis = board.create('axis', [[0, 0], [0, 1]], \n\t\t {name:'\u20ac', \n\t\t\twithLabel: true, \n\t\t\tlabel: {\n\t\t\t position: 'rt', // possible values are 'lft', 'rt', 'top', 'bot'\n\t\t\t offset: [-20, 0] // (in pixels)\n\t\t\t\t}\n\t\t\t});\n\n\nvar grafiek = board.create('functiongraph',\n [function(x){ return -a*x*x*x+b*x*x+(prijs-c)*x-d},0,s+v+5], \n{strokeColor:\"green\",setLabelText:'Winst',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} \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": "Determine the marginal cost function:
\n\\( MC(q) = \\) [[0]]
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\n\\( AC(q) = \\) [[0]]
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\nWhat is the profit function?
\n\\( W(q) = \\) [[0]]
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\n[[0]]
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\n[[0]]
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\n