\(\displaystyle {@FuncName@} : \mathbb{R}^2 \to \mathbb{R}\), \((x,y) \mapsto {@SelectedFunction@}\)

und ein Taylorpolynom \(T_{{@Order@}}{@FuncName@}\) {@NumberWords[Order+1]@} Grades von \({@FuncName@}\).

Die folgende Abbildung zeigt den Graphen von \({@FuncName@}\) (Graph □) und den Graphen von \(T_{{@Order@}}{@FuncName@}\) (Graph □) auf dem Einheitsquadrat \([-1,1]^2\) in der \(x_1 x_2\)-Ebene von \(\mathbb{R}^3\) aufgezeichnet gegen die \(x_3\)-Achse. Sie können das Funktionsargument in der \(x_1 x_2\)-Ebene verschieben (Punkt ●). Die zugehörigen Funktionswerte unter \({@FuncName@}\) und \(T_{@FuncName@}\) werden Ihnen durch Angabe der zugehörigen Punkte auf den Graphen angezeigt (Punkt ● bzw. Punkt ●).

[[jsxgraph input-ref-ans1="stateRef"]] const board = JXG.JSXGraph.initBoard(divid, { boundingbox: [-1.9, 1.9, 1.9, -1.9], axis: false, showCopyright: false, keepaspectratio: true, /* showNavigation: false, */ fixed: true }); /* ------------------------------------- */ /* --------------------------------------*/ /* ---- STACK to JSXGraph interface ---- */ var state = {#Argument#}; state[2] = {#SliderPos#}[0]; /* position of slider 'SliderAngleOfProj1' */ state[3] = {#SliderPos#}[1];; /* position of slider 'SliderAngleOfProj2' */ var stateInput = document.getElementById(stateRef); if (stateInput.value && stateInput.value != '') { state = JSON.parse(stateInput.value); } /* definition of (approximated) function */ var ExFunc = board.jc.snippet('{#SelectedFunction#}', true, 'x,y', true); /* definition of approximating function */ var StrApproxFunc = '{#ListOfFunctions#}'.substring(1, '{#ListOfFunctions#}'.length - 1); var SplitStrApproxFunc = StrApproxFunc.split(","); var MaxOrder = SplitStrApproxFunc.length - 1; /* order / degree of the polynomial approximation */ var Order = {#Order#}; /* initial coordinates of the point of development */ var PointOfDevInit = [{#PointOfDev#}[0], {#PointOfDev#}[1]]; /* resolution of the grid */ var ResOfGrid = {#ResOfGrid#}; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- colors ---- */ const Color1 = `hsl(${223}, ${100}%, ${70}%)`; /* #648FFF */ const Color2 = `hsl(${251}, ${83}%, ${65}%)`; /* #785EF0 */ const Color3 = `hsl(${331}, ${72}%, ${51}%)`; /* #DC267F */ const Color4 = `hsl(${23}, ${100}%, ${50}%)`; /* #FE6100 */ const Color5 = `hsl(${41}, ${100}%, ${50}%)`; /* #FFB000 */ /* ------------------------------------- */ /* --------------------------------------*/ /* draws slider controlling the order of the approximating function */ var SliderOrder = board.create('slider', [ [-1.3, -1.75], [1.2, -1.75], [0, Order, MaxOrder] ], { withLabel: true, highlight: false, name: 'order', snapWidth: 1, visible: false, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); /* order of the approximating function */ var Order = function(n) { return n <= SliderOrder.Value() }; /* definition of approximating function */ var ApproxFuncList = []; for (var i = 0; i <= MaxOrder; i++) { ApproxFuncList.push(board.jc.snippet(SplitStrApproxFunc[i], true, 'x,y,a,b', true)) } var ApproxFunc = function(x, y, a, b) { return ApproxFuncList[SliderOrder.Value()](x, y, a, b) }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- general operations on vectors and matrices ---- */ /* vector product */ var NormalVec = function(a, b) { return [ ((a[1] * b[2]) - (a[2] * b[1])), ((a[2] * b[0]) - (a[0] * b[2])), ((a[0] * b[1]) - (a[1] * b[0])) ] }; /* euclidean scalar product */ var ScalarProd = function(a, b) { return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]) }; /* euclidean norm of vector */ var EuclidNorm = function(a) { return Math.sqrt(ScalarProd(a, a)) }; /* multiplication of scalar vectors */ var ScalarMultVec = function(x, a) { return [x * a[0], x * a[1], x * a[2]] }; /* entrywise multiplication of vectors */ var VecMultVec = function(a, b) { return [a[0] * b[0], a[1] * b[1], a[2] * b[2]] }; /* sum of vectors */ var VecSumVec = function(a, b) { return [a[0] + b[0], a[1] + b[1], a[2] + b[2]] }; /* difference of vectors */ var VecSubVec = function(a, b) { return [a[0] - b[0], a[1] - b[1], a[2] - b[2]] }; /* euclidean distance of vectors */ var EuclidDist = function(a, b) { return EuclidNorm(VecSubVec(a, b)) } /* multiplication of scalar matrix*/ var ScalarMultMat = function(x, A) { return [ScalarMultVec(x, A[0]), ScalarMultVec(x, A[1]), ScalarMultVec(x, A[2])] } /* normalizes non-zero vector */ var UnitVec = function(a) { let x = Math.sqrt(ScalarProd(a, a)); let b = ScalarMultVec(x, a); return b; } /* determinant of matrix*/ var DetOfMat = function(A) { return ScalarProd(A[0], NormalVec(A[1], A[2])) }; /* transpose of matrix */ var TranspOfMat = function(A) { let B1 = [A[0][0], A[1][0], A[2][0]]; let B2 = [A[0][1], A[1][1], A[2][1]]; let B3 = [A[0][2], A[1][2], A[2][2]]; return [B1, B2, B3]; } /* pre-inverse of matrix */ var PreInvOfMat = function(A) { let B = TranspOfMat(A); let b1 = NormalVec(B[1], B[2]); let b2 = NormalVec(B[2], B[0]); let b3 = NormalVec(B[0], B[1]); let C = [b1, b2, b3]; return C; }; /* inverse of matrix */ var InvOfMat = function(A) { let B = PreInvOfMat(A); let b = 1 / DetOfMat(A); return ScalarMultMat(b, B); }; /* matrix applied as linear map to vector */ var MatMultVec = function(a, A) { let b1 = ScalarProd(A[0], a); let b2 = ScalarProd(A[1], a); let b3 = ScalarProd(A[2], a); return [b1, b2, b3]; } /* ------------------------------------- */ /* --------------------------------------*/ /* ---- sliders for controlling the camera ---- */ var SliderAngleOfProj1 = board.create('slider', [ [-1.75, -1.3], [-1.75, 1.3], [0.005, state[2], 0.5] ], { withLabel: true, highlight: false, name: 'el', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var SliderAngleOfProj2 = board.create('slider', [ [-1.3, -1.6], [1.2, -1.6], [-1, state[3], 1] ], { withLabel: true, highlight: false, name: 'az', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var AOP1 = function() { return SliderAngleOfProj1.Value() * Math.PI }; var AOP2 = function() { return (SliderAngleOfProj2.Value() + 1) * Math.PI }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- 3D to 2D projection ---- */ /* angles for 2D projection */ var cpsi = function() { return Math.cos(AOP2()) }; var spsi = function() { return Math.sin(AOP2()) }; var cphi = function() { return Math.cos(AOP1()) }; var sphi = function() { return Math.sin(AOP1()) }; /* matrix for 2D projection */ var ProjMat2D = function() { return [ [(-1) * cpsi(), spsi(), 0], [(-1) * spsi() * sphi(), (-1) * cpsi() * sphi(), cphi()], [0, 0, 1] ] }; /* returns 2D coordinates of 3D objects */ var CamProjection = function(a) { let Output = MatMultVec(a, ProjMat2D()); return [Output[0], Output[1]]; } var CamInv = function(a) { let Output = MatMultVec([a[0], a[1], a[2]], InvOfMat(ProjMat2D())); return [Output[0], Output[1], Output[2]]; } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- coordinate system ---- */ /* 3D coordinates of arrows in dicating the coordinate system */ var AxisCoords = function(h) { /* initial coordinates */ let ICs0 = [0, 0, h]; let ICs1 = [1.2, 0, h]; let ICs2 = [0, 1.2, h]; let ICs3 = [0, 0, 1.2 + h]; /* ---- */ let ICs4 = [-1.1, 0, h]; let ICs5 = [0, -1.1, h]; return [ CamProjection(ICs0), CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3), CamProjection(ICs4), CamProjection(ICs5), ] }; /* 3D coordinates of labels of coordinate system*/ var AxisLabelPos = function(h) { /* initial coordinates */ let ICs1 = [1.3, 0, h]; let ICs2 = [0, 1.3, h]; let ICs3 = [0, 0, 1.3 + h]; return [ CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3) ] }; /* labels of coordinate system*/ var AxisLabel = ['x_1', 'x_2', 'x_3']; /* draws arrows indicating the coordinates system */ for (let i = 0; i <= 2; i++) { board.create('arrow', [ [() => AxisCoords(0)[0][0], () => AxisCoords(0)[0][1]], [() => AxisCoords(0)[i + 1][0], () => AxisCoords(0)[i + 1][1]] ], { strokecolor: 'black', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(0)[i][0], () => AxisLabelPos(0)[i][1], AxisLabel[i]]); } /* draws arrows indicating the coordinates system in the domain */ for (let i = 0; i <= 1; i++) { board.create('arrow', [ [() => AxisCoords(-1)[i + 4][0], () => AxisCoords(-1)[i + 4][1]], [() => AxisCoords(-1)[i + 1][0], () => AxisCoords(-1)[i + 1][1]] ], { strokecolor: 'grey', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(-1)[i][0], () => AxisLabelPos(-1)[i][1], AxisLabel[i]], { strokeColor: 'grey', }); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- collection of functions that draw grids and graphs ---- */ /* function that returns 2D coordinates of given 'grid' */ var GridCoordsConv = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [CamProjection(G[0][i][0]), CamProjection(G[0][i][1])]; ICsY[i] = [CamProjection(G[1][i][0]), CamProjection(G[1][i][1])]; } return ([ICsX, ICsY]) } /* function that draws given 2D grid */ var DrawGrid = function(G) { for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('segment', [() => G()[0][i][0], () => G()[0][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); board.create('segment', [() => G()[1][i][0], () => G()[1][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); } } /* function that draws a line segment between points in 3D */ var DrawSegment = function(p, q, C, D) { board.create('segment', [() => CamProjection(p()), () => CamProjection(q())], { strokecolor: C, strokeWidth: 1, layer: 0, highlight: false, fixed: true, dash: D, }); } /* function that return parametrization of line segment between points in 3D */ var LineParam = function(p, q) { return function(t) { return VecSumVec(ScalarMultVec(t, p), ScalarMultVec((1 - t), q)) } } /* function that returns 3D coordinate lines as functions on [0,1] */ var GridParam = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = LineParam(G()[0][i][0], G()[0][i][1]); ICsY[i] = LineParam(G()[1][i][0], G()[1][i][1]); } return ([ICsX, ICsY]) } /* function that returns image of 3D coordinate lines as 2D functions on [0,1] */ var ImGridParam = function(P, F) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = function(t) { return CamProjection([P[0][i](t)[0], P[0][i](t)[1], F(P[0][i](t)[0], P[0][i](t)[1])]) }; ICsY[i] = function(t) { return CamProjection([P[1][i](t)[0], P[1][i](t)[1], F(P[1][i](t)[0], P[1][i](t)[1])]) }; } return ([ICsX, ICsY]) } /* function that draws image of given 3D grid under given function */ var DrawGraph = function(P, F, C) { let ICs = ImGridParam(P, F); for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('curve', [function(t) { return ICs[0][i](t)[0] }, function(t) { return ICs[0][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); board.create('curve', [function(t) { return ICs[1][i](t)[0] }, function(t) { return ICs[1][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); } } /* function that draws the point on the graph a given function for a given 3D point in the domain */ var DrawPoint = function(P, F, C) { let ICs3D = function() { return [P()[0], P()[1], F(P()[0], P()[1])] }; let ICs2D = function() { return CamProjection(ICs3D()) }; let IC = board.create('point', [() => ICs2D()[0], () => ICs2D()[1]], { color: Color4 }); IC.setAttribute({ showInfobox: false, layer: 3, color: C, highlight: false, name: function() { return '(' + JXG.toFixed( ICs3D()[0], 2) + ', ' + JXG.toFixed( ICs3D()[1], 2) + ', ' + JXG.toFixed( ICs3D()[2], 2) + ')' } }); DrawSegment(P, ICs3D, 'grey', 2); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- definition of grid in domain ---- */ var FidHigh = 600 / ResOfGrid; var FidLow = 200 / ResOfGrid; var SizeOfGrid = 1; var StepOfGrid = SizeOfGrid / ResOfGrid; /* 3D coordinates of grid in domain */ var GridDomain3D = function() { /* initial coordinates */ let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [ [-SizeOfGrid + i * StepOfGrid, -SizeOfGrid, -1], [-SizeOfGrid + i * StepOfGrid, SizeOfGrid, -1] ]; ICsY[i] = [ [-SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1], [SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1] ]; } return ([ICsX, ICsY]) } /* 2D coordinates of grid in domain */ var GridDomain = function() { return GridCoordsConv(GridDomain3D()) }; DrawGrid(GridDomain); var GridDomainParam = GridParam(GridDomain3D); /* ------------------------------------- */ /* ------------------------------------- */ /* draws point in domain at which the taylor polynomial is developed */ var PointOfDev = board.create('point', CamProjection([PointOfDevInit[0], PointOfDevInit[1], -1]),{visible: false}); var PointOfDev3D = function() { /* return CamInv([PointOfDev.X(), PointOfDev.Y(), -1]) */ return [PointOfDevInit[0], PointOfDevInit[1], -1] }; PointOfDev.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( PointOfDev3D()[0], 2) + ', ' + JXG.toFixed( PointOfDev3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ /* DrawSegment(PointOfDev3D, () => [PointOfDev3D()[0], 0, PointOfDev3D()[2]], 'grey', 2); */ /* DrawSegment(PointOfDev3D, () => [0, PointOfDev3D()[1], PointOfDev3D()[2]], 'grey', 2); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws arbitrary point in domain to visualize difference between (approximated) function and approximating function */ var ArbPointInDom = board.create('point', CamProjection([state[0], state[1], -1])); var ArbPointInDom3D = function() { return CamInv([ArbPointInDom.X(), ArbPointInDom.Y(), -1]) }; ArbPointInDom.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( ArbPointInDom3D()[0], 2) + ', ' + JXG.toFixed( ArbPointInDom3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ DrawSegment(ArbPointInDom3D, () => [ArbPointInDom3D()[0], 0, ArbPointInDom3D()[2]], 'grey', 2); DrawSegment(ArbPointInDom3D, () => [0, ArbPointInDom3D()[1], ArbPointInDom3D()[2]], 'grey', 2); /* ------------------------------------- */ /* ------------------------------------- */ var PointDefRad = board.create('point', CamProjection([0.3, 0.3, -1]), { showInfobox: false, layer: 4, color: Color3, name: false, visible : false, }); var PointDefRad3D = function() { return CamInv([PointDefRad.X(), PointDefRad.Y(), -1]) }; var RadiusOfCircle = function() { return EuclidDist(ArbPointInDom3D(), PointDefRad3D()) } PointDefRad.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return JXG.toFixed(RadiusOfCircle(), 2) } }); /* function that draws image of circle at given 3D point with given radius under given function */ var DrawCircleAtPoint = function(P, r, F, C) { let ICsX = function(t) { return P()[0] + r() * Math.cos(t) }; let ICsY = function(t) { return P()[1] + r() * Math.sin(t) }; let ICsZ = function(t) { return F(ICsX(t), ICsY(t)) }; let UCs = function(t) { return CamProjection([ICsX(t), ICsY(t), ICsZ(t)]) }; board.create('curve', [function(t) { return UCs(t)[0] }, function(t) { return UCs(t)[1] }, 0, 6.5 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh * 2, numberPointsLow: FidLow * 2, }); } /* draw circles at point in domain and their images */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, () => -1, Color3); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, ExFunc, Color4); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws images of points in domain */ /* DrawPoint(PointOfDev3D, ExFunc, 'Grey'); */ DrawPoint(ArbPointInDom3D, ExFunc, Color1); DrawPoint(ArbPointInDom3D, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); /* ------------------------------------- */ /* ------------------------------------- */ /* draw graph of (approximated) function */ DrawGraph(GridDomainParam, ExFunc, Color1) /* draw graph of approximating function */ DrawGraph(GridDomainParam, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4) /* ------------------------------------- */ /* ------------------------------------- */ /* returns point in domain close to given point */ var ReturnToDomain = function(a) { let Output = []; if (a[0] < -1) { Output[0] = -1; } else if (a[0] > 1) { Output[0] = 1; } else { Output[0] = a[0]; } if (a[1] < -1) { Output[1] = -1; } else if (a[1] > 1) { Output[1] = 1; } else { Output[1] = a[1]; } Output[2] = a[2]; return Output }; board.on('move', function() { board.suspendUpdate(); PointOfDev.moveTo(CamProjection(ReturnToDomain(PointOfDev3D()))); ArbPointInDom.moveTo(CamProjection(ReturnToDomain(ArbPointInDom3D()))); PointDefRad.moveTo(CamProjection(ReturnToDomain(PointDefRad3D()))); board.unsuspendUpdate(); }); /* ------------------------------------- */ /* ------------------------------------- */ var newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100, SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; var ClickIndicator = 0; var SendData = function(){ newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100,SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; stateInput.value = JSON.stringify(newState); stateInput.dispatchEvent(new Event('change')); } ArbPointInDom.on('drag', function () { SendData(); }); stateInput.style.display = 'none'; /* ------------------------------------- */ /* ------------------------------------- */ /* ensures that all draggable points (e.g. PointOfDev, ArbPointInDom) are rotatated accordingly when angles of projections are changed by sliders */ /* Buffer */ var PointOfDevBuffer = PointOfDev3D(); var ArbPointInDomBuffer = ArbPointInDom3D(); var PointDefRadBuffer = PointDefRad3D(); /* function that updates position of objects */ var UpdatePos = function(){ let ICs1 = CamProjection(PointOfDevBuffer); let ICs2 = CamProjection(ArbPointInDomBuffer); let ICs3 = CamProjection(PointDefRadBuffer); board.suspendUpdate(); PointOfDev.moveTo([ ICs1[0], ICs1[1] ]); ArbPointInDom.moveTo([ ICs2[0], ICs2[1] ]); PointDefRad.moveTo([ ICs3[0], ICs3[1] ]); board.unsuspendUpdate(); SendData(); } SliderAngleOfProj1.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj1.on('drag', function() { UpdatePos() }); }); SliderAngleOfProj2.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj2.on('drag', function() { UpdatePos() }); }); ArbPointInDom.on('down', function() { PointDefRadBuffer = VecSubVec(ArbPointInDom3D(), PointDefRad3D()); ArbPointInDom.on('drag', function() { let ICs = CamProjection(VecSubVec(ArbPointInDom3D(), PointDefRadBuffer)); board.suspendUpdate(); PointDefRad.moveTo([ ICs[0], ICs[1] ]); board.unsuspendUpdate(); }); }); /* ------------------------------------- */ /* ------------------------------------- */ [[/jsxgraph]]

Geben Sie mithilfe der obigen Abbildung einen Entwicklungspunkt \((a,b) \in [-1,1]^2\) des Taylorpolynoms \(T_{{@Order@}}{@FuncName@}\) von \({@FuncName@}\) an. Verschieben Sie dazu das Funktionsargument in der \(x_1 x_2\)-Ebene (Punkt ●) an die Position eines Entwicklungspunktes. [[input:ans1]]

[[validation:ans1]]

[[feedback:prt1]]

\(\displaystyle {@FuncName@} : \mathbb{R}^2 \to \mathbb{R}\), \((x,y) \mapsto {@SelectedFunction@}\)

und ein Taylorpolynom \(T_{{@Order@}}{@FuncName@}\) {@NumberWords1[Order+1]@} Grades von \({@FuncName@}\).

Die folgende Abbildung zeigt den Graphen von \({@FuncName@}\) (Graph □) und den Graphen von \(T_{{@Order@}}{@FuncName@}\) (Graph □) auf dem Einheitsquadrat \([-1,1]^2\) in der \(x_1 x_2\)-Ebene von \(\mathbb{R}^3\) aufgezeichnet gegen die \(x_3\)-Achse. Sie können das Funktionsargument in der \(x_1 x_2\)-Ebene verschieben (Punkt ●). Die zugehörigen Funktionswerte unter \({@FuncName@}\) und \(T_{@FuncName@}\) werden Ihnen durch Angabe der zugehörigen Punkte auf den Graphen angezeigt (Punkt ● bzw. Punkt ●).

[[jsxgraph input-ref-ans1="stateRef"]] [[jsxgraph input-ref-ans1="stateRef"]] const board = JXG.JSXGraph.initBoard(divid, { boundingbox: [-1.9, 1.9, 1.9, -1.9], axis: false, showCopyright: false, keepaspectratio: true, /* showNavigation: false, */ fixed: true }); /* ------------------------------------- */ /* --------------------------------------*/ /* ---- STACK to JSXGraph interface ---- */ var state = {#Argument#}; state[2] = {#SliderPos#}[0]; /* position of slider 'SliderAngleOfProj1' */ state[3] = {#SliderPos#}[1];; /* position of slider 'SliderAngleOfProj2' */ var stateInput = document.getElementById(stateRef); if (stateInput.value && stateInput.value != '') { state = JSON.parse(stateInput.value); } /* definition of (approximated) function */ var ExFunc = board.jc.snippet('{#SelectedFunction#}', true, 'x,y', true); /* definition of approximating function */ var StrApproxFunc = '{#ListOfFunctions#}'.substring(1, '{#ListOfFunctions#}'.length - 1); var SplitStrApproxFunc = StrApproxFunc.split(","); var MaxOrder = SplitStrApproxFunc.length - 1; /* order / degree of the polynomial approximation */ var Order = {#Order#}; /* initial coordinates of the point of development */ var PointOfDevInit = [{#PointOfDev#}[0], {#PointOfDev#}[1]]; /* resolution of the grid */ var ResOfGrid = {#ResOfGrid#}; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- colors ---- */ const Color1 = `hsl(${223}, ${100}%, ${70}%)`; /* #648FFF */ const Color2 = `hsl(${251}, ${83}%, ${65}%)`; /* #785EF0 */ const Color3 = `hsl(${331}, ${72}%, ${51}%)`; /* #DC267F */ const Color4 = `hsl(${23}, ${100}%, ${50}%)`; /* #FE6100 */ const Color5 = `hsl(${41}, ${100}%, ${50}%)`; /* #FFB000 */ /* ------------------------------------- */ /* --------------------------------------*/ /* draws slider controlling the order of the approximating function */ var SliderOrder = board.create('slider', [ [-1.3, -1.75], [1.2, -1.75], [0, Order, MaxOrder] ], { withLabel: true, highlight: false, name: 'order', snapWidth: 1, visible: false, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); /* order of the approximating function */ var Order = function(n) { return n <= SliderOrder.Value() }; /* definition of approximating function */ var ApproxFuncList = []; for (var i = 0; i <= MaxOrder; i++) { ApproxFuncList.push(board.jc.snippet(SplitStrApproxFunc[i], true, 'x,y,a,b', true)) } var ApproxFunc = function(x, y, a, b) { return ApproxFuncList[SliderOrder.Value()](x, y, a, b) }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- general operations on vectors and matrices ---- */ /* vector product */ var NormalVec = function(a, b) { return [ ((a[1] * b[2]) - (a[2] * b[1])), ((a[2] * b[0]) - (a[0] * b[2])), ((a[0] * b[1]) - (a[1] * b[0])) ] }; /* euclidean scalar product */ var ScalarProd = function(a, b) { return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]) }; /* euclidean norm of vector */ var EuclidNorm = function(a) { return Math.sqrt(ScalarProd(a, a)) }; /* multiplication of scalar vectors */ var ScalarMultVec = function(x, a) { return [x * a[0], x * a[1], x * a[2]] }; /* entrywise multiplication of vectors */ var VecMultVec = function(a, b) { return [a[0] * b[0], a[1] * b[1], a[2] * b[2]] }; /* sum of vectors */ var VecSumVec = function(a, b) { return [a[0] + b[0], a[1] + b[1], a[2] + b[2]] }; /* difference of vectors */ var VecSubVec = function(a, b) { return [a[0] - b[0], a[1] - b[1], a[2] - b[2]] }; /* euclidean distance of vectors */ var EuclidDist = function(a, b) { return EuclidNorm(VecSubVec(a, b)) } /* multiplication of scalar matrix*/ var ScalarMultMat = function(x, A) { return [ScalarMultVec(x, A[0]), ScalarMultVec(x, A[1]), ScalarMultVec(x, A[2])] } /* normalizes non-zero vector */ var UnitVec = function(a) { let x = Math.sqrt(ScalarProd(a, a)); let b = ScalarMultVec(x, a); return b; } /* determinant of matrix*/ var DetOfMat = function(A) { return ScalarProd(A[0], NormalVec(A[1], A[2])) }; /* transpose of matrix */ var TranspOfMat = function(A) { let B1 = [A[0][0], A[1][0], A[2][0]]; let B2 = [A[0][1], A[1][1], A[2][1]]; let B3 = [A[0][2], A[1][2], A[2][2]]; return [B1, B2, B3]; } /* pre-inverse of matrix */ var PreInvOfMat = function(A) { let B = TranspOfMat(A); let b1 = NormalVec(B[1], B[2]); let b2 = NormalVec(B[2], B[0]); let b3 = NormalVec(B[0], B[1]); let C = [b1, b2, b3]; return C; }; /* inverse of matrix */ var InvOfMat = function(A) { let B = PreInvOfMat(A); let b = 1 / DetOfMat(A); return ScalarMultMat(b, B); }; /* matrix applied as linear map to vector */ var MatMultVec = function(a, A) { let b1 = ScalarProd(A[0], a); let b2 = ScalarProd(A[1], a); let b3 = ScalarProd(A[2], a); return [b1, b2, b3]; } /* ------------------------------------- */ /* --------------------------------------*/ /* ---- sliders for controlling the camera ---- */ var SliderAngleOfProj1 = board.create('slider', [ [-1.75, -1.3], [-1.75, 1.3], [0.005, state[2], 0.5] ], { withLabel: true, highlight: false, name: 'el', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var SliderAngleOfProj2 = board.create('slider', [ [-1.3, -1.6], [1.2, -1.6], [-1, state[3], 1] ], { withLabel: true, highlight: false, name: 'az', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var AOP1 = function() { return SliderAngleOfProj1.Value() * Math.PI }; var AOP2 = function() { return (SliderAngleOfProj2.Value() + 1) * Math.PI }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- 3D to 2D projection ---- */ /* angles for 2D projection */ var cpsi = function() { return Math.cos(AOP2()) }; var spsi = function() { return Math.sin(AOP2()) }; var cphi = function() { return Math.cos(AOP1()) }; var sphi = function() { return Math.sin(AOP1()) }; /* matrix for 2D projection */ var ProjMat2D = function() { return [ [(-1) * cpsi(), spsi(), 0], [(-1) * spsi() * sphi(), (-1) * cpsi() * sphi(), cphi()], [0, 0, 1] ] }; /* returns 2D coordinates of 3D objects */ var CamProjection = function(a) { let Output = MatMultVec(a, ProjMat2D()); return [Output[0], Output[1]]; } var CamInv = function(a) { let Output = MatMultVec([a[0], a[1], a[2]], InvOfMat(ProjMat2D())); return [Output[0], Output[1], Output[2]]; } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- coordinate system ---- */ /* 3D coordinates of arrows in dicating the coordinate system */ var AxisCoords = function(h) { /* initial coordinates */ let ICs0 = [0, 0, h]; let ICs1 = [1.2, 0, h]; let ICs2 = [0, 1.2, h]; let ICs3 = [0, 0, 1.2 + h]; /* ---- */ let ICs4 = [-1.1, 0, h]; let ICs5 = [0, -1.1, h]; return [ CamProjection(ICs0), CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3), CamProjection(ICs4), CamProjection(ICs5), ] }; /* 3D coordinates of labels of coordinate system*/ var AxisLabelPos = function(h) { /* initial coordinates */ let ICs1 = [1.3, 0, h]; let ICs2 = [0, 1.3, h]; let ICs3 = [0, 0, 1.3 + h]; return [ CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3) ] }; /* labels of coordinate system*/ var AxisLabel = ['x_1', 'x_2', 'x_3']; /* draws arrows indicating the coordinates system */ for (let i = 0; i <= 2; i++) { board.create('arrow', [ [() => AxisCoords(0)[0][0], () => AxisCoords(0)[0][1]], [() => AxisCoords(0)[i + 1][0], () => AxisCoords(0)[i + 1][1]] ], { strokecolor: 'black', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(0)[i][0], () => AxisLabelPos(0)[i][1], AxisLabel[i]]); } /* draws arrows indicating the coordinates system in the domain */ for (let i = 0; i <= 1; i++) { board.create('arrow', [ [() => AxisCoords(-1)[i + 4][0], () => AxisCoords(-1)[i + 4][1]], [() => AxisCoords(-1)[i + 1][0], () => AxisCoords(-1)[i + 1][1]] ], { strokecolor: 'grey', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(-1)[i][0], () => AxisLabelPos(-1)[i][1], AxisLabel[i]], { strokeColor: 'grey', }); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- collection of functions that draw grids and graphs ---- */ /* function that returns 2D coordinates of given 'grid' */ var GridCoordsConv = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [CamProjection(G[0][i][0]), CamProjection(G[0][i][1])]; ICsY[i] = [CamProjection(G[1][i][0]), CamProjection(G[1][i][1])]; } return ([ICsX, ICsY]) } /* function that draws given 2D grid */ var DrawGrid = function(G) { for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('segment', [() => G()[0][i][0], () => G()[0][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); board.create('segment', [() => G()[1][i][0], () => G()[1][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); } } /* function that draws a line segment between points in 3D */ var DrawSegment = function(p, q, C, D) { board.create('segment', [() => CamProjection(p()), () => CamProjection(q())], { strokecolor: C, strokeWidth: 1, layer: 0, highlight: false, fixed: true, dash: D, }); } /* function that return parametrization of line segment between points in 3D */ var LineParam = function(p, q) { return function(t) { return VecSumVec(ScalarMultVec(t, p), ScalarMultVec((1 - t), q)) } } /* function that returns 3D coordinate lines as functions on [0,1] */ var GridParam = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = LineParam(G()[0][i][0], G()[0][i][1]); ICsY[i] = LineParam(G()[1][i][0], G()[1][i][1]); } return ([ICsX, ICsY]) } /* function that returns image of 3D coordinate lines as 2D functions on [0,1] */ var ImGridParam = function(P, F) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = function(t) { return CamProjection([P[0][i](t)[0], P[0][i](t)[1], F(P[0][i](t)[0], P[0][i](t)[1])]) }; ICsY[i] = function(t) { return CamProjection([P[1][i](t)[0], P[1][i](t)[1], F(P[1][i](t)[0], P[1][i](t)[1])]) }; } return ([ICsX, ICsY]) } /* function that draws image of given 3D grid under given function */ var DrawGraph = function(P, F, C) { let ICs = ImGridParam(P, F); for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('curve', [function(t) { return ICs[0][i](t)[0] }, function(t) { return ICs[0][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); board.create('curve', [function(t) { return ICs[1][i](t)[0] }, function(t) { return ICs[1][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); } } /* function that draws the point on the graph a given function for a given 3D point in the domain */ var DrawPoint = function(P, F, C) { let ICs3D = function() { return [P()[0], P()[1], F(P()[0], P()[1])] }; let ICs2D = function() { return CamProjection(ICs3D()) }; let IC = board.create('point', [() => ICs2D()[0], () => ICs2D()[1]], { color: Color4 }); IC.setAttribute({ showInfobox: false, layer: 3, color: C, highlight: false, name: function() { return '(' + JXG.toFixed( ICs3D()[0], 2) + ', ' + JXG.toFixed( ICs3D()[1], 2) + ', ' + JXG.toFixed( ICs3D()[2], 2) + ')' } }); DrawSegment(P, ICs3D, 'grey', 2); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- definition of grid in domain ---- */ var FidHigh = 600 / ResOfGrid; var FidLow = 200 / ResOfGrid; var SizeOfGrid = 1; var StepOfGrid = SizeOfGrid / ResOfGrid; /* 3D coordinates of grid in domain */ var GridDomain3D = function() { /* initial coordinates */ let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [ [-SizeOfGrid + i * StepOfGrid, -SizeOfGrid, -1], [-SizeOfGrid + i * StepOfGrid, SizeOfGrid, -1] ]; ICsY[i] = [ [-SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1], [SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1] ]; } return ([ICsX, ICsY]) } /* 2D coordinates of grid in domain */ var GridDomain = function() { return GridCoordsConv(GridDomain3D()) }; DrawGrid(GridDomain); var GridDomainParam = GridParam(GridDomain3D); /* ------------------------------------- */ /* ------------------------------------- */ /* draws point in domain at which the taylor polynomial is developed */ var PointOfDev = board.create('point', CamProjection([PointOfDevInit[0], PointOfDevInit[1], -1]),{visible: false}); var PointOfDev3D = function() { /* return CamInv([PointOfDev.X(), PointOfDev.Y(), -1]) */ return [PointOfDevInit[0], PointOfDevInit[1], -1] }; PointOfDev.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( PointOfDev3D()[0], 2) + ', ' + JXG.toFixed( PointOfDev3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ /* DrawSegment(PointOfDev3D, () => [PointOfDev3D()[0], 0, PointOfDev3D()[2]], 'grey', 2); */ /* DrawSegment(PointOfDev3D, () => [0, PointOfDev3D()[1], PointOfDev3D()[2]], 'grey', 2); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws arbitrary point in domain to visualize difference between (approximated) function and approximating function */ var ArbPointInDom = board.create('point', CamProjection([state[0], state[1], -1])); var ArbPointInDom3D = function() { return CamInv([ArbPointInDom.X(), ArbPointInDom.Y(), -1]) }; ArbPointInDom.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( ArbPointInDom3D()[0], 2) + ', ' + JXG.toFixed( ArbPointInDom3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ DrawSegment(ArbPointInDom3D, () => [ArbPointInDom3D()[0], 0, ArbPointInDom3D()[2]], 'grey', 2); DrawSegment(ArbPointInDom3D, () => [0, ArbPointInDom3D()[1], ArbPointInDom3D()[2]], 'grey', 2); /* ------------------------------------- */ /* ------------------------------------- */ var PointDefRad = board.create('point', CamProjection([0.3, 0.3, -1]), { showInfobox: false, layer: 4, color: Color3, name: false, visible : false, }); var PointDefRad3D = function() { return CamInv([PointDefRad.X(), PointDefRad.Y(), -1]) }; var RadiusOfCircle = function() { return EuclidDist(ArbPointInDom3D(), PointDefRad3D()) } PointDefRad.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return JXG.toFixed(RadiusOfCircle(), 2) } }); /* function that draws image of circle at given 3D point with given radius under given function */ var DrawCircleAtPoint = function(P, r, F, C) { let ICsX = function(t) { return P()[0] + r() * Math.cos(t) }; let ICsY = function(t) { return P()[1] + r() * Math.sin(t) }; let ICsZ = function(t) { return F(ICsX(t), ICsY(t)) }; let UCs = function(t) { return CamProjection([ICsX(t), ICsY(t), ICsZ(t)]) }; board.create('curve', [function(t) { return UCs(t)[0] }, function(t) { return UCs(t)[1] }, 0, 6.5 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh * 2, numberPointsLow: FidLow * 2, }); } /* draw circles at point in domain and their images */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, () => -1, Color3); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, ExFunc, Color4); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws images of points in domain */ /* DrawPoint(PointOfDev3D, ExFunc, 'Grey'); */ DrawPoint(ArbPointInDom3D, ExFunc, Color1); DrawPoint(ArbPointInDom3D, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); /* ------------------------------------- */ /* ------------------------------------- */ /* draw graph of (approximated) function */ DrawGraph(GridDomainParam, ExFunc, Color1) /* draw graph of approximating function */ DrawGraph(GridDomainParam, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4) /* ------------------------------------- */ /* ------------------------------------- */ board.on('move', function() { board.suspendUpdate(); /* returns point in domain close to given point */ const ReturnToDomain = function(a) { let Output = []; if (a[0] < -1) { Output[0] = -1; } else if (a[0] > 1) { Output[0] = 1; } else { Output[0] = a[0]; } if (a[1] < -1) { Output[1] = -1; } else if (a[1] > 1) { Output[1] = 1; } else { Output[1] = a[1]; } Output[2] = a[2]; return Output }; PointOfDev.moveTo(CamProjection(ReturnToDomain(PointOfDev3D()))); ArbPointInDom.moveTo(CamProjection(ReturnToDomain(ArbPointInDom3D()))); PointDefRad.moveTo(CamProjection(ReturnToDomain(PointDefRad3D()))); board.unsuspendUpdate(); }); /* ------------------------------------- */ /* ------------------------------------- */ var newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100, SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; var ClickIndicator = 0; var SendData = function(){ newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100,SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; stateInput.value = JSON.stringify(newState); stateInput.dispatchEvent(new Event('change')); } ArbPointInDom.on('drag', function () { SendData(); }); stateInput.style.display = 'none'; /* ------------------------------------- */ /* ------------------------------------- */ /* ensures that all draggable points (e.g. PointOfDev, ArbPointInDom) are rotatated accordingly when angles of projections are changed by sliders */ /* Buffer */ var PointOfDevBuffer = PointOfDev3D(); var ArbPointInDomBuffer = ArbPointInDom3D(); var PointDefRadBuffer = PointDefRad3D(); /* function that updates position of objects */ var UpdatePos = function(){ let ICs1 = CamProjection(PointOfDevBuffer); let ICs2 = CamProjection(ArbPointInDomBuffer); let ICs3 = CamProjection(PointDefRadBuffer); board.suspendUpdate(); PointOfDev.moveTo([ ICs1[0], ICs1[1] ]); ArbPointInDom.moveTo([ ICs2[0], ICs2[1] ]); PointDefRad.moveTo([ ICs3[0], ICs3[1] ]); board.unsuspendUpdate(); SendData(); } SliderAngleOfProj1.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj1.on('drag', function() { UpdatePos() }); }); SliderAngleOfProj2.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj2.on('drag', function() { UpdatePos() }); }); ArbPointInDom.on('down', function() { PointDefRadBuffer = VecSubVec(ArbPointInDom3D(), PointDefRad3D()); ArbPointInDom.on('drag', function() { let ICs = CamProjection(VecSubVec(ArbPointInDom3D(), PointDefRadBuffer)); board.suspendUpdate(); PointDefRad.moveTo([ ICs[0], ICs[1] ]); board.unsuspendUpdate(); }); }); /* ------------------------------------- */ /* ------------------------------------- */ [[/jsxgraph]]

Geben Sie mithilfe der obigen Abbildung einen Entwicklungspunkt \((a,b) \in [-1,1]^2\) des Taylorpolynoms \(T_{{@Order@}}{@FuncName@}\) von \({@FuncName@}\) an. Verschieben Sie dazu das Funktionsargument in der \(x_1 x_2\)-Ebene (Punkt ●) an die Position eines Entwicklungspunktes. [[input:ans1]]

[[validation:ans1]]

[[feedback:prt1]]

\(\displaystyle {@FuncName@} : \mathbb{R}^2 \to \mathbb{R}\), \((x,y) \mapsto {@SelectedFunction@}\)

und ein Taylorpolynom \(T_{{@Order@}}{@FuncName@}\) {@NumberWords1[Order+1]@} Grades von \({@FuncName@}\).

Die folgende Abbildung zeigt den Graphen von \({@FuncName@}\) (Graph □) und den Graphen von \(T_{{@Order@}}{@FuncName@}\) (Graph □) auf dem Einheitsquadrat \([-1,1]^2\) in der \(x_1 x_2\)-Ebene von \(\mathbb{R}^3\) aufgezeichnet gegen die \(x_3\)-Achse. Sie können das Funktionsargument in der \(x_1 x_2\)-Ebene verschieben (Punkt ●). Die zugehörigen Funktionswerte unter \({@FuncName@}\) und \(T_{@FuncName@}\) werden Ihnen durch Angabe der zugehörigen Punkte auf den Graphen angezeigt (Punkt ● bzw. Punkt ●).

[[jsxgraph input-ref-ans1="stateRef"]] [[jsxgraph input-ref-ans1="stateRef"]] const board = JXG.JSXGraph.initBoard(divid, { boundingbox: [-1.9, 1.9, 1.9, -1.9], axis: false, showCopyright: false, keepaspectratio: true, /* showNavigation: false, */ fixed: true }); /* ------------------------------------- */ /* --------------------------------------*/ /* ---- STACK to JSXGraph interface ---- */ var state = {#Argument#}; state[2] = {#SliderPos#}[0]; /* position of slider 'SliderAngleOfProj1' */ state[3] = {#SliderPos#}[1];; /* position of slider 'SliderAngleOfProj2' */ var stateInput = document.getElementById(stateRef); if (stateInput.value && stateInput.value != '') { state = JSON.parse(stateInput.value); } /* definition of (approximated) function */ var ExFunc = board.jc.snippet('{#SelectedFunction#}', true, 'x,y', true); /* definition of approximating function */ var StrApproxFunc = '{#ListOfFunctions#}'.substring(1, '{#ListOfFunctions#}'.length - 1); var SplitStrApproxFunc = StrApproxFunc.split(","); var MaxOrder = SplitStrApproxFunc.length - 1; /* order / degree of the polynomial approximation */ var Order = {#Order#}; /* initial coordinates of the point of development */ var PointOfDevInit = [{#PointOfDev#}[0], {#PointOfDev#}[1]]; /* resolution of the grid */ var ResOfGrid = {#ResOfGrid#}; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- colors ---- */ const Color1 = `hsl(${223}, ${100}%, ${70}%)`; /* #648FFF */ const Color2 = `hsl(${251}, ${83}%, ${65}%)`; /* #785EF0 */ const Color3 = `hsl(${331}, ${72}%, ${51}%)`; /* #DC267F */ const Color4 = `hsl(${23}, ${100}%, ${50}%)`; /* #FE6100 */ const Color5 = `hsl(${41}, ${100}%, ${50}%)`; /* #FFB000 */ /* ------------------------------------- */ /* --------------------------------------*/ /* draws slider controlling the order of the approximating function */ var SliderOrder = board.create('slider', [ [-1.3, -1.75], [1.2, -1.75], [0, Order, MaxOrder] ], { withLabel: true, highlight: false, name: 'order', snapWidth: 1, visible: false, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); /* order of the approximating function */ var Order = function(n) { return n <= SliderOrder.Value() }; /* definition of approximating function */ var ApproxFuncList = []; for (var i = 0; i <= MaxOrder; i++) { ApproxFuncList.push(board.jc.snippet(SplitStrApproxFunc[i], true, 'x,y,a,b', true)) } var ApproxFunc = function(x, y, a, b) { return ApproxFuncList[SliderOrder.Value()](x, y, a, b) }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- general operations on vectors and matrices ---- */ /* vector product */ var NormalVec = function(a, b) { return [ ((a[1] * b[2]) - (a[2] * b[1])), ((a[2] * b[0]) - (a[0] * b[2])), ((a[0] * b[1]) - (a[1] * b[0])) ] }; /* euclidean scalar product */ var ScalarProd = function(a, b) { return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]) }; /* euclidean norm of vector */ var EuclidNorm = function(a) { return Math.sqrt(ScalarProd(a, a)) }; /* multiplication of scalar vectors */ var ScalarMultVec = function(x, a) { return [x * a[0], x * a[1], x * a[2]] }; /* entrywise multiplication of vectors */ var VecMultVec = function(a, b) { return [a[0] * b[0], a[1] * b[1], a[2] * b[2]] }; /* sum of vectors */ var VecSumVec = function(a, b) { return [a[0] + b[0], a[1] + b[1], a[2] + b[2]] }; /* difference of vectors */ var VecSubVec = function(a, b) { return [a[0] - b[0], a[1] - b[1], a[2] - b[2]] }; /* euclidean distance of vectors */ var EuclidDist = function(a, b) { return EuclidNorm(VecSubVec(a, b)) } /* multiplication of scalar matrix*/ var ScalarMultMat = function(x, A) { return [ScalarMultVec(x, A[0]), ScalarMultVec(x, A[1]), ScalarMultVec(x, A[2])] } /* normalizes non-zero vector */ var UnitVec = function(a) { let x = Math.sqrt(ScalarProd(a, a)); let b = ScalarMultVec(x, a); return b; } /* determinant of matrix*/ var DetOfMat = function(A) { return ScalarProd(A[0], NormalVec(A[1], A[2])) }; /* transpose of matrix */ var TranspOfMat = function(A) { let B1 = [A[0][0], A[1][0], A[2][0]]; let B2 = [A[0][1], A[1][1], A[2][1]]; let B3 = [A[0][2], A[1][2], A[2][2]]; return [B1, B2, B3]; } /* pre-inverse of matrix */ var PreInvOfMat = function(A) { let B = TranspOfMat(A); let b1 = NormalVec(B[1], B[2]); let b2 = NormalVec(B[2], B[0]); let b3 = NormalVec(B[0], B[1]); let C = [b1, b2, b3]; return C; }; /* inverse of matrix */ var InvOfMat = function(A) { let B = PreInvOfMat(A); let b = 1 / DetOfMat(A); return ScalarMultMat(b, B); }; /* matrix applied as linear map to vector */ var MatMultVec = function(a, A) { let b1 = ScalarProd(A[0], a); let b2 = ScalarProd(A[1], a); let b3 = ScalarProd(A[2], a); return [b1, b2, b3]; } /* ------------------------------------- */ /* --------------------------------------*/ /* ---- sliders for controlling the camera ---- */ var SliderAngleOfProj1 = board.create('slider', [ [-1.75, -1.3], [-1.75, 1.3], [0.005, state[2], 0.5] ], { withLabel: true, highlight: false, name: 'el', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var SliderAngleOfProj2 = board.create('slider', [ [-1.3, -1.6], [1.2, -1.6], [-1, state[3], 1] ], { withLabel: true, highlight: false, name: 'az', postLabel: ' π', snapWidth : 0.01, moveOnUp: false, point1: {frozen: true}, point2: {frozen: true}, }); var AOP1 = function() { return SliderAngleOfProj1.Value() * Math.PI }; var AOP2 = function() { return (SliderAngleOfProj2.Value() + 1) * Math.PI }; /* ------------------------------------- */ /* --------------------------------------*/ /* ---- 3D to 2D projection ---- */ /* angles for 2D projection */ var cpsi = function() { return Math.cos(AOP2()) }; var spsi = function() { return Math.sin(AOP2()) }; var cphi = function() { return Math.cos(AOP1()) }; var sphi = function() { return Math.sin(AOP1()) }; /* matrix for 2D projection */ var ProjMat2D = function() { return [ [(-1) * cpsi(), spsi(), 0], [(-1) * spsi() * sphi(), (-1) * cpsi() * sphi(), cphi()], [0, 0, 1] ] }; /* returns 2D coordinates of 3D objects */ var CamProjection = function(a) { let Output = MatMultVec(a, ProjMat2D()); return [Output[0], Output[1]]; } var CamInv = function(a) { let Output = MatMultVec([a[0], a[1], a[2]], InvOfMat(ProjMat2D())); return [Output[0], Output[1], Output[2]]; } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- coordinate system ---- */ /* 3D coordinates of arrows in dicating the coordinate system */ var AxisCoords = function(h) { /* initial coordinates */ let ICs0 = [0, 0, h]; let ICs1 = [1.2, 0, h]; let ICs2 = [0, 1.2, h]; let ICs3 = [0, 0, 1.2 + h]; /* ---- */ let ICs4 = [-1.1, 0, h]; let ICs5 = [0, -1.1, h]; return [ CamProjection(ICs0), CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3), CamProjection(ICs4), CamProjection(ICs5), ] }; /* 3D coordinates of labels of coordinate system*/ var AxisLabelPos = function(h) { /* initial coordinates */ let ICs1 = [1.3, 0, h]; let ICs2 = [0, 1.3, h]; let ICs3 = [0, 0, 1.3 + h]; return [ CamProjection(ICs1), CamProjection(ICs2), CamProjection(ICs3) ] }; /* labels of coordinate system*/ var AxisLabel = ['x_1', 'x_2', 'x_3']; /* draws arrows indicating the coordinates system */ for (let i = 0; i <= 2; i++) { board.create('arrow', [ [() => AxisCoords(0)[0][0], () => AxisCoords(0)[0][1]], [() => AxisCoords(0)[i + 1][0], () => AxisCoords(0)[i + 1][1]] ], { strokecolor: 'black', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(0)[i][0], () => AxisLabelPos(0)[i][1], AxisLabel[i]]); } /* draws arrows indicating the coordinates system in the domain */ for (let i = 0; i <= 1; i++) { board.create('arrow', [ [() => AxisCoords(-1)[i + 4][0], () => AxisCoords(-1)[i + 4][1]], [() => AxisCoords(-1)[i + 1][0], () => AxisCoords(-1)[i + 1][1]] ], { strokecolor: 'grey', strokeWidth: 1.2, layer: 1, highlight: false, fixed: true, }); /* draws labels of coordinates system */ board.create('text', [() => AxisLabelPos(-1)[i][0], () => AxisLabelPos(-1)[i][1], AxisLabel[i]], { strokeColor: 'grey', }); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- collection of functions that draw grids and graphs ---- */ /* function that returns 2D coordinates of given 'grid' */ var GridCoordsConv = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [CamProjection(G[0][i][0]), CamProjection(G[0][i][1])]; ICsY[i] = [CamProjection(G[1][i][0]), CamProjection(G[1][i][1])]; } return ([ICsX, ICsY]) } /* function that draws given 2D grid */ var DrawGrid = function(G) { for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('segment', [() => G()[0][i][0], () => G()[0][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); board.create('segment', [() => G()[1][i][0], () => G()[1][i][1]], { strokecolor: 'lightgrey', strokeWidth: 1, layer: 0, highlight: false, fixed: true, }); } } /* function that draws a line segment between points in 3D */ var DrawSegment = function(p, q, C, D) { board.create('segment', [() => CamProjection(p()), () => CamProjection(q())], { strokecolor: C, strokeWidth: 1, layer: 0, highlight: false, fixed: true, dash: D, }); } /* function that return parametrization of line segment between points in 3D */ var LineParam = function(p, q) { return function(t) { return VecSumVec(ScalarMultVec(t, p), ScalarMultVec((1 - t), q)) } } /* function that returns 3D coordinate lines as functions on [0,1] */ var GridParam = function(G) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = LineParam(G()[0][i][0], G()[0][i][1]); ICsY[i] = LineParam(G()[1][i][0], G()[1][i][1]); } return ([ICsX, ICsY]) } /* function that returns image of 3D coordinate lines as 2D functions on [0,1] */ var ImGridParam = function(P, F) { let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = function(t) { return CamProjection([P[0][i](t)[0], P[0][i](t)[1], F(P[0][i](t)[0], P[0][i](t)[1])]) }; ICsY[i] = function(t) { return CamProjection([P[1][i](t)[0], P[1][i](t)[1], F(P[1][i](t)[0], P[1][i](t)[1])]) }; } return ([ICsX, ICsY]) } /* function that draws image of given 3D grid under given function */ var DrawGraph = function(P, F, C) { let ICs = ImGridParam(P, F); for (let i = 0; i <= 2 * ResOfGrid; i++) { board.create('curve', [function(t) { return ICs[0][i](t)[0] }, function(t) { return ICs[0][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); board.create('curve', [function(t) { return ICs[1][i](t)[0] }, function(t) { return ICs[1][i](t)[1] }, 0, 1 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh, numberPointsLow: FidLow, }); } } /* function that draws the point on the graph a given function for a given 3D point in the domain */ var DrawPoint = function(P, F, C) { let ICs3D = function() { return [P()[0], P()[1], F(P()[0], P()[1])] }; let ICs2D = function() { return CamProjection(ICs3D()) }; let IC = board.create('point', [() => ICs2D()[0], () => ICs2D()[1]], { color: Color4 }); IC.setAttribute({ showInfobox: false, layer: 3, color: C, highlight: false, name: function() { return '(' + JXG.toFixed( ICs3D()[0], 2) + ', ' + JXG.toFixed( ICs3D()[1], 2) + ', ' + JXG.toFixed( ICs3D()[2], 2) + ')' } }); DrawSegment(P, ICs3D, 'grey', 2); } /* ------------------------------------- */ /* ------------------------------------- */ /* ---- definition of grid in domain ---- */ var FidHigh = 600 / ResOfGrid; var FidLow = 200 / ResOfGrid; var SizeOfGrid = 1; var StepOfGrid = SizeOfGrid / ResOfGrid; /* 3D coordinates of grid in domain */ var GridDomain3D = function() { /* initial coordinates */ let ICsX = []; let ICsY = []; for (let i = 0; i <= 2 * ResOfGrid; i++) { ICsX[i] = [ [-SizeOfGrid + i * StepOfGrid, -SizeOfGrid, -1], [-SizeOfGrid + i * StepOfGrid, SizeOfGrid, -1] ]; ICsY[i] = [ [-SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1], [SizeOfGrid, -SizeOfGrid + i * StepOfGrid, -1] ]; } return ([ICsX, ICsY]) } /* 2D coordinates of grid in domain */ var GridDomain = function() { return GridCoordsConv(GridDomain3D()) }; DrawGrid(GridDomain); var GridDomainParam = GridParam(GridDomain3D); /* ------------------------------------- */ /* ------------------------------------- */ /* draws point in domain at which the taylor polynomial is developed */ var PointOfDev = board.create('point', CamProjection([PointOfDevInit[0], PointOfDevInit[1], -1]),{visible: false}); var PointOfDev3D = function() { /* return CamInv([PointOfDev.X(), PointOfDev.Y(), -1]) */ return [PointOfDevInit[0], PointOfDevInit[1], -1] }; PointOfDev.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( PointOfDev3D()[0], 2) + ', ' + JXG.toFixed( PointOfDev3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ /* DrawSegment(PointOfDev3D, () => [PointOfDev3D()[0], 0, PointOfDev3D()[2]], 'grey', 2); */ /* DrawSegment(PointOfDev3D, () => [0, PointOfDev3D()[1], PointOfDev3D()[2]], 'grey', 2); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws arbitrary point in domain to visualize difference between (approximated) function and approximating function */ var ArbPointInDom = board.create('point', CamProjection([state[0], state[1], -1])); var ArbPointInDom3D = function() { return CamInv([ArbPointInDom.X(), ArbPointInDom.Y(), -1]) }; ArbPointInDom.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return '(' + JXG.toFixed( ArbPointInDom3D()[0], 2) + ', ' + JXG.toFixed( ArbPointInDom3D()[1], 2) + ')' } }); /* draws dashed lines to indicate position of PointOfDev and PointOnGraph*/ DrawSegment(ArbPointInDom3D, () => [ArbPointInDom3D()[0], 0, ArbPointInDom3D()[2]], 'grey', 2); DrawSegment(ArbPointInDom3D, () => [0, ArbPointInDom3D()[1], ArbPointInDom3D()[2]], 'grey', 2); /* ------------------------------------- */ /* ------------------------------------- */ var PointDefRad = board.create('point', CamProjection([0.3, 0.3, -1]), { showInfobox: false, layer: 4, color: Color3, name: false, visible : false, }); var PointDefRad3D = function() { return CamInv([PointDefRad.X(), PointDefRad.Y(), -1]) }; var RadiusOfCircle = function() { return EuclidDist(ArbPointInDom3D(), PointDefRad3D()) } PointDefRad.setAttribute({ showInfobox: false, layer: 4, color: Color3, name: function() { return JXG.toFixed(RadiusOfCircle(), 2) } }); /* function that draws image of circle at given 3D point with given radius under given function */ var DrawCircleAtPoint = function(P, r, F, C) { let ICsX = function(t) { return P()[0] + r() * Math.cos(t) }; let ICsY = function(t) { return P()[1] + r() * Math.sin(t) }; let ICsZ = function(t) { return F(ICsX(t), ICsY(t)) }; let UCs = function(t) { return CamProjection([ICsX(t), ICsY(t), ICsZ(t)]) }; board.create('curve', [function(t) { return UCs(t)[0] }, function(t) { return UCs(t)[1] }, 0, 6.5 ], { strokecolor: C, strokeWidth: 1, layer: 2, highlight: false, fixed: true, doAdvancedPlot: false, numberPointsHigh: FidHigh * 2, numberPointsLow: FidLow * 2, }); } /* draw circles at point in domain and their images */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, () => -1, Color3); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, ExFunc, Color4); */ /* DrawCircleAtPoint(ArbPointInDom3D, RadiusOfCircle, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); */ /* ------------------------------------- */ /* ------------------------------------- */ /* draws images of points in domain */ /* DrawPoint(PointOfDev3D, ExFunc, 'Grey'); */ DrawPoint(ArbPointInDom3D, ExFunc, Color1); DrawPoint(ArbPointInDom3D, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4); /* ------------------------------------- */ /* ------------------------------------- */ /* draw graph of (approximated) function */ DrawGraph(GridDomainParam, ExFunc, Color1) /* draw graph of approximating function */ DrawGraph(GridDomainParam, function(x, y) { return ApproxFunc(x, y, PointOfDev3D()[0], PointOfDev3D()[1]) }, Color4) /* ------------------------------------- */ /* ------------------------------------- */ board.on('move', function() { board.suspendUpdate(); /* returns point in domain close to given point */ const ReturnToDomain = function(a) { let Output = []; if (a[0] < -1) { Output[0] = -1; } else if (a[0] > 1) { Output[0] = 1; } else { Output[0] = a[0]; } if (a[1] < -1) { Output[1] = -1; } else if (a[1] > 1) { Output[1] = 1; } else { Output[1] = a[1]; } Output[2] = a[2]; return Output }; PointOfDev.moveTo(CamProjection(ReturnToDomain(PointOfDev3D()))); ArbPointInDom.moveTo(CamProjection(ReturnToDomain(ArbPointInDom3D()))); PointDefRad.moveTo(CamProjection(ReturnToDomain(PointDefRad3D()))); board.unsuspendUpdate(); }); /* ------------------------------------- */ /* ------------------------------------- */ var newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100, SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; var ClickIndicator = 0; var SendData = function(){ newState = [Math.round(ArbPointInDom3D()[0]*100)/100, Math.round(ArbPointInDom3D()[1]*100)/100,SliderAngleOfProj1.Value(),SliderAngleOfProj2.Value()]; stateInput.value = JSON.stringify(newState); stateInput.dispatchEvent(new Event('change')); } ArbPointInDom.on('drag', function () { SendData(); }); stateInput.style.display = 'none'; /* ------------------------------------- */ /* ------------------------------------- */ /* ensures that all draggable points (e.g. PointOfDev, ArbPointInDom) are rotatated accordingly when angles of projections are changed by sliders */ /* Buffer */ var PointOfDevBuffer = PointOfDev3D(); var ArbPointInDomBuffer = ArbPointInDom3D(); var PointDefRadBuffer = PointDefRad3D(); /* function that updates position of objects */ var UpdatePos = function(){ let ICs1 = CamProjection(PointOfDevBuffer); let ICs2 = CamProjection(ArbPointInDomBuffer); let ICs3 = CamProjection(PointDefRadBuffer); board.suspendUpdate(); PointOfDev.moveTo([ ICs1[0], ICs1[1] ]); ArbPointInDom.moveTo([ ICs2[0], ICs2[1] ]); PointDefRad.moveTo([ ICs3[0], ICs3[1] ]); board.unsuspendUpdate(); SendData(); } SliderAngleOfProj1.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj1.on('drag', function() { UpdatePos() }); }); SliderAngleOfProj2.on('down', function() { PointOfDevBuffer = PointOfDev3D(); ArbPointInDomBuffer = ArbPointInDom3D(); PointDefRadBuffer = PointDefRad3D(); SliderAngleOfProj2.on('drag', function() { UpdatePos() }); }); ArbPointInDom.on('down', function() { PointDefRadBuffer = VecSubVec(ArbPointInDom3D(), PointDefRad3D()); ArbPointInDom.on('drag', function() { let ICs = CamProjection(VecSubVec(ArbPointInDom3D(), PointDefRadBuffer)); board.suspendUpdate(); PointDefRad.moveTo([ ICs[0], ICs[1] ]); board.unsuspendUpdate(); }); }); /* ------------------------------------- */ /* ------------------------------------- */ [[/jsxgraph]]

Geben Sie mithilfe der obigen Abbildung einen Entwicklungspunkt \((a,b) \in [-1,1]^2\) des Taylorpolynoms \(T_{{@Order@}}{@FuncName@}\) von \({@FuncName@}\) an. Verschieben Sie dazu das Funktionsargument in der \(x_1 x_2\)-Ebene (Punkt ●) an die Position eines Entwicklungspunktes. [[input:ans1]]

[[validation:ans1]]

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