1. Comprendere attraverso i punti di una funzione quando essa cresce o decresce
In figura è rappresentato il grafico del volume d’aria presente nei polmoni di una persona durante una respirazione regolare.
{
"cssWidth": "95%",
"align": "center",
"xlim": [0,5],
"ylim": [2,3],
"boxPlot": true,
"boxNumbersInside": true,
"aspectRatio": "3:2",
"axisLabels": ["t","V"],
"axisUnitMeasures": ["s", "L"],
"axisLabelStyle": "italic",
"showXTicks": true,
"showYTicks": true,
"interactive": true,
"constrainView": true,
"pointSelection": true,
"padding": 55,
"data": [
{ "fn": "0.6*sin(pi*x/2.5)^4+2.2", "color": "red1", "domain": [0,6], "derivativeColor": "black1" },
{ "points": [[0.25,"0.6*sin(pi*0.25/2.5)^4+2.2"],[0.75,"0.6*sin(pi*0.75/2.5)^4+2.2"],[1,"0.6*sin(pi*1/2.5)^4+2.2"],[1.25,"0.6*sin(pi*1.25/2.5)^4+2.2"],[1.5,"0.6*sin(pi*1.5/2.5)^4+2.2"],[1.75,"0.6*sin(pi*1.75/2.5)^4+2.2"],[2,"0.6*sin(pi*2/2.5)^4+2.2"],[2.25,"0.6*sin(pi*2.25/2.5)^4+2.2"]], "fillColor": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
2. Retta secante, retta tangente, sua pendenza
{
"cssWidth": "95%",
"align": "center",
"xlim": [0,5],
"ylim": [2,3],
"boxPlot": true,
"boxNumbersInside": true,
"aspectRatio": "3:2",
"axisLabels": ["t","V"],
"axisUnitMeasures": ["s", "L"],
"axisLabelStyle": "italic",
"showXTicks": true,
"showYTicks": true,
"interactive": true,
"constrainView": true,
"pointSelection": true,
"slopeSelection": true,
"tangentSelection": true,
"padding": 55,
"data": [
{ "fn": "0.6*sin(pi*x/2.5)^4+2.2", "color": "red1", "domain": [0,6], "derivativeColor": "black1" },
{ "points": [[0.25,"0.6*sin(pi*0.25/2.5)^4+2.2"],[0.75,"0.6*sin(pi*0.75/2.5)^4+2.2"],[1,"0.6*sin(pi*1/2.5)^4+2.2"],[1.25,"0.6*sin(pi*1.25/2.5)^4+2.2"],[1.5,"0.6*sin(pi*1.5/2.5)^4+2.2"],[1.75,"0.6*sin(pi*1.75/2.5)^4+2.2"],[2,"0.6*sin(pi*2/2.5)^4+2.2"],[2.25,"0.6*sin(pi*2.25/2.5)^4+2.2"]], "fillColor": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
3. Costruzione della funzione derivata di funzioni polinomiali
{
"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 10,
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"legend": true,
"data": [
{ "fn": "3x", "label": "$y = 3x$", "derivativeColor": "summer3" },
{ "points": [[-3,-9],[-1,-3],[0,0],[1,3],[3,9]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
Derivata:
$$
y = 3
$$
{
"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 10,
"ylim": [-1.9,17.9],
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"derivativeYLim": [-9.9,9.9],
"legend": true,
"data": [
{ "fn": "x^2", "label": "$y = x^2$", "derivativeColor": "summer3" },
{ "points": [[-3,9],[-2,4],[-1,1],[0,0],[1,1],[2,4],[3,9]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
Derivata:
$$
y = 2x
$$
{
"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 10,
"ylim": [-9.9,9.9],
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"derivativeYLim": [-9.9,9.9],
"legend": true,
"data": [
{ "fn": "x^3", "label": "$y = x^3$", "derivativeColor": "summer3" },
{ "points": [[-2,-8],[-1,-1],[0,0],[1,1],[2,8]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
Derivata:
$$
y = 3x^2
$$
{
"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 0,
"xlim": [-8,8],
"ylim": [-3.9,3.9],
"xStep": "pi/2",
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"derivativeYLim": [-4.9,4.9],
"legend": true,
"data": [
{ "fn": "sin(x)", "label": "$y = \\sin(x)$", "derivativeColor": "summer3" },
{ "points": [["-2*pi","sin(-2*pi)"],[-1,"sin(-1)"],[0,0],[1,"sin(1)"],["1.5*pi","sin(1.5*pi)"],["pi",0]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
Derivata:
$$
y = cos(x)
$$
{
"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 10,
"xlim": [-9.9,4.9],
"ylim": [-0.9,9.9],
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"derivativeYLim": [-9.9,9.9],
"legend": true,
"data": [
{ "fn": "e^x", "label": "$y = e^x$", "derivativeColor": "summer3" },
{ "points": [[-5,"e^(-5)"],[-2,"e^(-2)"],[0,1],[1,"e"],[2,"e^2"]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}
Derivata:
$$
y = e^x
$$
4. Mettiti alla prova
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"aspectRatio": "3:2",
"cssWidth": "95%",
"align": "center",
"marginBottom": 10,
"xlim": [-3.4,3.4],
"ylim": [-1.9,1.9],
"interactive": true,
"slopeSelection": true,
"tangentSelection": true,
"addDerivativePlot": true,
"showDerivative": false,
"showDerivativeFunction": false,
"traceDerivative": true,
"derivativeToggle": true,
"showDerivativePoint": false,
"derivativeYLim": [-2.9,2.9],
"legend": true,
"data": [
{ "fn": "0.5*sin(0.5*pi*x)^2+0.5*cos(0.75*pi*x)", "derivativeColor": "summer3" },
{ "points": [[-2.8,0.927782507],[-2,0],[-1.6,-0.231762746],[-0.8,0.297745751],[0,0.5],[0.8,0.297745751],[1.6,-0.231762746],[2,0],[2.8,0.927782507]], "color": "red1", "radius": 3, "strokeColor": "#fff", "strokeWidth": 2 }
]
}