Jesús Avalos Rodríguez

JESÚS AVALOS RODRÍGUEZ

Docente del Departamento Académico de Matemáticas

Facultad de Ciencias Físicas y Matemáticas

Universidad Nacional de Trujillo

E-mail: javalos@unitru.edu.pe

EJEMPLO DE PENDIENTES

EJEMPLO DE DERIVABILIDAD

EJEMPLO DE CONTINUIDAD

TEOREMA SOBRE CONTINUIDAD

TEOREMA: DERIVABILIDAD IMPLICA CONTINUIDAD

RULES OF DIFFERENTIATION

DERIVADA

ANIMATION SINUS

=====================================

movie = Animate[Plot[Sin[x + a], {x, 0, 10}], {a, 0, 5}];

Export["movie.swf", movie]

% Save on documents

CÓNICAS

TRANSFORMACIÓN DE FUNCIONES

CÓDIGO en Mathematica

============================================================

Manipulate[
Module[{function, buttonText, color1 = Red, color2 = Blue},
If[reset == True, {a, b, c, d, xval} = {1, 1, 0, 0, 0};
reset = False];
function[x_] := {x, x^2, 1/5 x (x - 1) (x + 3), Sin[x], Exp[x],
Piecewise[{{3, x < -3}, {Abs[x], -3 <= x < 1}, {-(x - 2)^2 + 4,
x >= 1}}]};
Show[{
Graphics[{
If[MemberQ[options, sp],
Text[
ToString[
Style[{Round[xval, .01], function[Round[xval, .01]][[fff]]},
color1], TraditionalForm], {4.75, -4.25}, {1, 0}], color1],
If[MemberQ[options, sp],
Text[ToString[
Style[If[
b =!= 0., {1/b Round[xval, .01] + c,
a*function[Round[xval, .01]][[fff]] + d}, ""], color2],
TraditionalForm], {4.75, -4.75}, {1, 0}], color1],
AbsolutePointSize[9],
color2, Dashing[{0.01}],
If[b =!= 0., Point[{1/b xval + c, a*function[xval][[fff]] + d}],
color2],
If[b =!= 0.,
Line[{{0, a*function[xval][[fff]] + d}, {1/b xval + c,
a*function[xval][[fff]] + d}}], color2],
If[b =!= 0.,
Line[{{1/b xval + c, 0}, {1/b xval + c,
a*function[xval][[fff]] + d}}], color2],
color1, Dashing[{0.02}],
Point[{xval, function[xval][[fff]]}],
Line[{{0, function[xval][[fff]]}, {xval,
function[xval][[fff]]}}],
Line[{{xval, 0}, {xval, function[xval][[fff]]}}]
}],
Plot[a function[b (x - c)][[fff]] + d,
{x, Which[b < 0., (1/b xval + c), b == 0., -5, b > 0., -5],
Which[b < 0., 5, b == 0., 5, b > 0., 1/b xval + c]},
PlotStyle -> {color2, AbsoluteThickness[3]}, PlotRange -> All],
Plot[a function[b (x - c)][[fff]] + d,
{x, Which[b < 0., -5, b == 0., -5, b > 0., (1/b xval + c)],
Which[b < 0., 1/b xval + c, b == 0., 5, b > 0., 5]},
PlotStyle -> color2, PlotRange -> All],
Plot[function[x][[fff]], {x, -5, xval},
PlotStyle -> {color1, AbsoluteThickness[3]}, PlotRange -> All],
Plot[function[x][[fff]], {x, xval, 5}, PlotStyle -> color1,
PlotRange -> All]
},
PlotRange -> {{-5, 5}, {-5, 5}}, Axes -> True,
AxesStyle ->
AbsoluteThickness[If[MemberQ[options, grid], 1.25, 1]],
PlotLabel ->
If[MemberQ[options, pl],
Grid[{{ToString[Style[HoldForm[f][HoldForm[x]], color1],
TraditionalForm]}, {ToString[
Style[If[Round[a, .1] == 1., 1,
If[Round[a, .1] == 0., 0, a]] HoldForm[f][
If[Round[b, .1] == 1., 1,
If[Round[b, .1] == 0., 0, b]] (HoldForm[x] -
If[Round[c, .1] == 0., 0, c])] +
If[Round[d, .1] == 0., 0, d], color2], TraditionalForm]}}]],
ImageSize -> If[MemberQ[options, format], 500, 380],
GridLines ->
If[MemberQ[options, grid], {Range[-5, 5], Range[-5, 5]}, None]
]],
{{a, 1}, -4.9, 4.9, 0.1,
ImageSize -> If[MemberQ[options, format], Medium, Tiny]},
{{b, 1}, -4.9, 4.9, 0.1,
ImageSize -> If[MemberQ[options, format], Medium, Tiny]},
{{c, 0}, -4.9, 4.9, 0.1,
ImageSize -> If[MemberQ[options, format], Medium, Tiny]},
{{d, 1}, -4.9, 4.9, 0.1,
ImageSize -> If[MemberQ[options, format], Medium, Tiny]},
Delimiter,
{{xval, 0, "valor x"}, -4.9, 4.9, .1,
ImageSize -> If[MemberQ[options, format], Medium, Tiny]},
Delimiter,
{{reset, False, "reiniciar"}, {False, True}},
Delimiter,
{{fff, 1, " "}, Evaluate@functionButtons,
ControlType -> RadioButtonBar, Appearance -> "Vertical"},
Delimiter,
{{options, {sp, pl}, ""}, {pl -> " ver función",
sp -> " ubicación de x", grid -> " grilla de líneas",
format -> " tamaño grande"},
ControlType -> CheckboxBar, Appearance -> "Vertical"},
ControlPlacement -> Left,
AutorunSequencing -> {{1, 3}, {3, 3}, {5, 3}, {7, 6}},
Initialization :> {
buttonText = {"lineal", "cuadrática", "cúbica", "seno",
"exponencial", "por partes"};
functionButtons =
Map[#[[1]] -> #[[2]] &,
Transpose[{Range[Length[buttonText]], buttonText}]];
}]

POLINOMIO DE TAYLOR DE UNA FUNCIÓN DE UNA VARIABLE

CÓDIGO en Mathematica

=================================================================

Manipulate[
checkfcn = TrueQ[f[x] === Log[1 + x]];
If[checkfcn && (xpt <= -1), xpt = -0.9999];
If[checkfcn && (errx <= -1 || xpt <= -1), xpt = -0.9999];
If[errx < -xmax, errx = -xmax];
If[errx > xmax, errx = xmax];
If[xpt < -xmax, xpt = -xmax];
If[xpt > xmax, xpt = xmax];
Switch[opts, 1, showpoly = True; showerror = False, 2,
showpoly = False; showerror = True, 3, showpoly = False;
showerror = False];

graphlist =
Evaluate@Intersection[
Table[Normal[Series[f[x], {x, xpt, n}]], {n, 0, terms}]];
T = Function[{x}, Evaluate@Normal[Series[f[x], {x, xpt, terms}]]];

Pane[
Column[{
Column[{
Row[{Text@
Style[Row[{Style["f", Italic], "(", Style["x", Italic],
") = ", ToString[f[x], TraditionalForm]}], 20, Red]
}],

Dynamic@
Item[Show[
maingraph =
Plot[f[x], {x, -xmax, xmax}, PlotStyle -> Red,
PlotRange -> {{-xmax, xmax}, {-ymax, ymax}}],
If[showall,
Plot[graphlist, {x, -xmax, xmax},
PlotRange -> {{-xmax, xmax}, {-ymax, ymax}}], {}],
Plot[Last[graphlist], {x, -xmax, xmax},
PlotRange -> {{-xmax, xmax}, {-ymax, ymax}},
PlotStyle -> Black],
Graphics[{
If[showerror, {Darker[Green, 0.3],
{{PointSize[0.02], Point[{errx, T[errx]}]},
{PointSize[0.02], Point[{errx, f[errx]}]},
{Line[{{errx, T[errx]}, {errx, f[errx]}}]}
}}, {}],
{Purple, PointSize[0.02], Point[{xpt, f[xpt]}]}
}],
ImageSize -> {450, 250},
PlotRange -> {{-xmax, xmax}, {-ymax, ymax}},
ImagePadding -> 20, AspectRatio -> 250/450],
ItemSize -> {Automatic, 25}],

Item[If[showerror,
Column[{
Dynamic@Text@Style[Column[{

Row[{Style["f", Italic], "(", errx, ") = ",
TraditionalForm@f[errx]}],

Row[{Subscript[Style["T", Italic], terms], "(", errx,
") = ", T[errx]}],
Row[{"error = ", TraditionalForm[f[errx] - T[errx]]}],

Style[Row[{"|error| = ", N@Abs[f[errx] - T[errx]]}],
16, Bold]
}], 14, Darker[Green, 0.5]],

If[showerror,
Row[{Style["error point ", "Label", 10],
If[checkfcn, Control[{{errx, 0}, -0.9999, xmax}],
Control[{{errx, 0}, -xmax, xmax}]]}], ""]}],
If[showpoly,
Pane[
Text@Style[
Row[{Subscript[Style["T", Italic], terms], "(",
Style["x", Italic], ") = ",
Normal[Series[f[x], {x, xpt, terms}]] /.
x -> Style["x", Italic]}], 12], 400], ""]],
ItemSize -> {Automatic, 11}]

}, Alignment -> Top]
}],
{525, 405}, Alignment -> Center],

Row[{Grid[{{Pane[Grid[{
{"expansion point",
PaneSelector[
{True ->
Control[{{xpt, 0}, -0.9999, xmax,
Appearance -> "Labeled", ImageSize -> Tiny}],

False ->
Control[{{xpt, 0}, -xmax, xmax, Appearance -> "Labeled",
ImageSize -> Tiny}]},
Dynamic[checkfcn]
]},
{"terms (highest degree)",
Control[{{terms, 2}, 0, 10, 1, Appearance -> "Labeled",
ImageSize -> Tiny}]},
{"x axis",
Control[{{xmax, 10}, 1.01, 20, Appearance -> "Labeled",
ImageSize -> Tiny}]}

}, Alignment -> {{Right, Left}, Center}], 300],
Column[{
Row[{"f(x) = ",
PopupMenu[Dynamic[f], Map[MapAt[TraditionalForm, #, 2] &,
{Cos -> Cos[x], Sin -> Sin[x], Tan -> Tan[x],
Exp -> Exp[x], ArcTan -> ArcTan[x],
Log[1 + #] & -> Log[1 + x], 1/(# + 1) & -> 1/(x + 1)}]]}],
Row[{"graph all polynomials ",
Control[{{showall, True}, {True, False}}]}],
Row[{"show ",
Control[{{opts, 1}, {1 -> " polynomial ", 2 -> " error ",
3 -> " neither"}, ControlType -> RadioButtonBar}]}]
}]
}}, Dividers -> {{False, Gray, False}, None},
Spacings -> {1, Automatic}]}],

{{f, Cos, "f(x) ="}, Map[MapAt[TraditionalForm, #, 2] &,
{Cos -> Cos[x], Sin -> Sin[x], Tan -> Tan[x], Exp -> Exp[x],
ArcTan -> ArcTan[x], Log[1 + #] & -> Log[1 + x],
1/(# + 1) & -> 1/(x + 1)}], None},
{{ymax, 3, "y axis"}, 1, 20, VerticalSlider,
ControlPlacement -> Left},
{{graphlist, {1}}, ControlType -> None},
{{T, Function[{x}, 1]}, ControlType -> None},
{{checkfcn, False}, ControlType -> None},
AutorunSequencing -> {1, 5, 6}, TrackedSymbols -> True
]

GRAPHING DERIVATIVES

CHANGING COEFICIENT IN POLYNOMIALS OF LOW DEGREE

This Demonstration illustrates how changing a coefficient affects the shape of the graph of a poynomial and its first and second derivatives. Observe the variation in the location and number of critical and inflection points.

CUBIC EQUATION

UNDERSTANDING CONCAVITY

A differentiable function f on some interval is said to be concave up if f' is increasing and concave down if f' is decreasing. If f' is constant, then the function has no concavity. Points where a function changes concavity are called inflection points.

The red line is the tangent to the curve at and the dashed blue line is the tangent to the curve a little to the right of P.

SNOWBOARDING OVER DERIVATIVES

This Demonstration shows the geometric nature of the first and second derivative using a snowboarder. Imagine watching a snowboarder using a telescope. The inclination of the snowboard gives a numeric value for the first derivative, as read from the calibrated scale on the edge of the telescope. Here we assume that the snowboarder is not jumping, so that the board is always tangent to the slope. The second derivative is represented geometrically by how the front tip of the snowboard rotates upwards or backwards. You can make flags show in places where the first/second derivative are zero and you can choose different courses for practice. The author has found this Demonstration useful with his students, explaining using one course and letting them try to determine sign diagrams for the other courses.

FINITE LIMIT AT FINITE POINT

TANGENT LINE PROBLEM

DERIVATIVE AS A FUNCTION

As a pointwise limit, the derivative gives the slope of the tangent to the curve. Varying the point and plotting the tangent slope as a function of x yields a new curve, the derivative as a function. It is possible to guess at a formula for the derivative from this curve.