CoCalc Public FilesExistenceUnicite.sagews
Author: Juan Carlos Bustamante
Views : 49
Compute Environment: Ubuntu 18.04 (Deprecated)

## Théorème

Soit l'équation différentielle du premier ordre avec condition initiale $\begin{array}{ccl} y'(x) &= &f(x,y)\\ y(x_0) & = &y_0 \end{array}$

Si les fonctions $f(x,y)$ et $\frac{\partial f}{\partial y} f(x,y)$ sont continues dans un rectangle contenant $(x_0,y_0)$ en son intérieur, alors il esiste un réel $\delta > 0$ et une fonction $y(x)$ ayant une dérivée continue dans $]x_0-\delta,\ x_0 + \delta[$ qui satisfait l'équation (et la condition initiale donnée) sur $]x_0-\delta,\ x_0 + \delta[$

### Exemple

Considérons le problème à valeur initiale

(Share server only supports KaTeX; open in CoCalc to see this formula.)

Les courbes solution de l'Équation différentielle, sans conditions initiale sont données par $y^2 - 2y = x^3 + 2x^2+2x+c$, $c$ étant une constante (c'est une équation à variables séparables). Si on tient compte de la condition initiale on trouve $c=3$.

• Cette relation definit deux fonctions $y = 1 \pm \sqrt{x^3+2x^2+2x+4}$ tant que $x>-2$ (pour que l'expression sous la racine soit positive)
• Noter qu'une seule des deux fonctions satisfait à la condition initiale donnée (avec le signe $-$).

### Variante

On considère maintenant la condition initiale $y(0)=1$

• On peut résoudre comme avant (séparation des variables)
• Cette fois on a bel et bien deux solutions.
• Remarquons que sur la droite $y=1$, la fonction $f(x,y)$ n'est pas continue, il n'y a donc pas de contradiciton au théorème d'existence unicité.
var('y,x')
f(y) = (3*x^2 + 4 *x + 2)/(2*(y-1))
courbes = [implicit_plot(y^2 - 2*y == x^3 +2*x^2 +2*x +k , (x,-3,3),(y,-3,3), color='grey') for k in sxrange(-20,20,2) ]
Sol = implicit_plot(y^2 - 2*y == x^3 +2*x^2 +2*x +3 , (x,-3,3),(y,-3,3), color='red')
Sol2= implicit_plot(y^2 - 2*y == x^3 +2*x^2 +2*x -1 , (x,-3,3),(y,-3,3), color='green')
show(sum(courbes)+Sol+Sol2, figsize=6, aspect_ratio=1, axes_labels=['$x$','$y(x)$'], axes = True)

(y, x)

### Exemple

On considère maintenant la problème

(Share server only supports KaTeX; open in CoCalc to see this formula.)

Il se trouve que toute fonction de la forme \[y_a(x) = \left{\begin{array}{ll}0 & \text{ pour } 0\leqslant x

Encore une fois, la fonction $f(x,y)= y^{1/3}$ ne vérifie pas les conditions du théorème lorsque $y_0 = 0$. Par contre, si $y_0 \ne 0$, on voit bien l'unicité de la solution.


︠88a8b7cb-fdef-4b6e-ba64-f02c867c50fbs︠
var('y,x')
f(x,y) = sign(y)*abs(y)^(1/3)
Champ = plot_slope_field(f(x,y), (x, 0, 2), (y, -1, 1))
Courbes0 = [plot(0,(x,0,k), color="red", thickness=2) for k in sxrange(0.3,2,0.2) ]
Courbes1 = [plot(((2/3)*(x-k))^(3/2), (x,k,2), color="red", thickness=2) for k in sxrange(0.2,2,0.2) ]
Courbes2 = [plot(-((2/3)*(x-k))^(3/2), (x,k,2), color="red", thickness=2) for k in sxrange(0.2,2,0.2) ]
show(Champ + sum(Courbes0) +sum(Courbes1) + sum(Courbes2),ymin=-1, ymax=1,figsize=8)

(y, x)
xrange?

File:
Unable to read docstring (source code not available)

plot?

File: /projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/misc/decorators.py
Signature : plot(funcs, exclude=None, fillalpha=0.5, fillcolor='automatic', detect_poles=False, plot_points=200, thickness=1, adaptive_tolerance=0.01, rgbcolor=(0, 0, 1), adaptive_recursion=5, aspect_ratio='automatic', alpha=1, legend_label=None, fill=False, *args, **kwds)
Docstring :
Use plot by writing

"plot(X, ...)"

where X is a Sage object (or list of Sage objects) that either is
callable and returns numbers that can be coerced to floats, or has
a plot method that returns a "GraphicPrimitive" object.

There are many other specialized 2D plot commands available in
Sage, such as "plot_slope_field", as well as various graphics
primitives like "Arrow"; type "sage.plot.plot?" for a current list.

Type "plot.options" for a dictionary of the default options for
plots. You can change this to change the defaults for all future
plots. Use "plot.reset()" to reset to the default options.

PLOT OPTIONS:

* "plot_points" - (default: 200) the minimal number of plot
points.

* "adaptive_recursion" - (default: 5) how many levels of
recursion to go before giving up when doing adaptive refinement.
Setting this to 0 disables adaptive refinement.

* "adaptive_tolerance" - (default: 0.01) how large a difference
should be before the adaptive refinement code considers it
significant.  See the documentation further below for more
information, starting at "the algorithm used to insert".

* "base" - (default: 10) the base of the logarithm if a
logarithmic scale is set. This must be greater than 1. The base
can be also given as a list or tuple "(basex, basey)". "basex"
sets the base of the logarithm along the horizontal axis and
"basey" sets the base along the vertical axis.

* "scale" -- (default: ""linear"") string. The scale of the axes.
Possible values are ""linear"", ""loglog"", ""semilogx"",
""semilogy"".

The scale can be also be given as single argument that is a list
or tuple "(scale, base)" or "(scale, basex, basey)".

The ""loglog"" scale sets both the horizontal and vertical axes
to logarithmic scale. The ""semilogx"" scale sets the horizontal
axis to logarithmic scale. The ""semilogy"" scale sets the
vertical axis to logarithmic scale. The ""linear"" scale is the
default value when "Graphics" is initialized.

* "xmin" - starting x value in the rendered figure. This
parameter is passed directly to the "show" procedure and it could
be overwritten.

* "xmax" - ending x value in the rendered figure. This parameter
is passed directly to the "show" procedure and it could be
overwritten.

* "ymin" - starting y value in the rendered figure. This
parameter is passed directly to the "show" procedure and it could
be overwritten.

* "ymax" - ending y value in the rendered figure. This parameter
is passed directly to the "show" procedure and it could be
overwritten.

* "color" - an RGB tuple (r,g,b) with each of r,g,b between 0 and
1, or a color name as a string (e.g., 'purple'), or an HTML color
such as '#aaff0b'.

* "detect_poles" - (Default: False) If set to True poles are
detected. If set to "show" vertical asymptotes are drawn.

* "legend_color" - the color of the text for this item in the
legend

* "legend_label" - the label for this item in the legend

Note: * If the "scale" is ""linear"", then irrespective of what

"base" is set to, it will default to 10 and will remain unused.

* If you want to limit the plot along the horizontal axis in
the final rendered figure, then pass the "xmin" and "xmax"
keywords to the "show()" method. To limit the plot along the
vertical axis, "ymin" and "ymax" keywords can be provided to
either this "plot" command or to the "show" command.

* For the other keyword options that the "plot" function can
take, refer to the method "show()".

APPEARANCE OPTIONS:

The following options affect the appearance of the line through the
points on the graph of X (these are the same as for the line
function):

INPUT:

* "alpha" - How transparent the line is

* "thickness" - How thick the line is

* "rgbcolor" - The color as an RGB tuple

* "hue" - The color given as a hue

Any MATPLOTLIB line option may also be passed in.  E.g.,

* "linestyle" - (default: "-") The style of the line, which is
one of

* ""-"" or ""solid""

* ""--"" or ""dashed""

* ""-."" or ""dash dot""

* "":"" or ""dotted""

* ""None"" or "" "" or """" (nothing)

The linestyle can also be prefixed with a drawing style (e.g.,
""steps--"")

* ""default"" (connect the points with straight lines)

* ""steps"" or ""steps-pre"" (step function; horizontal line
is to the left of point)

* ""steps-mid"" (step function; points are in the middle of
horizontal lines)

* ""steps-post"" (step function; horizontal line is to the
right of point)

* "marker"  - The style of the markers, which is one of

* ""None"" or "" "" or """" (nothing) -- default

* "","" (pixel), ""."" (point)

* ""_"" (horizontal line), ""|"" (vertical line)

* ""o"" (circle), ""p"" (pentagon), ""s"" (square), ""x""
(x), ""+"" (plus), ""*"" (star)

* ""D"" (diamond), ""d"" (thin diamond)

* ""H"" (hexagon), ""h"" (alternative hexagon)

* ""<"" (triangle left), "">"" (triangle right), ""^""
(triangle up), ""v"" (triangle down)

* ""1"" (tri down), ""2"" (tri up), ""3"" (tri left), ""4""
(tri right)

* "0" (tick left), "1" (tick right), "2" (tick up), "3"
(tick down)

* "4" (caret left), "5" (caret right), "6" (caret up), "7"
(caret down), "8" (octagon)

* ""$...$"" (math TeX string)

* "(numsides, style, angle)" to create a custom, regular
symbol

* "numsides" -- the number of sides

* "style" -- "0" (regular polygon), "1" (star shape), "2"
(asterisk), "3" (circle)

* "angle" -- the angular rotation in degrees

* "markersize" - the size of the marker in points

* "markeredgecolor" -- the color of the marker edge

* "markerfacecolor" -- the color of the marker face

* "markeredgewidth" - the size of the marker edge in points

* "exclude" - (Default: None) values which are excluded from the
plot range. Either a list of real numbers, or an equation in one
variable.

FILLING OPTIONS:

* "fill" - (Default: False) One of:

* "axis" or True: Fill the area between the function and the
x-axis.

* "min": Fill the area between the function and its minimal
value.

* "max": Fill the area between the function and its maximal
value.

* a number c: Fill the area between the function and the
horizontal line y = c.

* a function g: Fill the area between the function that is
plotted and g.

* a dictionary "d" (only if a list of functions are plotted):
The keys of the dictionary should be integers. The value of
"d[i]" specifies the fill options for the i-th function in the
list. If "d[i] == [j]": Fill the area between the i-th and the
j-th function in the list.  (But if "d[i] == j": Fill the area
between the i-th function in the list and the horizontal line y
= j.)

* "fillcolor" - (default: 'automatic') The color of the fill.
Either 'automatic' or a color.

* "fillalpha" - (default: 0.5) How transparent the fill is. A
number between 0 and 1.

Note: * this function does NOT simply sample equally spaced
points

between xmin and xmax. Instead it computes equally spaced
points and adds small perturbations to them. This reduces the
possibility of, e.g., sampling sin only at multiples of 2pi,
which would yield a very misleading graph.

* if there is a range of consecutive points where the function
has no value, then those points will be excluded from the plot.
See the example below on automatic exclusion of points.

EXAMPLES:

We plot the sin function:

sage: P = plot(sin, (0,10)); print P
Graphics object consisting of 1 graphics primitive
sage: len(P)     # number of graphics primitives
1
sage: len(P[0])  # how many points were computed (random)
225
sage: P          # render
Graphics object consisting of 1 graphics primitive

sage: P = plot(sin, (0,10), plot_points=10); print P
Graphics object consisting of 1 graphics primitive
sage: len(P[0])  # random output
32
sage: P          # render
Graphics object consisting of 1 graphics primitive

We plot with "randomize=False", which makes the initial sample
points evenly spaced (hence always the same). Adaptive plotting
might insert other points, however, unless "adaptive_recursion=0".

sage: p=plot(1, (x,0,3), plot_points=4, randomize=False, adaptive_recursion=0)
sage: list(p[0])
[(0.0, 1.0), (1.0, 1.0), (2.0, 1.0), (3.0, 1.0)]

Some colored functions:

sage: plot(sin, 0, 10, color='purple')
Graphics object consisting of 1 graphics primitive
sage: plot(sin, 0, 10, color='#ff00ff')
Graphics object consisting of 1 graphics primitive

We plot several functions together by passing a list of functions
as input:

sage: plot([sin(n*x) for n in [1..4]], (0, pi))
Graphics object consisting of 4 graphics primitives

We can also build a plot step by step from an empty plot:

sage: a = plot([]); a       # passing an empty list returns an empty plot (Graphics() object)
Graphics object consisting of 0 graphics primitives
sage: a += plot(x**2); a    # append another plot
Graphics object consisting of 1 graphics primitive
sage: a += plot(x**3); a    # append yet another plot
Graphics object consisting of 2 graphics primitives

The function sin(1/x) wiggles wildly near 0. Sage adapts to this
and plots extra points near the origin.

sage: plot(sin(1/x), (x, -1, 1))
Graphics object consisting of 1 graphics primitive

Via the matplotlib library, Sage makes it easy to tell whether a
graph is on both sides of both axes, as the axes only cross if the
origin is actually part of the viewing area:

sage: plot(x^3,(x,0,2))  # this one has the origin
Graphics object consisting of 1 graphics primitive
sage: plot(x^3,(x,1,2))  # this one does not
Graphics object consisting of 1 graphics primitive

Another thing to be aware of with axis labeling is that when the
labels have quite different orders of magnitude or are very large,
scientific notation (the e notation for powers of ten) is used:

sage: plot(x^2,(x,480,500))  # this one has no scientific notation
Graphics object consisting of 1 graphics primitive
sage: plot(x^2,(x,300,500))  # this one has scientific notation on y-axis
Graphics object consisting of 1 graphics primitive

You can put a legend with "legend_label" (the legend is only put
once in the case of multiple functions):

sage: plot(exp(x), 0, 2, legend_label='$e^x$')
Graphics object consisting of 1 graphics primitive

Sage understands TeX, so these all are slightly different, and you
can choose one based on your needs:

sage: plot(sin, legend_label='sin')
Graphics object consisting of 1 graphics primitive
sage: plot(sin, legend_label='$sin$')
Graphics object consisting of 1 graphics primitive
sage: plot(sin, legend_label='$sin$')
Graphics object consisting of 1 graphics primitive

It is possible to use a different color for the text of each label:

sage: p1 = plot(sin, legend_label='sin', legend_color='red')
sage: p2 = plot(cos, legend_label='cos', legend_color='green')
sage: p1 + p2
Graphics object consisting of 2 graphics primitives

Note that the independent variable may be omitted if there is no
ambiguity:

sage: plot(sin(1/x), (-1, 1))
Graphics object consisting of 1 graphics primitive

Plotting in logarithmic scale is possible for 2D plots.  There are
two different syntaxes supported:

sage: plot(exp, (1, 10), scale='semilogy') # log axis on vertical
Graphics object consisting of 1 graphics primitive

sage: plot_semilogy(exp, (1, 10)) # same thing
Graphics object consisting of 1 graphics primitive

sage: plot_loglog(exp, (1, 10))   # both axes are log
Graphics object consisting of 1 graphics primitive

sage: plot(exp, (1, 10), scale='loglog', base=2) # long time # base of log is 2
Graphics object consisting of 1 graphics primitive

We can also change the scale of the axes in the graphics just
before displaying:

sage: G = plot(exp, 1, 10) # long time
sage: G.show(scale=('semilogy', 2)) # long time

The algorithm used to insert extra points is actually pretty
simple. On the picture drawn by the lines below:

sage: p = plot(x^2, (-0.5, 1.4)) + line([(0,0), (1,1)], color='green')
sage: p += line([(0.5, 0.5), (0.5, 0.5^2)], color='purple')
sage: p += point(((0, 0), (0.5, 0.5), (0.5, 0.5^2), (1, 1)), color='red', pointsize=20)
sage: p += text('A', (-0.05, 0.1), color='red')
sage: p += text('B', (1.01, 1.1), color='red')
sage: p += text('C', (0.48, 0.57), color='red')
sage: p += text('D', (0.53, 0.18), color='red')
sage: p.show(axes=False, xmin=-0.5, xmax=1.4, ymin=0, ymax=2)

You have the function (in blue) and its approximation (in green)
passing through the points A and B. The algorithm finds the
midpoint C of AB and computes the distance between C and D. If that
distance exceeds the "adaptive_tolerance" threshold (*relative* to
the size of the initial plot subintervals), the point D is added to
the curve.  If D is added to the curve, then the algorithm is
applied recursively to the points A and D, and D and B. It is
repeated "adaptive_recursion" times (5, by default).

The actual sample points are slightly randomized, so the above
plots may look slightly different each time you draw them.

We draw the graph of an elliptic curve as the union of graphs of 2
functions.

sage: def h1(x): return abs(sqrt(x^3  - 1))
sage: def h2(x): return -abs(sqrt(x^3  - 1))
sage: P = plot([h1, h2], 1,4)
sage: P          # show the result
Graphics object consisting of 2 graphics primitives

It is important to mention that when we draw several graphs at the
same time, parameters "xmin", "xmax", "ymin" and "ymax" are just
passed directly to the "show" procedure. In fact, these parameters
would be overwritten:

sage: p=plot(x^3, x, xmin=-1, xmax=1,ymin=-1, ymax=1)
sage: q=plot(exp(x), x, xmin=-2, xmax=2, ymin=0, ymax=4)
sage: (p+q).show()

As a workaround, we can perform the trick:

sage: p1 = line([(a,b) for a,b in zip(p[0].xdata,p[0].ydata) if (b>=-1 and b<=1)])
sage: q1 = line([(a,b) for a,b in zip(q[0].xdata,q[0].ydata) if (b>=0 and b<=4)])
sage: (p1+q1).show()

We can also directly plot the elliptic curve:

sage: E = EllipticCurve([0,-1])
sage: plot(E, (1, 4), color=hue(0.6))
Graphics object consisting of 1 graphics primitive

We can change the line style as well:

sage: plot(sin(x), (x, 0, 10), linestyle='-.')
Graphics object consisting of 1 graphics primitive

If we have an empty linestyle and specify a marker, we can see the
points that are actually being plotted:

sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker='.')
Graphics object consisting of 1 graphics primitive

The marker can be a TeX symbol as well:

sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker=r'$checkmark$')
Graphics object consisting of 1 graphics primitive

Sage currently ignores points that cannot be evaluated

sage: set_verbose(-1)
sage: plot(-x*log(x), (x,0,1))  # this works fine since the failed endpoint is just skipped.
Graphics object consisting of 1 graphics primitive
sage: set_verbose(0)

This prints out a warning and plots where it can (we turn off the
warning by setting the verbose mode temporarily to -1.)

sage: set_verbose(-1)
sage: plot(x^(1/3), (x,-1,1))
Graphics object consisting of 1 graphics primitive
sage: set_verbose(0)

Plotting the real cube root function for negative input requires
avoiding the complex numbers one would usually get. The easiest way
is to use absolute value:

sage: plot(sign(x)*abs(x)^(1/3), (x,-1,1))
Graphics object consisting of 1 graphics primitive

We can also use the following:

sage: plot(sign(x)*(x*sign(x))^(1/3), (x,-4,4))
Graphics object consisting of 1 graphics primitive

A way that points to how to plot other functions without symbolic
variants is using lambda functions:

sage: plot(lambda x : RR(x).nth_root(3), (x,-1, 1))
Graphics object consisting of 1 graphics primitive

We can detect the poles of a function:

sage: plot(gamma, (-3, 4), detect_poles = True).show(ymin = -5, ymax = 5)

We draw the Gamma-Function with its poles highlighted:

sage: plot(gamma, (-3, 4), detect_poles = 'show').show(ymin = -5, ymax = 5)

The basic options for filling a plot:

sage: p1 = plot(sin(x), -pi, pi, fill = 'axis')
sage: p2 = plot(sin(x), -pi, pi, fill = 'min')
sage: p3 = plot(sin(x), -pi, pi, fill = 'max')
sage: p4 = plot(sin(x), -pi, pi, fill = 0.5)
sage: graphics_array([[p1, p2], [p3, p4]]).show(frame=True, axes=False) # long time

sage: plot([sin(x), cos(2*x)*sin(4*x)], -pi, pi, fill = {0: 1}, fillcolor = 'red', fillalpha = 1)
Graphics object consisting of 3 graphics primitives

A example about the growth of prime numbers:

sage: plot(1.13*log(x), 1, 100, fill = lambda x: nth_prime(x)/floor(x), fillcolor = 'red')
Graphics object consisting of 2 graphics primitives

Fill the area between a function and its asymptote:

sage: f = (2*x^3+2*x-1)/((x-2)*(x+1))
sage: plot([f, 2*x+2], -7,7, fill = {0: [1]}, fillcolor='#ccc').show(ymin=-20, ymax=20)

Fill the area between a list of functions and the x-axis:

sage: def b(n): return lambda x: bessel_J(n, x)
sage: plot([b(n) for n in [1..5]], 0, 20, fill = 'axis')
Graphics object consisting of 10 graphics primitives

Note that to fill between the ith and jth functions, you must use
dictionary key-value pairs "i:[j]"; key-value pairs like "i:j" will
fill between the ith function and the line y=j:

sage: def b(n): return lambda x: bessel_J(n, x) + 0.5*(n-1)
sage: plot([b(c) for c in [1..5]], 0, 40, fill = dict([(i, [i+1]) for i in [0..3]]))
Graphics object consisting of 9 graphics primitives
sage: plot([b(c) for c in [1..5]], 0, 40, fill = dict([(i, i+1) for i in [0..3]])) # long time
Graphics object consisting of 9 graphics primitives

Extra options will get passed on to "show()", as long as they are
valid:

sage: plot(sin(x^2), (x, -3, 3), title='Plot of $sin(x^2)$', axes_labels=['$x$','$y$']) # These labels will be nicely typeset
Graphics object consisting of 1 graphics primitive
sage: plot(sin(x^2), (x, -3, 3), title='Plot of sin(x^2)', axes_labels=['x','y']) # These will not
Graphics object consisting of 1 graphics primitive
sage: plot(sin(x^2), (x, -3, 3), axes_labels=['x','y'], axes_labels_size=2.5) # Large axes labels (w.r.t. the tick marks)
Graphics object consisting of 1 graphics primitive

sage: plot(sin(x^2), (x, -3, 3), figsize=[8,2])
Graphics object consisting of 1 graphics primitive
sage: plot(sin(x^2), (x, -3, 3)).show(figsize=[8,2]) # These are equivalent

This includes options for custom ticks and formatting.  See
documentation for "show()" for more details.

sage: plot(sin(pi*x), (x, -8, 8), ticks=[[-7,-3,0,3,7],[-1/2,0,1/2]])
Graphics object consisting of 1 graphics primitive
sage: plot(2*x+1,(x,0,5),ticks=[[0,1,e,pi,sqrt(20)],2],tick_formatter="latex")
Graphics object consisting of 1 graphics primitive

This is particularly useful when setting custom ticks in multiples
of pi.

sage: plot(sin(x),(x,0,2*pi),ticks=pi/3,tick_formatter=pi)
Graphics object consisting of 1 graphics primitive

You can even have custom tick labels along with custom positioning.

sage: plot(x**2, (x,0,3), ticks=[[1,2.5],[0.5,1,2]], tick_formatter=[["$x_1$","$x_2$"],["$y_1$","$y_2$","$y_3$"]])
Graphics object consisting of 1 graphics primitive

You can force Type 1 fonts in your figures by providing the
relevant option as shown below. This also requires that LaTeX,
dvipng and Ghostscript be installed:

sage: plot(x, typeset='type1') # optional - latex

A example with excluded values:

sage: plot(floor(x), (x, 1, 10), exclude = [1..10])
Graphics object consisting of 11 graphics primitives

We exclude all points where "PrimePi" makes a jump:

sage: jumps = [n for n in [1..100] if prime_pi(n) != prime_pi(n-1)]
sage: plot(lambda x: prime_pi(x), (x, 1, 100), exclude = jumps)
Graphics object consisting of 26 graphics primitives

Excluded points can also be given by an equation:

sage: g(x) = x^2-2*x-2
sage: plot(1/g(x), (x, -3, 4), exclude = g(x) == 0, ymin = -5, ymax = 5) # long time
Graphics object consisting of 3 graphics primitives

"exclude" and "detect_poles" can be used together:

sage: f(x) = (floor(x)+0.5) / (1-(x-0.5)^2)
sage: plot(f, (x, -3.5, 3.5), detect_poles = 'show', exclude = [-3..3], ymin = -5, ymax = 5)
Graphics object consisting of 12 graphics primitives

Regions in which the plot has no values are automatically excluded.
The regions thus excluded are in addition to the exclusion points
present in the "exclude" keyword argument.:

sage: set_verbose(-1)
sage: plot(arcsec, (x, -2, 2))  # [-1, 1] is excluded automatically
Graphics object consisting of 2 graphics primitives

sage: plot(arcsec, (x, -2, 2), exclude=[1.5])  # x=1.5 is also excluded
Graphics object consisting of 3 graphics primitives

sage: plot(arcsec(x/2), -2, 2)  # plot should be empty; no valid points
Graphics object consisting of 0 graphics primitives

sage: plot(sqrt(x^2-1), -2, 2)  # [-1, 1] is excluded automatically
Graphics object consisting of 2 graphics primitives

sage: plot(arccsc, -2, 2)       # [-1, 1] is excluded automatically
Graphics object consisting of 2 graphics primitives

sage: set_verbose(0)

TESTS:

We do not randomize the endpoints:

sage: p = plot(x, (x,-1,1))
sage: p[0].xdata[0] == -1
True
sage: p[0].xdata[-1] == 1
True

We check to make sure that the x/y min/max data get set correctly
when there are multiple functions.

sage: d = plot([sin(x), cos(x)], 100, 120).get_minmax_data()
sage: d['xmin']
100.0
sage: d['xmax']
120.0

We check various combinations of tuples and functions, ending with
tests that lambda functions work properly with explicit variable
declaration, without a tuple.

sage: p = plot(lambda x: x,(x,-1,1))
sage: p = plot(lambda x: x,-1,1)
sage: p = plot(x,x,-1,1)
sage: p = plot(x,-1,1)
sage: p = plot(x^2,x,-1,1)
sage: p = plot(x^2,xmin=-1,xmax=2)
sage: p = plot(lambda x: x,x,-1,1)
sage: p = plot(lambda x: x^2,x,-1,1)
sage: p = plot(lambda x: 1/x,x,-1,1)
sage: f(x) = sin(x+3)-.1*x^3
sage: p = plot(lambda x: f(x),x,-1,1)

We check to handle cases where the function gets evaluated at a
point which causes an 'inf' or '-inf' result to be produced.

sage: p = plot(1/x, 0, 1)
sage: p = plot(-1/x, 0, 1)

Bad options now give better errors:

sage: P = plot(sin(1/x), (x,-1,3), foo=10)
Traceback (most recent call last):
...
RuntimeError: Error in line(): option 'foo' not valid.
sage: P = plot(x, (x,1,1)) # trac ticket #11753
Traceback (most recent call last):
...
ValueError: plot start point and end point must be different

We test that we can plot f(x)=x (see
http://trac.sagemath.org/10246):

sage: f(x)=x; f
x |--> x
sage: plot(f,(x,-1,1))
Graphics object consisting of 1 graphics primitive

Check that http://trac.sagemath.org/15030 is fixed:

sage: plot(abs(log(x)), x)
Graphics object consisting of 1 graphics primitive

Check that if excluded points are less than xmin then the exclusion
still works for polar and parametric plots. The following should
show two excluded points:

sage: set_verbose(-1)
sage: polar_plot(sin(sqrt(x^2-1)), (x,0,2*pi), exclude=[1/2,2,3])
Graphics object consisting of 3 graphics primitives

sage: parametric_plot((sqrt(x^2-1),sqrt(x^2-1/2)), (x,0,5), exclude=[1,2,3])
Graphics object consisting of 3 graphics primitives

sage: set_verbose(0)

(-8)^(1/3).n()

1.00000000000000 + 1.73205080756888*I