How to graph functions in Wolfram. How to graph functions in Wolfram Wolfram alpha commands

In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic medium with the names of all registered expedition participants.

Registration of participants is open. Get your ticket to Mars using this link.

If this post solved your problem or you just liked it, share the link to it with your friends on social networks.

One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."

In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic medium with the names of all registered expedition participants.

Registration of participants is open. Get your ticket to Mars using this link.

If this post solved your problem or you just liked it, share the link to it with your friends on social networks.

One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."

Let's start by plotting a simple 2-dimensional graph: plot sin(sqrt(7)x)+19cos(x) for x from -20 to 20

If we replace 7 with (-7), we get graphs of the real and imaginary parts of the function: plot sin(sqrt(-7)x)+19cos(x) for x from -5 to 5

In the two previous examples, we specified the range of values ​​for the argument x. What will happen if we do not specify the range of values ​​of x?

One of the unique features of Wolfram | Alpha is the automatic selection of an appropriate range of x for plotting functions of one and two variables, for example, as in plotting this function containing Bessel functions:

To Wolfram | Alpha, to plot a function, we always use the plot prefix. If we introduce any one-dimensional expression without the plot prefix, we will receive, in addition to the graph of the function in rectangular Cartesian coordinates, a lot of other information about this function.

Compare:

Additionally, the plotted graph image will be larger if you use the plot prefix.

Simultaneously in Wolfram | Alpha can plot graphs of several functions.

If you hover your mouse over the lower left corner of the image, two links become available: Save as image and Copyable planetext. Consider this graph:

The first Save as image link, which opens in the lower left corner of the image, allows you to save the constructed graph as a picture on the user’s computer - when you click on Save as image, the image will automatically start downloading:

Now let's look at how in Wolfram | Alpha construct graphs of functions of two variables. Let's start with the function y^2 cos(x) for x from -6 to 6 and y from -2 to 2

As in the one-dimensional case, Wolfram | Alpha automatically determines the appropriate range of argument values ​​where the function has the most characteristic form. In case Wolfram | Alpha cannot find a suitable range, this is most likely because the system was unable to determine the range where the function has the most interesting behavior. In this case, we can set the range manually, as was done above. Check out the following examples:
But what if you want to plot several graphs of functions of two variables at the same time?

Wolfram | Alpha plots a separate graph for each feature in the list. Here are some more examples:
New Wolfram Feature | Alpha is the ability to plot the real and imaginary parts of complex-valued functions of two variables:
In all the examples discussed above, Wolfram | Alpha also produced contour plots (level lines) in addition to 3D plots (surfaces). To see the connection between three-dimensional and contour graphs, you need to click the “Show contour lines” button. Note that both 3D and contour plots use the same range of arguments.

All three-dimensional plots are plotted using the plot3d function of Mathematica. Contour plots were made using ContourPlot. In both cases, to see the Mathematica code for generating the image, you need to click the Copyable planetext link in the lower left corner of the desired image.

More information on using Wolfram|Alpha can be found in the blog

Request to Wolfram|Alpha: global maxima sin(x)

:

You can also build multiple graphs by asking the appropriate queries:

Request to Wolfram|Alpha: Plot sin(x), sin(2x), sin(3x)

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Plot sin(x), sin(sin(x)), sin(sin(sin(x))), sin(sin(sin(sin(x)))), sin(sin(sin(sin(sin(x) )))))

Code in Wolfram Language (Mathematica):

?

Request to Wolfram|Alpha: Contourplot sin(x/|y| - y/|x|) from x = -pi to pi and y = -pi to pi

Code in Wolfram Language (Mathematica):

?

Request to Wolfram|Alpha: Plot3d sin(x - y) / sin(x + y) from x = -2pi to 2pi and y = -2pi to 2pi

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Polar plot r = 1 + sin(100 theta)

Code in Wolfram Language (Mathematica):

The result of the last query obtained in Wolfram|Alpha looks like this:

You can also easily construct more complex expressions that depend on the sine function, for example:
, where (y) is the fractional part of the number y.

Request to Wolfram|Alpha: Plot frac(1/frac(1/sin(x)))

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Plot sin(x!)! from x = -3 to 3

Code in Wolfram Language (Mathematica):

. (total 101 members)

Request to Wolfram|Alpha: Plot nestlist(sin, 1., 100)

Code in Wolfram Language (Mathematica):

x=max(sin(t), cos(pi t)), y=max(cos(t), sin(pi t))) t = 0 to 100

Request to Wolfram|Alpha: Parametric plot (max(sin(t), cos(pi t)), max(cos(t), sin(pi t))) from t = 0 to 100

Code in Wolfram Language (Mathematica):

sin(sin(x + i y)) for x =-? before? and y =-? before?

Request to Wolfram|Alpha: Plot3d sin(sin(x + i y)) from x=-pi to pi and y =-pi to pi

Code in Wolfram Language (Mathematica):

r = min(sin(x), sin(sqrt(2) x), sin(sqrt(3) x), sin(sqrt(5) x)) for x = 0 to 100 ?

Request to Wolfram|Alpha: Polar plot min(sin(x), sin(sqrt(2) x), sin(sqrt(3) x), sin(sqrt(5) x)) from x = 0 to 100 pi

Code in Wolfram Language (Mathematica):

r = exp(sin(theta)) - 2 cos(4 theta) + sin^5(theta/12 - pi/24)

Request to Wolfram|Alpha: Polar plot r = exp(sin(theta)) - 2 cos(4 theta) + sin^5(theta/12 - pi/24)

Code in Wolfram Language (Mathematica):

(sin(s + pi/2) + sin(s + pi/2)sin (t + pi/2)/2, sin(s) + sin(s)sin (t + pi/2)/2, sin (t)/2) for s = 0 to 2? and t = 0 to 2?

Request to Wolfram|Alpha: Parametric plot3D (sin(s + pi/2) + sin(s + pi/2)sin (t + pi/2)/2, sin(s) + sin(s)sin (t + pi/2)/2 , sin (t)/2) from s = 0 to 2pi and t = 0 to 2pi

Code in Wolfram Language (Mathematica):

You can also use Mathematica syntax to define many expressions and process them:
(Re(sin(x + iy)), Im(Sin(x + iy)))

Request to Wolfram|Alpha: StreamDensityPlot[(Re], Im]), (x, -Pi, Pi), (y, -Pi, Pi), ColorFunction -> “ThermometerColors”]

Code in Wolfram Language (Mathematica):

at k=0,1,2,3,...,30 purple

Request to Wolfram|Alpha: Plot, (k, 0, 30)],(x, 0, Pi/2)] in purple

Code in Wolfram Language (Mathematica):

We can calculate some specific values ​​of the sine function, say

Request to Wolfram|Alpha: sin(pi/88)

Code in Wolfram Language (Mathematica):

You can ask Wolfam|Alpha if any expressions have certain properties or forms:
?

Request to Wolfram|Alpha: Is sin(2/3) algebraic?

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Toradicals(sin(pi/(2^4 3 5)))

Code in Wolfram Language (Mathematica):

You can search for periods of various functions:

Request to Wolfram|Alpha: Period of sin(x)

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Period of sin(x)+2sin(2x)+3sin(3x)

Code in Wolfram Language (Mathematica):

And also find the maxima and minima of functions containing the sine function:

Request to Wolfram|Alpha: Minimize sinx + |x|

Code in Wolfram Language (Mathematica):

Find the maximum of the function (sin(x)/x)^2 located between the points? and 4?

Request to Wolfram|Alpha: Maximize (sin(x)/x)^2 between pi and 4 pi

Code in Wolfram Language (Mathematica):

We can also construct both 2D and 3D Lissajous figures:
(sin(11t), sin(13t))

Request to Wolfram|Alpha: Parametric plot (sin(11t), sin(13t))

Code in Wolfram Language (Mathematica):

(sin(2t), sin(3t), sin(5t)) from t = 0 to 2pi

Request to Wolfram|Alpha: Parametric plot (sin(2t), sin(3t), sin(5t)) from t = 0 to 2pi

Code in Wolfram Language (Mathematica):

The result of the Parametric plot (sin(11t), sin(13t)) query mentioned above:

You can not only construct curves, but also calculate their curvature:
(sin(3t), sin(4t)) at point t = 1

Request to Wolfram|Alpha: Curvature of (sin(3t), sin(4t)) at t = 1

Code in Wolfram Language (Mathematica):

Find the coordinates of the inflection points of the curve:
(sin(t), sin(2t)) at t = 0 to?

Request to Wolfram|Alpha: Arc length (sin(t), sin(2t)) from t = 0 to pi

Code in Wolfram Language (Mathematica):

Arc length of the polar curve r = phi sin(phi) for phi = 0 to 12?

Request to Wolfram|Alpha: Arc length r = phi sin(phi) from phi = 0 to 12pi

Code in Wolfram Language (Mathematica):

Below is the result that Wolfram|Alpha produces for the previous query about the length of a curve defined in a polar coordinate system:

You can find the cusp points of a function:

Request to Wolfram|Alpha: Corners |sin(x)|

Code in Wolfram Language (Mathematica):

Or check the function for periodicity:
?

Request to Wolfram|Alpha: Periodicity sin(4x + pi/3)

Code in Wolfram Language (Mathematica):

There are many mathematical formulas that may be required. Let's look at a few specific examples:

Request to Wolfram|Alpha: Trig reduce sin(x)^10

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Trig expand sin(10x)

Code in Wolfram Language (Mathematica):

In the same way, you can obtain the basic formulas of trigonometry:

Request to Wolfram|Alpha: Half-angle formulas sinx

Code in Wolfram Language (Mathematica):

Request to Wolfram|Alpha: Double-angle formulas sinx

Code in Wolfram Language (Mathematica).

In July 2020, NASA launches an expedition to Mars. The spacecraft will deliver to Mars an electronic medium with the names of all registered expedition participants.

Registration of participants is open. Get your ticket to Mars using this link.

If this post solved your problem or you just liked it, share the link to it with your friends on social networks.

One of these code options needs to be copied and pasted into the code of your web page, preferably between tags and or immediately after the tag. According to the first option, MathJax loads faster and slows down the page less. But the second option automatically monitors and loads the latest versions of MathJax. If you insert the first code, it will need to be updated periodically. If you insert the second code, the pages will load more slowly, but you will not need to constantly monitor MathJax updates.

The easiest way to connect MathJax is in Blogger or WordPress: in the site control panel, add a widget designed to insert third-party JavaScript code, copy the first or second version of the download code presented above into it, and place the widget closer to the beginning of the template (by the way, this is not at all necessary , since the MathJax script is loaded asynchronously). That's all. Now learn the markup syntax of MathML, LaTeX, and ASCIIMathML, and you are ready to insert mathematical formulas into your site's web pages.

Another New Year's Eve... frosty weather and snowflakes on the window glass... All this prompted me to write again about... fractals, and what Wolfram Alpha knows about it. There is an interesting article on this subject, which contains examples of two-dimensional fractal structures. Here we will look at more complex examples of three-dimensional fractals.

A fractal can be visually represented (described) as a geometric figure or body (meaning that both are a set, in this case, a set of points), the details of which have the same shape as the original figure itself. That is, this is a self-similar structure, examining the details of which when magnified, we will see the same shape as without magnification. Whereas in the case of an ordinary geometric figure (not a fractal), upon magnification we will see details that have a simpler shape than the original figure itself. For example, at a high enough magnification, part of an ellipse looks like a straight line segment. This does not happen with fractals: with any increase in them, we will again see the same complex shape, which will be repeated again and again with each increase.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article Fractals and Art in the Name of Science: “Fractals are geometric shapes that are as complex in their details as in their overall form. That is, if part of the fractal will be enlarged to the size of the whole, it will appear as a whole, either exactly, or perhaps with a slight deformation."