Image:Graph of sliding derivative line.gif
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[edit] Summary
| Description | Boulder, Colorado
Graph of sliding derivative line |
|---|---|
| Source |
self-made |
| Date |
February 19, 2008 |
| Author | |
| Permission (Reusing this image) |
See below. |
[edit] Licensing
 |
I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide. In case this is not legally possible, |
I made this with SAGE, an open-source math package, and release it to the world.
A version of the source code lives at this link.
A wikipedian asked, so the SAGE source code I used to generate the image follows, verbose documentation and all.
Below. The current image has different constants than the original code I posted,
number_of_plot_points = 50
and
figure_size = 2.5
Anyone wishing to use/modify/shoot at this source code is free to:
#***************************************************************************** # Copyright (C) 2008 Dean Moore < deanlorenmoore@gmail.com >, # < dino@boulder.net > # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #***************************************************************************** ################################################################################# # # # What this program does, needed input, boilerplate info & disclaimers: # # # # This program was written for the mathematical package SAGE (by implication, # # using the Python programming language), and animates a sliding derivative # # line to a given one-variable single-valued function, defined as *f* (below). # # # # A picture paints a thousand words, and it's best to look at some sample # # output. In short, it's an animated illustration of a standard Calc I # # concept. # # # # Once function *f* is defined, parameters *a* and *b* define the top & bottom # # x-values, as the interval [a, b]. The program is designed to churn without # # further input, doing all needed calculations after definition of the # # function & interval; the user may wish to vary a few other parameters as: # # # # The number of frames in the final "movie," the length, thickness & color of # # the function's curve, the length, thickness & color of the tangent line, and # # a few others (below documentation). # # # # Caveat: don't use unbounded functions, e.g., f(x) = 1/x on (0,1]. No # # functions with unbounded first derivatives, e.g., f(x) = x^(1/3) about x = 0. # # # # Doubtless there are other ways to break the code of which I have not # # thought. If the user finds it amusing, fine with me. # # # # User-input material: # # # # f: The function. No unbounded functions; no unbounded first derivatives. # # # # a: lower x-bound. # # b: upper x-bound. # # # # number_of_plot_points: Number of frames in final "movie." # # # # thickness_of_curve: Thickness of function's curve. # # color_of_curve: By name. # # # # length_of_deriv_line: Length of line that slides along, always tangent to # # the function *f*; its slope always the derivative of # # *f* at point of tangency. # # thickness_deriv_line: Thickness of derivative line. # # color_of_deriv_line: By name. # # # # sliding_point_size: By name, how big is the tracing point See output. # # color_of_sliding_point: A point traces the curve; this is its color. # # # # figure_size: Regulates size of final image. # # # # delay_between_frames: Delay between frames of final "movie." # # # # The code isn't that error-checking, assuming certain numbers positive, # # assuming b > a, assuming *f* "nice" with a "nice" derivative. The program # # isn't designed to be idiot-proof. # # # ################################################################################# ################################################################################# # # # Why did I write it? A challenge to learn more, to do it in SAGE. I looked # # at < http://en.wikipedia.org/wiki/Derivative > & thought, "Gee, maybe some # # more lively images. It's 2008." # # # # The code's not perfect. The sliding line runs faster where the curve is # # steep, and kind of slow where it's flat. This is an unavoidable consequence # # of breaking interval [a, b] into equal subintervals. I can scope in my head # # one way to make the sliding line run at a somewhat consistent rate, but it # # would be CPU-intensive & bothersome. # # # # Written by Dean Moore, February, 2008. # # # ################################################################################# # User-defined material: f = x*sin(x^2) # The function. a = -1 # Bottom x-value b = 3 # Top x-value number_of_plot_points = 125 # Number of points at which the function & derivative # are sampled: the more, the smoother runs the final # "movie" -- and the slower the program runs, the # slower the "movie" & the more bytes to the image. thickness_of_curve = 0.75 # Thickness of function's curve. color_of_curve = (0, 0, 1) # By name. length_of_deriv_line = 3 # Length of "sliding" derivative line. thickness_deriv_line = 1 # Thickness of derivative line. color_of_deriv_line = (1, 0, 0) # By name. sliding_point_size = 20 # By name; how big is the tracing point. color_of_sliding_point = (0, 0, 0) # A point traces the curve; this is its color. figure_size = 4 # Regulates size of final picture. The bigger, the # more bytes / image, and the slower it runs. delay_between_frames = 0 # Time between "frames" of final "movie." # End user-defined material. # The rest of the program churns without user input. The disinterested user # may safely ignore the rest, unless s/he wishes to modify the code -- or I # made some error I didn't catch. # Top matter, no nice way to classify. Stuff that needs done: v = [] # This will hold graphic image of function's curve. length_of_deriv_line /= 2 # We draw half below, half # above. User doesn't need to know this. # We use the same loop several times, from bottom of interval at *a* to # to top at *b* by an obvious step; put the range in a parameter. Note # we include the endpoint: loop_range = srange(a, b + (b - a)/number_of_plot_points, (b - a)/number_of_plot_points) # Done with this. # We use each function value three times; place them # in a dictionary for efficiency. function_values = {} # Blank dictionary to hold function values. for i in loop_range: function_values[i] = f(i) # Done filling dictionary. f_prime = f.derivative() # Derivative of function *f* (above). def deriv_line(x0,x): # A function of two variables: return f_prime(x0)*x + ( f(x0) - f_prime(x0)*x0 ) # Draws a line through (x0, f(x0)), # slope (df/dx)(x0). The derivative # and function are evaluated at # "fixed" *x0*; the "real" independent # variable is *x*. Parens are only # for clarity, y = mx + b form. # End top matter. # The next code is more involved. We need to make a sliding derivative # line of constant length. While experimenting I tried from, say, one # below & one above on the x-axis, and it looked awful. # Define four dictionaries that will hold coordinates of line segments: bottom_x_points = {} bottom_y_points = {} top_x_points = {} top_y_points = {} # Next loop populates the dictionaries with coordinates of line segments of # constant length *length_of_deriv_line* (well, its original value, twice # its current value): for i in loop_range: theta = RDF(arctan(f_prime(i))) # Angle of derivative line. # Use trig & basic calc # to verify next block: c = RDF(cos(theta)) # Both of these are s = RDF(sin(theta)) # used twice, never # outside this block. # Note sign patterns in next: bottom_x_points[i] = i - length_of_deriv_line*c # Bottom x-value of deriv line bottom_y_points[i] = function_values[i] - length_of_deriv_line*s # Bottom y-value of deriv line. top_x_points[i] = i + length_of_deriv_line*c # Top x-value of deriv line top_y_points[i] = function_values[i] + length_of_deriv_line*s # Top y-value of deriv line. # Done computing line segments that comprise the sliding derivative line. # We find max & min x-values for the graph, so the display looks nice: max_x_val = max(max(top_x_points.values()), max(bottom_x_points.values())) # Not perfect, but if parameter min_x_val = min(min(top_x_points.values()), min(bottom_x_points.values())) # *number_of_plot_points* is large # enough for a fairly fine "mesh," # good enough. # We find max & min y-values for graph: max_y_val = max(max(top_y_points.values()), max(bottom_y_points.values())) # See above documentation of min_y_val = min(min(top_y_points.values()), min(bottom_y_points.values())) # *max_x_val* & *min_x_val*. # Graph the sliding derivative line: sliding_deriv_line = animate(line([(bottom_x_points[i], bottom_y_points[i]), (top_x_points[i], top_y_points[i] )], thickness = thickness_deriv_line, rgbcolor = color_of_deriv_line) for i in loop_range ) # Graph the point that slides along the graph. sliding_points = animate([point((i,function_values[i]), rgbcolor = color_of_sliding_point, pointsize = sliding_point_size) for i in loop_range ], xmin = min_x_val, ymin = min_y_val, xmax = max_x_val, ymax = max_y_val, figsize = [figure_size,figure_size] ) # Of note on next: when I first did the graph, it was "wiggly." This # led to a note to sage-support, which led to a ticket, which someone # fixed (thanks!), but the fix won't come out until the next version # of SAGE. So for this code I use a work-around. # Animate the graph: graph = (plot(f, [a, b], thickness = thickness_of_curve, rgbcolor = color_of_curve)) for i in loop_range: v.append(graph) # End of i-loop. curve = animate(v) # Must be done. # Now show the final "movie": (curve + sliding_points + sliding_deriv_line).show(delay = delay_between_frames) # Done with program.
--- Done source code.
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This file is an exact duplicate of another file from the Wikimedia Commons. Unless it is currently protected from editing, this media file may qualify for speedy deletion if it satisfies these conditions. |  |
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Dimensions | User | Comment | |
|---|---|---|---|---|
| current | 03:40, 22 February 2008 | 250×250 (202 KB) | Oleg Alexandrov (Talk | contribs) | (Reverted to version as of 02:46, 22 February 2008, my image is bigger than this) |
| 03:37, 22 February 2008 | 174×200 (294 KB) | Oleg Alexandrov (Talk | contribs) | (Crop a bit) | |
| 02:46, 22 February 2008 | 250×250 (202 KB) | Dino (Talk | contribs) | (Fewer bytes, about 208 kb) | |
| 01:41, 21 February 2008 | 500×500 (842 KB) | Dino (Talk | contribs) | (Got rid of "wiggling"; thanks to some person on sage-support for fixing this ticket; I did a work-around.) | |
| 17:16, 20 February 2008 | 400×400 (550 KB) | Dino (Talk | contribs) | ||
| 02:32, 20 February 2008 | 400×400 (547 KB) | Dino (Talk | contribs) | ({{Information |Description=Graph of sliding derivative line |Source=self-made |Date=February 19, 2008 |Location=Boulder, Colorado |Author=~~~ |other_versions= }}) |
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