Differences between revisions 40 and 52 (spanning 12 versions)
⇤ ← Revision 40 as of 2011-06-17 01:43:57 →
Size: 56871
Editor: JohnTravis
Comment:
← Revision 52 as of 2012-04-07 01:19:41 → ⇥
Size: 57661
Editor: jason
Comment: make interacts live.
Deletions are marked like this. Additions are marked like this.
Line 6: Line 6:
Line 9: Line 10:
{{{ {{{#!sagecell
Line 58: Line 59:
{{{ {{{#!sagecell
Line 100: Line 101:
{{{ {{{#!sagecell
Line 116: Line 117:
{{{ {{{#!sagecell
Line 136: Line 137:
{{{ {{{#!sagecell
Line 164: Line 165:
{{{ {{{#!sagecell
Line 237: Line 238:
{{{ {{{#!sagecell
Line 250: Line 251:
{{{ {{{#!sagecell
Line 348: Line 349:
{{{ {{{#!sagecell
Line 454: Line 455:
{{{ {{{#!sagecell
Line 496: Line 497:
{{{ {{{#!sagecell
Line 519: Line 520:
{{{ {{{#!sagecell
Line 546: Line 547:
{{{ {{{#!sagecell
Line 572: Line 573:
{{{ {{{#!sagecell
Line 587: Line 588:
{{{ {{{#!sagecell
Line 633: Line 634:
{{{
from scipy.special.orthogonal import p_roots		 {{{#!sagecell
import scipy
import numpy
from scipy.special.orthogonal import p_roots, t_roots, u_roots
Line 643: Line 646:
            'Chebyshev': {'w': 1/sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': t_roots},
                'Chebyshev2': {'w': sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': u_roots},
                'Trapezoid': {'w': 1, 'xmin': -1, 'xmax': 1, 'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[2.0r]*(n-2)+[1.0r])*1.0r/n)},
                'Simpson': {'w': 1, 'xmin': -1, 'xmax': 1, 'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[4.0r,2.0r]*int((n-3.0r)/2.0r)+[4.0r,1.0r])*2.0r/(3.0r*n))}}		      'Chebyshev': {'w': 1/sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': t_roots},
     'Chebyshev2': {'w': sqrt(1-x**2), 'xmin': -1, 'xmax': 1, 'func': u_roots},
     'Trapezoid': {'w': 1, 'xmin': -1, 'xmax': 1,          'func': lambda n: (linspace(-1r,1,n), numpy.array([1.0r]+[2.0r]*(n-2)+[1.0r])*1.0r/n)},
     'Simpson': {'w': 1, 'xmin': -1, 'xmax': 1,          'func': lambda n: (linspace(-1r,1,n),
            numpy.array([1.0r]+[4.0r,2.0r]*int((n-3.0r)/2.0r)+[4.0r,1.0r])*2.0r/(3.0r*n))}}
Line 650: Line 656:
    return polygon([(center-width2,0),(center+width2,0),(center+width2,height),(center-width2,height)],**kwds)     return polygon([(center-width2,0),
        (center+width2,0),(center+width2,height),(center-width2,height)],**kwds)
Line 654: Line 661:
def weights(n=slider(1,30,1,default=10),f=input_box(default=3*x+cos(10*x)),show_method=["Legendre", "Chebyshev", "Chebyshev2", "Trapezoid","Simpson"]): def weights(n=slider(1,30,1,default=10),f=input_box(default=3*x+cos(10*x),type=SR),
    show_method=["Legendre", "Chebyshev", "Chebyshev2", "Trapezoid","Simpson"]):
Line 663: Line 671:
    scaled_ff = fast_float(scaled_func)     scaled_ff = fast_float(scaled_func, 'x')
Line 671: Line 679:
    stems = sum(line([(x,0),(x,scaled_ff(x))],rgbcolor=(1-y,1-y,1-y),thickness=2,markersize=6,alpha=y) for x,y in coords_scaled)
    points = sum([point([(x,0),(x,scaled_ff(x))],rgbcolor='black',pointsize=30) for x,_ in coords])		     stems = sum(line([(x,0),(x,scaled_ff(x))],rgbcolor=(1-y,1-y,1-y),
        thickness=2,markersize=6,alpha=y) for x,y in coords_scaled)
    points = sum([point([(x,0),
        (x,scaled_ff(x))],rgbcolor='black',pointsize=30) for x,_ in coords])
Line 677: Line 687:
    show(graph,xmin=plot_min,xmax=plot_max)     show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto")
Line 685: Line 695:
    html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,latex(f.subs(x="x_i")), approximation, integral, latex(scaled_func)))     html("$$\sum_{i=1}^{i=%s}w_i\left(%s\\right)= %s\\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,
        latex(f), approximation, integral, latex(scaled_func)))
Line 698: Line 709:
{{{ {{{#!sagecell
Line 810: Line 821:
{{{ {{{#!sagecell
Line 936: Line 947:
{{{ http://www.sagenb.org/home/pub/2828/

{{{#!sagecell
Line 944: Line 957:
## An updated version of this worksheet may be available at http://sagenb.mc.edu
Line 970: Line 984:
{{{ {{{#!sagecell
Line 1007: Line 1021:
{{{ {{{#!sagecell
Line 1060: Line 1074:
{{{
%hide
%auto		 {{{#!sagecell
Line 1121: Line 1133:
{{{ {{{#!sagecell
Line 1159: Line 1171:
{{{ {{{#!sagecell
Line 1197: Line 1209:
{{{ http://www.sagenb.org/home/pub/2829/

{{{#!sagecell
Line 1201: Line 1215:
##
##
## An updated version of this worksheet may be available at http://sagenb.mc.edu
Line 1299: Line 1316:
{{{ http://www.sagenb.org/home/pub/2826/

{{{#!sagecell
Line 1356: Line 1375:
{{{ {{{#!sagecell
Line 1375: Line 1394:
{{{ http://www.sagenb.org/home/pub/2827/

{{{#!sagecell
Line 1382: Line 1403:
##
## An updated version of this worksheet may be available at http://sagenb.mc.edu

Sage Interactions - Calculus

goto interact main page

Contents

  1. Sage Interactions - Calculus
    1. Root Finding Using Bisection
    2. Newton's Method
    3. A contour map and 3d plot of two inverse distance functions
    4. A simple tangent line grapher
    5. Numerical integrals with the midpoint rule
    6. Numerical integrals with various rules
    7. Some polar parametric curves
    8. Function tool
    9. Newton-Raphson Root Finding
    10. Coordinate Transformations
    11. Taylor Series
    12. Illustration of the precise definition of a limit
    13. A graphical illustration of sin(x)/x -> 1 as x-> 0
    14. Quadric Surface Plotter
    15. The midpoint rule for numerically integrating a function of two variables
    16. Gaussian (Legendre) quadrature
    17. Vector Calculus, 2-D Motion
    18. Vector Calculus, 3-D Motion
    19. Multivariate Limits by Definition
    20. Directional Derivatives
    21. 3D graph with points and curves
    22. Approximating function in two variables by differential
    23. Taylor approximations in two variables
    24. Volumes over non-rectangular domains
    25. Lateral Surface Area
    26. Parametric surface example
    27. Line Integrals in 3D Vector Field

Root Finding Using Bisection

by William Stein

bisect.png

Newton's Method

Note that there is a more complicated Newton's method below.

by William Stein

http://sagenb.org/home/pub/2824/

newton.png

A contour map and 3d plot of two inverse distance functions

by William Stein

http://sagenb.org/home/pub/2823/

mountains.png

A simple tangent line grapher

by Marshall Hampton

tangents.png

Numerical integrals with the midpoint rule

by Marshall Hampton

num_int.png

Numerical integrals with various rules

by Nick Alexander (based on the work of Marshall Hampton)

num_int2.png

Some polar parametric curves

by Marshall Hampton. This is not very general, but could be modified to show other families of polar curves.

polarcurves1.png

Function tool

Enter symbolic functions f, g, and a, a range, then click the appropriate button to compute and plot some combination of f, g, and a along with f and g. This is inspired by the Matlab funtool GUI.

funtool.png

Newton-Raphson Root Finding

by Neal Holtz

This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued.

newtraph.png

Coordinate Transformations

by Jason Grout

coordinate-transform-1.png coordinate-transform-2.png

Taylor Series

by Harald Schilly

taylor_series_animated.gif

Illustration of the precise definition of a limit

by John Perry

I'll break tradition and put the image first. Apologies if this is Not A Good Thing.

snapshot_epsilon_delta.png

A graphical illustration of sin(x)/x -> 1 as x-> 0

by Wai Yan Pong

sinelimit.png

Quadric Surface Plotter

by Marshall Hampton. This is pretty simple, so I encourage people to spruce it up. In particular, it isn't set up to show all possible types of quadrics.

quadrics.png

The midpoint rule for numerically integrating a function of two variables

by Marshall Hampton

numint2d.png

Gaussian (Legendre) quadrature

by Jason Grout

The output shows the points evaluated using Gaussian quadrature (using a weight of 1, so using Legendre polynomials). The vertical bars are shaded to represent the relative weights of the points (darker = more weight). The error in the trapezoid, Simpson, and quadrature methods is both printed out and compared through a bar graph. The "Real" error is the error returned from scipy on the definite integral.

quadrature1.png quadrature2.png

Vector Calculus, 2-D Motion

By Rob Beezer

A fast_float() version is available in a worksheet

motion2d.png

Vector Calculus, 3-D Motion

by Rob Beezer

Available as a worksheet

motion3d.png

Multivariate Limits by Definition

by John Travis

http://www.sagenb.org/home/pub/2828/

3D_Limit_Defn.png

3D_Limit_Defn_Contours.png

Directional Derivatives

This interact displays graphically a tangent line to a function, illustrating a directional derivative (the slope of the tangent line).

directional derivative.png

3D graph with points and curves

By Robert Marik

This sagelet is handy when showing local, constrained and absolute maxima and minima in two variables. Available as a worksheet

3Dgraph_with_points.png

Approximating function in two variables by differential

by Robert Marik

3D_differential.png

Taylor approximations in two variables

by John Palmieri

This displays the nth order Taylor approximation, for n from 1 to 10, of the function sin(x2 + y2) cos(y) exp(-(x2+y2)/2).

taylor-3d.png

Volumes over non-rectangular domains

by John Travis

http://www.sagenb.org/home/pub/2829/

3D_Irregular_Volume.png

Lateral Surface Area

by John Travis

http://www.sagenb.org/home/pub/2826/

Lateral_Surface.png

Parametric surface example

by Marshall Hampton

parametric_surface.png

Line Integrals in 3D Vector Field

by John Travis

http://www.sagenb.org/home/pub/2827/

3D_Line_Integral.png

interact/calculus (last edited 2020-08-11 14:10:09 by kcrisman)