Differences between revisions 45 and 58 (spanning 13 versions)

Sage Interactions - Calculus

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Contents

Sage Interactions - Calculus

Root Finding Using Bisection

by William Stein

Newton's Method

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

by William Stein

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

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

by William Stein

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

A simple tangent line grapher

by Marshall Hampton

Numerical integrals with the midpoint rule

by Marshall Hampton

Numerical integrals with various rules

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

Some polar parametric curves

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

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.

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.

Coordinate Transformations

by Jason Grout

Taylor Series

by Harald Schilly

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.

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

by Wai Yan Pong

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.

The midpoint rule for numerically integrating a function of two variables

by Marshall Hampton

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.

Vector Calculus, 2-D Motion FIXME

By Rob Beezer

A fast_float() version is available in a worksheet

Vector Calculus, 3-D Motion

by Rob Beezer

Available as a worksheet

Multivariate Limits by Definition FIXME

by John Travis

http://sagenb.mc.edu/home/pub/97/

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

Approximating function in two variables by differential

by Robert Marik

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).

Volumes over non-rectangular domains

by John Travis

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

Lateral Surface Area

by John Travis

http://sagenb.mc.edu/home/pub/89/

Parametric surface example

by Marshall Hampton

Line Integrals in 3D Vector Field

by John Travis

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

-  ⇤ ← Revision 45 as of 2012-03-16 06:47:41 → 
  Size: 57859
  Editor: jason
  Comment:
+   ← Revision 58 as of 2013-04-24 18:48:01 → ⇥
  Size: 61047
  Editor: travis
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 6:
-{{{#!html
<script type="text/javascript" src="http://aleph.sagemath.org/static/jquery.min.js"></script>
<script type="text/javascript" src="http://aleph.sagemath.org/embedded_sagecell.js"></script>
<script type="text/javascript">alert('hi');</script>
    <script>
$(function() {
    var makecells = function() {
        sagecell.makeSagecell({
            inputLocation: '#interact1',
            evalButtonText: 'Interact'});
    }
    sagecell.init(makecells);
})</script>

}}}
-Line 25:
+Line 10:
-{{{#!html
<div id="interact1"><script type="text/code">
+{{{#!sagecell
-Line 65:
+Line 49:
-</script></div>
-Line 76:
+Line 59:
-{{{
+{{{#!sagecell
-Line 118:
+Line 101:
-{{{
+{{{#!sagecell
-Line 134:
+Line 117:
-{{{
+{{{#!sagecell
-Line 154:
+Line 137:
-{{{
+{{{#!sagecell
-Line 182:
+Line 165:
-{{{
+{{{#!sagecell
-Line 255:
+Line 238:
-{{{
+{{{#!sagecell
-Line 268:
+Line 251:
-{{{
+{{{#!sagecell
-Line 366:
+Line 349:
-{{{
+{{{#!sagecell
-Line 472:
+Line 455:
-{{{
+{{{#!sagecell
-Line 474:
+Line 457:
+# polar coordinates
#(x,y)=(u*cos(v),u*sin(v)); (u_range,v_range)=([0..6],[0..2*pi,step=pi/12])

# weird example
(x,y)=(u^2-v^2,u*v+cos(u*v)); (u_range,v_range)=([-5..5],[-5..5])

thickness=4
square_length=.05
-Line 477:
+Line 469:
-def trans(x=input_box(u^2-v^2, label="x=",type=SR), \
         y=input_box(u*v+cos(u*v), label="y=",type=SR), \
         t_val=slider(0,10,0.2,6, label="Length of curves"), \
+def trans(x=input_box(x, label="x",type=SR),
         y=input_box(y, label="y",type=SR),
-Line 482:
+Line 473:
-         u_range=input_box(range(-5,5,1), label="u lines"),
         v_range=input_box(range(-5,5,1), label="v lines")):
     thickness=4
     u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
     v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
     t_min = -t_val
     t_max = t_val
     g1=sum([parametric_plot((i,v), (v,t_min,t_max), rgbcolor=(1,0,0)) for i in u_range])
     g2=sum([parametric_plot((u,i), (u,t_min,t_max), rgbcolor=(0,0,1)) for i in v_range])
     vline_straight=parametric_plot((u,v_val), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
     uline_straight=parametric_plot((u_val, v), (v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
 
     (g1+g2+vline_straight+uline_straight).save("uv_coord.png",aspect_ratio=1, figsize=[5,5], axes_labels=['$u$','$v$'])
     xuv = fast_float(x,'u','v')
     yuv = fast_float(y,'u','v')
     xvu = fast_float(x,'v','u')
     yvu = fast_float(y,'v','u')
     g3=sum([parametric_plot((partial(xuv,i),partial(yuv,i)), (v,t_min,t_max), rgbcolor=(1,0,0)) for i in u_range])
     g4=sum([parametric_plot((partial(xvu,i),partial(yvu,i)), (u,t_min,t_max), rgbcolor=(0,0,1)) for i in v_range])
     uline=parametric_plot((partial(xuv,u_val),partial(yuv,u_val)),(v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
     vline=parametric_plot((partial(xvu,v_val),partial(yvu,v_val)), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
     (g3+g4+vline+uline).save("xy_coord.png", aspect_ratio=1, figsize=[5,5], axes_labels=['$x$','$y$'])
     print jsmath("x=%s, \: y=%s"%(latex(x), latex(y)))
     print "<html><table><tr><td><img src='cell://uv_coord.png'/></td><td><img src='cell://xy_coord.png'/></td></tr></table></html>"
}}}
+         t_val=slider(0,10,0.2,6, label="Length"),
         u_range=input_box(u_range, label="u lines"),
         v_range=input_box(v_range, label="v lines")):

    x(u,v)=x
    y(u,v)=y
    u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
    v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
    t_min = -t_val
    t_max = t_val
    uvplot=sum([parametric_plot((i,v), (v,t_min,t_max), color='red',axes_labels=['u','v'],figsize=[5,5]) for i in u_range])
    uvplot+=sum([parametric_plot((u,i), (u,t_min,t_max), color='blue',axes_labels=['u','v']) for i in v_range])
    uvplot+=parametric_plot((u,v_val), (u,t_min,t_max), rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
    uvplot+=parametric_plot((u_val, v), (v,t_min,t_max),rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
    pt=vector([u_val,v_val])
    du=vector([(t_max-t_min)*square_length,0])
    dv=vector([0,(t_max-t_min)*square_length])
    uvplot+=polygon([pt,pt+dv,pt+du+dv,pt+du],color='purple',alpha=0.7)
    uvplot+=line([pt,pt+dv,pt+du+dv,pt+du],color='green')

    T(u,v)=(x,y)
    xuv = fast_float(x,'u','v')
    yuv = fast_float(y,'u','v')
    xvu = fast_float(x,'v','u')
    yvu = fast_float(y,'v','u')
    xyplot=sum([parametric_plot((partial(xuv,i),partial(yuv,i)), (v,t_min,t_max), color='red', axes_labels=['x','y'],figsize=[5,5]) for i in u_range])
    xyplot+=sum([parametric_plot((partial(xvu,i),partial(yvu,i)), (u,t_min,t_max), color='blue') for i in v_range])
    xyplot+=parametric_plot((partial(xuv,u_val),partial(yuv,u_val)),(v,t_min,t_max),color='red', linestyle='-',thickness=thickness)
    xyplot+=parametric_plot((partial(xvu,v_val),partial(yvu,v_val)), (u,t_min,t_max), color='blue', linestyle='-',thickness=thickness)
    jacobian=abs(T.diff().det()).simplify_full()
    t_vals=[0..1,step=t_val*.01]
    vertices=[(x(*c),y(*c)) for c in [pt+t*dv for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+dv+t*du for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+(1-t)*dv+du for t in t_vals]]
    vertices+=[(x(*c),y(*c)) for c in [pt+(1-t)*du for t in t_vals]]
    xyplot+=polygon(vertices,color='purple',alpha=0.7)
    xyplot+=line(vertices,color='green')
    html("$T(u,v)=%s$"%(latex(T(u,v))))
    html("Jacobian: $%s$"%latex(jacobian(u,v)))
    html("A very small region in $xy$ plane is approximately %0.4g times the size of the corresponding region in the $uv$ plane"%jacobian(u_val,v_val).n())
    html.table([[uvplot,xyplot]])}}}
-Line 514:
+Line 521:
-{{{
+{{{#!sagecell
-Line 537:
+Line 544:
-{{{
+{{{#!sagecell
-Line 564:
+Line 571:
-{{{
+{{{#!sagecell
-Line 590:
+Line 597:
-{{{
+{{{#!sagecell
-Line 605:
+Line 612:
-{{{
+{{{#!sagecell
-Line 651:
+Line 658:
-{{{
from scipy.special.orthogonal import p_roots
+{{{#!sagecell
import scipy
import numpy
from scipy.special.orthogonal import p_roots, t_roots, u_roots
-Line 661:
+Line 670:
-                '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 668:
+Line 680:
-    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 672:
+Line 685:
-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 681:
+Line 695:
-    scaled_ff = fast_float(scaled_func)
+    scaled_ff = fast_float(scaled_func, 'x')
-Line 689:
+Line 703:
-    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 695:
+Line 711:
-    show(graph,xmin=plot_min,xmax=plot_max)
+    show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto")
-Line 703:
+Line 719:
-    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 711:
+Line 728:
-== Vector Calculus, 2-D Motion ==
+== Vector Calculus, 2-D Motion FIXME ==
-Line 716:
+Line 733:
-{{{
+{{{#!sagecell
-Line 828:
+Line 845:
-{{{
+{{{#!sagecell
-Line 865:
+Line 882:
-velocity = derivative( position(t) )
acceleration = derivative(velocity(t))
+velocity = derivative( position(t), t)
acceleration = derivative(velocity(t), t)
-Line 868:
+Line 885:
-speed_deriv = derivative(speed)
+speed_deriv = derivative(speed, t)
-Line 870:
+Line 887:
-dT = derivative(tangent(t))
+dT = derivative(tangent(t), t)
-Line 873:
+Line 890:
-## dB = derivative(binormal(t))
+## dB = derivative(binormal(t), t)
-Line 951:
+Line 968:
-== Multivariate Limits by Definition ==
+== Multivariate Limits by Definition FIXME ==
-Line 954:
+Line 971:
-http://www.sagenb.org/home/pub/2828/

{{{
+http://sagenb.mc.edu/home/pub/97/

{{{#!sagecell
-Line 964:
+Line 981:
-##  An updated version of this worksheet may be available at http://sagenb.mc.edu
-Line 969:
+Line 985:
+var('x,y,z')
Rmin=1/10
-Line 971:
+Line 988:
-@interact
def _(f=input_box(default=(x^3-y^3)/(x^2+y^2)),R=slider(0.1/10,Rmax,1/10,2),x0=(0),y0=(0)):
+@interact(layout=dict(top=[['f'],['x0'],['y0']], 
bottom=[['in_3d','curves','R','graphjmol']]))
def _(f=input_box((x^2-y^2)/(x^2+y^2),width=30,label='$f(x)$'),
        R=slider(Rmin,Rmax,1/10,Rmax,label=',   $R$'),
        x0=input_box(0,width=10,label='$x_0$'),
        y0=input_box(0,width=10,label='$y_0$'),
        curves=checkbox(default=false,label='Show curves'),
        in_3d=checkbox(default=false,label='3D'),
        graphjmol=checkbox(default=true,label='Interactive graph')):
    if graphjmol:
        view_method = 'jmol'
    else:
        view_method = 'tachyon'
-Line 980:
+Line 1008:
-Line 982:
+Line 1010:
-    limit = plot3d(g,(t,0,2*pi),(r,1/100,R),transformation=cylinder,rgbcolor=(0,1,0))
+    collapsing_surface = plot3d(g,(t,0,2*pi),(r,1/100,R),transformation=cylinder,rgbcolor=(0,1,0))
-Line 984:
+Line 1012:
-    show(surface+limit)
    print html('Enter $(x_0 ,y_0 )$ above and see what happens as R approaches zero.')
    print html('The surface has a limit as $(x,y)$ approaches ('+str(x0)+','+str(y0)+') if the green region collapses to a point')
+    G = surface+collapsing_surface
    html('Enter $(x_0 ,y_0 )$ above and see what happens as $ R \\rightarrow 0 $.')
    html('The surface has a limit as $(x,y) \\rightarrow $ ('+str(x0)+','+str(y0)+') if the green region collapses to a point.')

#   If checked, add a couple of curves on the surface corresponding to limit as x->x0 for y=x^(3/5),
#   and as y->y0 for x=y^(3/5).  Should make this more robust but perhaps using 
#   these relatively obtuse curves could eliminate problems.

    if curves:
        curve_x = parametric_plot3d([x0-t,y0-t^(3/5),f(x=x0-t,y=y0-t^(3/5))],(t,Rmin,Rmax),color='red',thickness=10)
        curve_y = parametric_plot3d([x0+t^(3/5),y0+t,f(x=x0+t^(3/5),y=y0+t)],(t,Rmin,Rmax),color='red',thickness=10)
        R2 = Rmin/4
        G += arrow((x0-Rmin,y0-Rmin^(3/5),f(x=x0-Rmin,y=y0-Rmin^(3/5))),(x0-R2,y0-R2^(3/5),f(x=x0-R2,y=y0-R2^(3/5))),size=30 )
        G += arrow((x0+Rmin^(3/5),y0+Rmin,f(x=x0+Rmin^(3/5),y=y0+Rmin)),(x0+R2^(3/5),y0+R2,f(x=x0+R2^(3/5),y=y0+R2)),size=30 )  

        limit_x = limit(f(x=x0-t,y=y0-t^(3/5)),t=0)
        limit_y = limit(f(x=x0+t^(3/5),y=y0+t),t=0)
        text_x = text3d(limit_x,(x0,y0,limit_x))
        text_y = text3d(limit_y,(x0,y0,limit_y))
        G += curve_x+curve_y+text_x+text_y 
 
    
        html('The red curves represent a couple of trajectories on the surface.  If they do not meet, then')
        html('there is also no limit.  (If computer hangs up, likely the computer can not do these limits.)')
        html('\n<center><font color="red">$\lim_{(x,?)\\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+'  and <font color="red">$\lim_{(?,y)\\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y))
        
    if in_3d:
        show(G,stereo="redcyan",viewer=view_method)
    else:
        show(G,perspective_depth=true,viewer=view_method)
-Line 991:
+Line 1046:
-{{{
+{{{#!sagecell
-Line 1028:
+Line 1083:
-{{{
+{{{#!sagecell
-Line 1081:
+Line 1136:
-{{{
%hide
%auto
+{{{#!sagecell
-Line 1142:
+Line 1195:
-{{{
+{{{#!sagecell
-Line 1180:
+Line 1233:
-{{{
+{{{#!sagecell
-Line 1220:
+Line 1273:
-{{{
+{{{#!sagecell
-Line 1325:
+Line 1378:
-http://www.sagenb.org/home/pub/2826/

{{{
+http://sagenb.mc.edu/home/pub/89/

{{{#!sagecell
-Line 1332:
+Line 1385:
+##
-Line 1334:
+Line 1388:
-@interact
def _(f=input_box(default=6-4*x^2-y^2*2/5,label='$f(x,y) = $'),
        g=input_box(default=-2+sin(x)+sin(y),label='$g(x,y) = $'),
        u=input_box(default=cos(t),label='$u(t) = $'),
        v=input_box(default=2*sin(t),label='$v(t) = $'),
        a=input_box(default=0,label='$a = $'),
        b=input_box(default=3*pi/2,label='$b = $'),
+@interact(layout=dict(top=[['f','u'],['g','v']], 
left=[['a'],['b'],['in_3d'],['smoother']],
bottom=[['xx','yy']]))
def _(f=input_box(default=6-4*x^2-y^2*2/5,label='Top = $f(x,y) = $',width=30),
        g=input_box(default=-2+sin(x)+sin(y),label='Bottom = $g(x,y) = $',width=30),
        u=input_box(default=cos(t),label='   $ x = u(t) = $',width=20),
        v=input_box(default=2*sin(t),label='   $ y = v(t) = $',width=20),
        a=input_box(default=0,label='$a = $',width=10),
        b=input_box(default=3*pi/2,label='$b = $',width=10),
-Line 1343:
+Line 1399:
-        smoother=checkbox(default=false)):
+        in_3d = checkbox(default=true,label='3D'),
        smoother=checkbox(default=false),
        auto_update=true):
-Line 1345:
+Line 1403:
-    ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2)
+    ds = sqrt(derivative(u,t)^2+derivative(v,t)^2)
-Line 1349:
+Line 1407:
-    A = (f(x=u(t),y=v(t))-g(x=u(t),y=v(t)))*ds.simplify_trig().simplify()
+    A = (f(x=u,y=v)-g(x=u,y=v))*ds.simplify_trig().simplify()
-Line 1354:
+Line 1412:
-    line_integral = integral(A,t,a,b)
+#   If you want Sage to try, uncomment the lines below.

#    line_integral = integrate(A,t,a,b)
#    html(r'<align=center size=+1>Lateral Surface Area = $ %s $ </font>'%latex(line_integral))
-Line 1356:
+Line 1418:
-    html(r'<h4 align=center>Lateral Surface Area = $ %s $ </h4>'%latex(line_integral))

    html(r'<h4 align=center>Lateral Surface $ \approx $ %s</h2>'%str(line_integral_approx))
+    html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx))
-Line 1366:
+Line 1426:
-    G += parametric_plot3d([u,v,g(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')
    G += parametric_plot3d([u,v,f(x=u(t),y=v(t))],(t,a,b),thickness=2,color='red')
+    G += parametric_plot3d([u,v,g(x=u,y=v)],(t,a,b),thickness=2,color='red')
    G += parametric_plot3d([u,v,f(x=u,y=v)],(t,a,b),thickness=2,color='red')
-Line 1376:
+Line 1436:
-        G += parametric_plot3d([u(w),v(w),s*f(x=u(w),y=v(w))+(1-s)*g(x=u(w),y=v(w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9)
    show(G,spin=true)
+        G += parametric_plot3d([u(t=w),v(t=w),s*f(x=u(t=w),y=v(t=w))+(1-s)*g(x=u(t=w),y=v(t=w))],(s,0,1),thickness=lat_thick,color='yellow',opacity=0.9)
             if in_3d:
        show(G,stereo='redcyan',spin=true)
    else:
        show(G,perspective_depth=true,spin=true)
-Line 1384:
+Line 1448:
-{{{
+{{{#!sagecell
-Line 1405:
+Line 1469:
-{{{
+{{{#!sagecell

Diff for "interact/calculus"

Sage Interactions - Calculus

Root Finding Using Bisection

Newton's Method

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

A simple tangent line grapher

Numerical integrals with the midpoint rule

Numerical integrals with various rules

Some polar parametric curves

Function tool

Newton-Raphson Root Finding

Coordinate Transformations

Taylor Series

Illustration of the precise definition of a limit

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

Quadric Surface Plotter

The midpoint rule for numerically integrating a function of two variables

Gaussian (Legendre) quadrature

Vector Calculus, 2-D Motion FIXME

Vector Calculus, 3-D Motion

Multivariate Limits by Definition FIXME

Directional Derivatives

3D graph with points and curves

Approximating function in two variables by differential

Taylor approximations in two variables

Volumes over non-rectangular domains

Lateral Surface Area

Parametric surface example

Line Integrals in 3D Vector Field