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Sage Interactions - Calculus

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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

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2824-Double%20Precision%20Root%20Finding%20Using%20Newton's%20Method.sagews

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

by William Stein

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews

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)

{{{#!sagecellvar('x') @interact def midpoint(f = input_box(default = sin(x^2) + 2, type = SR),

interval=range_slider(0, 10, 1, default=(0, 4), label="Interval"), number_of_subdivisions = slider(1, 20, 1, default=4, label="Number of boxes"), endpoint_rule = selector(['Midpoint', 'Left', 'Right', 'Upper', 'Lower'], nrows=1, label="Endpoint rule")): a, b = map(QQ, interval) t = var('t') func = fast_callable(f(x=t), RDF, vars=[t]) dx = ZZ(b-a)/ZZ(number_of_subdivisions) xs = [] ys = [] for q in range(number_of_subdivisions):
- if endpoint_rule == 'Left':
  - xs.append(q*dx + a)
  elif endpoint_rule == 'Midpoint':
  - xs.append(q*dx + a + dx/2)
  elif endpoint_rule == 'Right':
  - xs.append(q*dx + a + dx)
  elif endpoint_rule == 'Upper':
  - x = find_local_maximum(func, q*dx + a, q*dx + dx + a)[1] xs.append(x)
  elif endpoint_rule == 'Lower':
  - x = find_local_minimum(func, q*dx + a, q*dx + dx + a)[1] xs.append(x)
ys = [ func(x) for x in xs ] rects = Graphics() for q in range(number_of_subdivisions):
- xm = q*dx + dx/2 + a x = xs[q] y = ys[q]
  rects += line(xm-dx/2,0],[xm-dx/2,y],[xm+dx/2,y],[xm+dx/2,0, rgbcolor = (1,0,0)) rects += point((x, y), rgbcolor = (1,0,0))
min_y = min(0, find_local_minimum(func,a,b)[0]) max_y = max(0, find_local_maximum(func,a,b)[0])
# html('<h3>Numerical integrals with the midpoint rule</h3>') show(plot(func,a,b) + rects, xmin = a, xmax = b, ymin = min_y, ymax = max_y) def cap(x):
- # print only a few digits of precision
  if x < 1e-4:
  - return 0
  return RealField(20)(x)
sum_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ "f(%s)" % cap(i) for i in xs ])) num_html = "%s \cdot \\left[ %s \\right]" % (dx, ' + '.join([ str(cap(i)) for i in ys ])) numerical_answer = integral_numerical(func,a,b,max_points = 200)[0] estimated_answer = dx * sum([ ys[q] for q in range(number_of_subdivisions)])
pretty_print(html(r <div class="math"> \begin{align*} \int_{a}^{b} {f(x) \, dx} & = %s \\\ \sum_{i=1}^{%s} {f(x_i) \, \Delta x} & = %s \\\ & = %s \\\ & = %s . \end{align*} </div>
- % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)))

}}}

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 (FIXME in Jupyter)

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

By Rob Beezer

A fast_float() version is available in a worksheet

Vector Calculus, 3-D Motion (FIXME)

by Rob Beezer

Available as a worksheet

Multivariate Limits by Definition

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

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews

Lateral Surface Area (FIXME in Jupyter)

by John Travis

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

Parametric surface example (FIXME in Jupyter)

by Marshall Hampton

Line Integrals in 3D Vector Field

by John Travis

https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2827-$%20%5Cint_%7BC%7D%20%5Cleft%20%5Clangle%20M,N,P%20%5Cright%20%5Crangle%20dr%20$%20=%20$%20%25s%20$.sagews

-  ⇤ ← Revision 55 as of 2012-04-26 17:01:56 → 
  Size: 58516
  Editor: jason
  Comment:
+   ← Revision 89 as of 2017-03-18 15:09:23 → ⇥
  Size: 62874
  Editor: mforets
  Comment: fix typesetting error and a DeprecationWarning in Numerical integrals with various rules
-Deletions are marked like this.
+Additions are marked like this.
 Line 29:
-html("<h1>Double Precision Root Finding Using Bisection</h1>")
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16..-1))):
+pretty_print(html("<h1>Double Precision Root Finding Using Bisection</h1>"))
@interact
def _(f = cos(x) - x, a = float(0), b = float(1), eps=(-3,(-16, -1))):
 Line 35:
-         time c, intervals = bisect_method(f, a, b, eps)
+         c, intervals = bisect_method(f, a, b, eps)
 Line 41:
-         print "f(c) = %r"%f(c)
+         print "f(c) = %r"%f(x=c)
 Line 57:
-http://sagenb.org/home/pub/2824/
+https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2824-Double%20Precision%20Root%20Finding%20Using%20Newton's%20Method.sagews
 Line 77:
-html("<h1>Double Precision Root Finding Using Newton's Method</h1>")
@interact
def _(f = x^2 - 2, c = float(0.5), eps=(-3,(-16..-1)), interval=float(0.5)):
+pretty_print(html("<h1>Double Precision Root Finding Using Newton's Method</h1>"))
@interact
def _(f = x^2 - 2, c = float(0.5), eps=(-3,(-16, -1)), interval=float(0.5)):
 Line 82:
-     time z, iterates = newton_method(f, c, eps)
+     z, iterates = newton_method(f, c, eps)
 Line 87:
-     html(iterates)
+     pretty_print(html(iterates))
 Line 99:
-http://sagenb.org/home/pub/2823/
+https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews
 Line 118:
-html('<h2>Tangent line grapher</h2>')
+pretty_print(html('<h2>Tangent line grapher</h2>'))
 Line 125:
-    tanf = f(x0i) + df(x0i)*(x-x0i)
+    tanf = f(x=x0i) + df(x=x0i)*(x-x0i)
 Line 129:
-    fmax = f.find_maximum_on_interval(prange[0], prange[1])[0]
    fmin = f.find_minimum_on_interval(prange[0], prange[1])[0]
+    fmax = f.find_local_maximum(prange[0], prange[1])[0]
    fmin = f.find_local_minimum(prange[0], prange[1])[0]
 Line 146:
-    midys = [func(x_val) for x_val in midxs]
+    midys = [func(x=x_val) for x_val in midxs]
 Line 152:
-    min_y = find_minimum_on_interval(func,a,b)[0]
    max_y = find_maximum_on_interval(func,a,b)[0]
    html('<h3>Numerical integrals with the midpoint rule</h3>')
    html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')
+    min_y = min(0, find_local_minimum(func,a,b)[0])
    max_y = max(0, find_local_maximum(func,a,b)[0])
    pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>'))
    pretty_print(html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$'))
 Line 165:
-{{{#!sagecell
# by Nick Alexander (based on the work of Marshall Hampton)

var('x')
+{{{#!sagecellvar('x')
-Line 176:
+Line 173:
-    t = sage.calculus.calculus.var('t')
+    t = var('t')
-Line 190:
+Line 187:
-            x = find_maximum_on_interval(func, q*dx + a, q*dx + dx + a)[1]
+            x = find_local_maximum(func, q*dx + a, q*dx + dx + a)[1]
-Line 193:
+Line 190:
-            x = find_minimum_on_interval(func, q*dx + a, q*dx + dx + a)[1]
+            x = find_local_minimum(func, q*dx + a, q*dx + dx + a)[1]
-Line 204:
+Line 201:
-    min_y = min(0, find_minimum_on_interval(func,a,b)[0])
    max_y = max(0, find_maximum_on_interval(func,a,b)[0])
+    min_y = min(0, find_local_minimum(func,a,b)[0])
    max_y = max(0, find_local_maximum(func,a,b)[0])
-Line 221:
+Line 218:
-    html(r'''
    <div class="math">
    \begin{align*}
      \int_{a}^{b} {f(x) \, dx} & = %s \\\
      \sum_{i=1}^{%s} {f(x_i) \, \Delta x}
      & = %s \\\
      & = %s \\\
      & = %s .
    \end{align*}
    </div>
    ''' % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer))
+    pretty_print(html(r'''
    <div class="math"> 
    \begin{align*}      \int_{a}^{b} {f(x) \, dx} & = %s \\\      \sum_{i=1}^{%s} {f(x_i) \, \Delta x} & = %s \\\      & = %s \\\      & = %s . \end{align*} </div>''' 
                      % (numerical_answer, number_of_subdivisions, sum_html, num_html, estimated_answer)))
-Line 242:
+Line 236:
-    html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$')
+    pretty_print(html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$'))
-Line 329:
+Line 323:
-    html('<center><font color="red">$f = %s$</font></center>'%latex(f))
    html('<center><font color="green">$g = %s$</font></center>'%latex(g))
    html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h)))
+    pretty_print(html('<center><font color="red">$f = %s$</font></center>'%latex(f)))
    pretty_print(html('<center><font color="green">$g = %s$</font></center>'%latex(g)))
    pretty_print(html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h))))
-Line 451:
+Line 445:
-== Coordinate Transformations ==
+== Coordinate Transformations (FIXME in Jupyter) ==
-Line 510:
+Line 504:
-    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]])}}}
+    pretty_print(html("$T(u,v)=%s$"%(latex(T(u,v)))))
    pretty_print(html("Jacobian: $%s$"%latex(jacobian(u,v))))
    pretty_print(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()))
    pretty_print(table([[uvplot,xyplot]]))
}}}
-Line 531:
+Line 526:
-    html('$f(x)\;=\;%s$'%latex(f))
    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
+    pretty_print(html('$f(x)\;=\;%s$'%latex(f)))
    pretty_print(html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)))
-Line 545:
+Line 540:
-html("<h2>Limits: <i>ε-δ</i></h2>")
html("This allows you to estimate which values of <i>δ</i> guarantee that <i>f</i> is within <i>ε</i> units of a limit.")
html("<ul><li>Modify the value of <i>f</i> to choose a function.</li>")
html("<li>Modify the value of <i>a</i> to change the <i>x</i>-value where the limit is being estimated.</li>")
html("<li>Modify the value of <i>L</i> to change your guess of the limit.</li>")
html("<li>Modify the values of <i>δ</i> and <i>ε</i> to modify the rectangle.</li></ul>")
html("If the blue curve passes through the pink boxes, your values for <i>δ</i> and/or <i>ε</i> are probably wrong.")
+pretty_print(html("<h2>Limits: <i>ε-δ</i></h2>"))
pretty_print(html("This allows you to estimate which values of <i>δ</i> guarantee that <i>f</i> is within <i>ε</i> units of a limit."))
pretty_print(html("<ul><li>Modify the value of <i>f</i> to choose a function.</li>"))
pretty_print(html("<li>Modify the value of <i>a</i> to change the <i>x</i>-value where the limit is being estimated.</li>"))
pretty_print(html("<li>Modify the value of <i>L</i> to change your guess of the limit.</li>"))
pretty_print(html("<li>Modify the values of <i>δ</i> and <i>ε</i> to modify the rectangle.</li></ul>"))
pretty_print(html("If the blue curve passes through the pink boxes, your values for <i>δ</i> and/or <i>ε</i> are probably wrong."))
-Line 575:
+Line 570:
-    html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>')
    html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|')
    html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.') 
    html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.')
    html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.')
    html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.')
+    pretty_print(html('<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
    pretty_print(html('Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
    pretty_print(html('and the length of the <font color=blue>blue line</font> is |tan(x)| where x is the length of the arc.'))
    pretty_print(html('From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
    pretty_print(html('It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
    pretty_print(html('As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))
-Line 631:
+Line 626:
-sin,cos = math.sin,math.cos 
html("<h1>The midpoint rule for a function of two variables</h1>")
+pretty_print(html("<h1>The midpoint rule for a function of two variables</h1>"))
-Line 647:
+Line 642:
-    html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$")
    html('<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+'</p>')
+    pretty_print(html("$$\int_{"+str(R16(y_start))+"}^{"+str(R16(y_end))+"} "+ "\int_{"+str(R16(x_start))+"}^{"+str(R16(x_end))+"} "+func+"\ dx \ dy$$"))
    pretty_print(html('<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+'</p>'))
-Line 719:
+Line 714:
-    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)))
+    pretty_print(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 764:
+Line 759:
-velocity = derivative( position(t) )
acceleration = derivative(velocity(t))
+velocity = derivative(position(t), t)
acceleration = derivative(velocity(t), t)
-Line 767:
+Line 762:
-speed_deriv = derivative(speed)
+speed_deriv = derivative(speed, t)
-Line 769:
+Line 764:
-dT = derivative(tangent(t))
+dT = derivative(tangent(t), t)
-Line 840:
+Line 835:
-== Vector Calculus, 3-D Motion ==
+== Vector Calculus, 3-D Motion (FIXME) ==
-Line 971:
+Line 966:
-http://www.sagenb.org/home/pub/2828/
+http://sagenb.mc.edu/home/pub/97/
-Line 981:
+Line 976:
-##  An updated version of this worksheet may be available at http://sagenb.mc.edu
-Line 986:
+Line 980:
+var('x,y,z')
Rmin=1/10
-Line 988:
+Line 983:
-@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 997:
+Line 1003:
-Line 999:
+Line 1005:
-    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 1001:
+Line 1007:
-    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
    pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as $ R \\rightarrow 0 $.'))
    pretty_print(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 
 
    
        pretty_print(html('The red curves represent a couple of trajectories on the surface.  If they do not meet, then'))
        pretty_print(html('there is also no limit.  (If computer hangs up, likely the computer can not do these limits.)'))
        pretty_print(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 1009:
+Line 1042:
-Line 1025:
+Line 1059:
-Rmax=2
@interact
def _(f=input_box(default=(x^3-y^3)/(x^2+y^2)),
      N=slider(5,100,1,10,label='Number of Contours'),
      x0=(0),y0=(0)):

    print html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels increases.')
    print html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.')
+var('x,y,z,u')
@interact(layout=dict(top=[['f'],['x0'],['y0']], 
bottom=[['N'],['R']]))
def _(f=input_box(default=(x*y^2)/(x^2+y^4),width=30,label='$f(x)$'),
        N=slider(5,100,1,10,label='Number of Contours'),
        R=slider(0.1,1,0.01,1,label='Radius of circular neighborhood'),
        x0=input_box(0,width=10,label='$x_0$'),
        y0=input_box(0,width=10,label='$y_0$')):

    pretty_print(html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty $.'))
    pretty_print(html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.'))

#   Need to make certain the min and max contour lines are not huge due to asymptotes.  If so, clip and start contours at some reasonable
#   values so that there are a nice collection of contours to show around the desired point.
-Line 1035:
+Line 1075:
+    surface += parametric_plot([R*cos(u),R*sin(u)],[0,2*pi],color='black')
#      Nice to use if f=x*y^2/(x^2 + y^4)    
#    var('u')
#    surface += parametric_plot([u^2,u],[u,-1,1],color='black')
-Line 1036:
+Line 1080:
-    show(limit_point+surface)}}}
+#    show(limit_point+surface)
    pretty_print(table([[surface],['hi']]))
}}}
-Line 1113:
+Line 1159:
- html(r'Function $ f(x,y)=%s$ '%latex(f(x,y)))
+ pretty_print(html(r'Function $ f(x,y)=%s$ '%latex(f(x,y))))
-Line 1127:
+Line 1173:
-              html(r'<tr><td>$\quad f(%s,%s)\quad $</td><td>$\quad %s$</td>\
              </tr>'%(latex(x0),latex(y0),z0.n()))
+              pretty_print(html(r'<tr><td>$\quad f(%s,%s)\quad $</td><td>$\quad %s$</td>\
              </tr>'%(latex(x0),latex(y0),z0.n())))
-Line 1161:
+Line 1207:
-html('Points x0 and y0 are values where the exact value of the function \
+pretty_print(html('Points x0 and y0 are values where the exact value of the function \
-Line 1163:
+Line 1209:
-and approximation by differential at shifted point are compared.')
+and approximation by differential at shifted point are compared.'))
-Line 1181:
+Line 1227:
-  html(r'Function $ f(x,y)=%s \approx %s $ '%(latex(f(x,y)),latex(tangent(x,y))))
  html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori)))
  html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay))))
  html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay))
  html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value))
  html(r'Error is $%s$'%latex(abs_error))
+  pretty_print(html(r'Function $ f(x,y)=%s \approx %s $ '%(latex(f(x,y)),latex(tangent(x,y)))))
  pretty_print(html(r' $f %s = %s$'%(latex((x0,y0)),latex(exact_value_ori))))
  pretty_print(html(r'Shifted point $%s$'%latex(((x0+deltax),(y0+deltay)))))
  pretty_print(html(r'Value of the function in shifted point is $%s$'%f(x0+deltax,y0+deltay)))
  pretty_print(html(r'Value on the tangent plane in shifted point is $%s$'%latex(approx_value)))
  pretty_print(html(r'Error is $%s$'%latex(abs_error)))
-Line 1223:
+Line 1269:
-    html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$')
+    pretty_print(html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$'))
-Line 1233:
+Line 1279:
-http://www.sagenb.org/home/pub/2829/
+https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews
-Line 1336:
+Line 1382:
-== Lateral Surface Area ==
+== Lateral Surface Area (FIXME in Jupyter) ==
-Line 1340:
+Line 1386:
-http://www.sagenb.org/home/pub/2826/
+http://sagenb.mc.edu/home/pub/89/
-Line 1347:
+Line 1393:
+##
-Line 1349:
+Line 1396:
-@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 1358:
+Line 1407:
-        smoother=checkbox(default=false)):
+        in_3d = checkbox(default=true,label='3D'),
        smoother=checkbox(default=false),
        auto_update=true):
-Line 1360:
+Line 1411:
-    ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2)
+    ds = sqrt(derivative(u,t)^2+derivative(v,t)^2)
-Line 1364:
+Line 1415:
-    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 1369:
+Line 1420:
-    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 1371:
+Line 1426:
-    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))
+    pretty_print(html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx)))
-Line 1381:
+Line 1434:
-    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 1391:
+Line 1444:
-        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 1397:
+Line 1454:
-== Parametric surface example ==
+== Parametric surface example (FIXME in Jupyter) ==
-Line 1418:
+Line 1475:
-http://www.sagenb.org/home/pub/2827/
+https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2827-$%20%5Cint_%7BC%7D%20%5Cleft%20%5Clangle%20M,N,P%20%5Cright%20%5Crangle%20dr%20$%20=%20$%20%25s%20$.sagews
-Line 1454:
+Line 1511:
-    html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral))
+    pretty_print(html(r'<h2 align=center>$ \int_{C} \left \langle M,N,P \right \rangle dr $ = $ %s $ </h2>'%latex(line_integral)))

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 (FIXME in Jupyter)

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

Vector Calculus, 3-D Motion (FIXME)

Multivariate Limits by Definition

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 (FIXME in Jupyter)

Parametric surface example (FIXME in Jupyter)

Line Integrals in 3D Vector Field