Differences between revisions 77 and 116 (spanning 39 versions)
Revision 77 as of 2016-09-21 10:17:02
Size: 62595
Editor: jdemeyer
Comment:
Revision 116 as of 2020-08-11 14:08:17
Size: 63129
Editor: kcrisman
Comment:
Deletions are marked like this. Additions are marked like this.
Line 27: Line 27:
            raise ValueError, "f must have a sign change in the interval (%s,%s)"%(a,b)             raise ValueError("f must have a sign change in the interval (%s,%s)"%(a,b))
Line 29: 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 33: Line 33:
     print "eps = %s"%float(eps)      print("eps = %s" % float(eps))
Line 37: Line 37:
         print "f must have opposite sign at the endpoints of the interval"          print("f must have opposite sign at the endpoints of the interval")
Line 40: Line 40:
         print "root =", c
         print "f(c) = %r"%f(
c)
         print "iterations =", len(intervals)
         print("root =", c)
         print("f(c) = %r" % f(x=c))
         print(
"iterations =", len(intervals))
Line 69: Line 69:
    for i in xrange(maxiter):     for i in range(maxiter):
Line 77: 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 81: Line 81:
     print "eps = %s"%float(eps)      print("eps = %s"%float(eps))
Line 83: Line 83:
     print "root =", z
     print "f(c) = %r"%f(x=z)
     print("root = {}".format(z))
     print("f(c) = %r" % f(x=z))
Line 86: Line 86:
     print "iterations =", n
     html(iterates)
     print("iterations = {}".format(n))
     pretty_print(html(iterates))
Line 118: Line 118:
html('<h2>Tangent line grapher</h2>') pretty_print(html('<h2>Tangent line grapher</h2>'))
Line 125: Line 125:
    tanf = f(x0i) + df(x0i)*(x-x0i)     tanf = f(x=x0i) + df(x=x0i)*(x-x0i)
Line 127: Line 127:
    print 'Tangent line is y = ' + tanf._repr_()     print('Tangent line is y = ' + tanf._repr_())
Line 138: Line 138:
#find_maximum_on_interval and find_minimum_on_interval are deprecated
#use find_local_maximum find_local_minimum instead
#see http://trac.sagemath.org/2607 for details -RRubalcaba
Line 150: Line 146:
    midys = [func(x_val) for x_val in midxs]     midys = [func(x=x_val) for x_val in midxs]
Line 156: Line 152:
    min_y = min(0, sage.numerical.optimize.find_local_minimum(func,a,b)[0])
    max_y = max(0, sage.numerical.optimize.find_local_maximum(func,a,b)[0])
    html('<h3>Numerical integrals with the midpoint rule</h3>')
    html('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')
    print "\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0])
    print "Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)])))
    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(r'$\int_{a}^{b}{f(x) dx} {\approx} \sum_i{f(x_i) \Delta x}$'))
    print("\n\nSage numerical answer: " + str(integral_numerical(func,a,b,max_points = 200)[0]))
    print("Midpoint estimated answer: " + str(RDF(dx*sum([midys[q] for q in range(n)]))))
Line 170: Line 166:
# by Nick Alexander (based on the work of Marshall Hampton)
#find_maximum_on_interval and find_minimum_on_interval are deprecated
#use find_local_maximum find_local_minimum instead
#see http://trac.sagemath.org/2607 for details -RRubalcaba
Line 183: Line 174:
    t = sage.calculus.calculus.var('t')     t = var('t')
Line 214: Line 205:
    # html('<h3>Numerical integrals with the midpoint rule</h3>')     pretty_print(html('<h3>Numerical integrals with the midpoint rule</h3>'))
Line 222: Line 213:
    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 ]))
    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 ]))
Line 228: Line 219:
    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 249: Line 237:
    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 269: Line 257:
    except TypeError, msg:
        print msg[-200:]
        print "Unable to make sense of f,g, or a as symbolic expressions."
    except TypeError as msg:
        print(msg[-200:])
        print("Unable to make sense of f,g, or a as symbolic expressions.")
Line 336: Line 324:
    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 386: Line 374:
                     vertical_alignment="bottom" if f(x0) < 0 else "top" )                      vertical_alignment="bottom" if f(x=x0) < 0 else "top" )
Line 402: Line 390:
        fi = RR(f(xi))
        fpi = RR(df(xi))
        fi = RR(f(x=xi))
        fpi = RR(df(x=xi))
Line 438: Line 426:
                             vertical_alignment="bottom" if f(xip1) < 0 else "top" )                              vertical_alignment="bottom" if f(x=xip1) < 0 else "top" )
Line 454: Line 442:
            html( t )             pretty_print(html( t ))
Line 478: Line 466:
         u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7),
         v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7),
         u_percent=slider(0,1,0.05,label="u", default=.7),
         v_percent=slider(0,1,0.05,label="v", default=.7),
Line 509: Line 497:
    jacobian=abs(T.diff().det()).simplify_full()     jacobian(u,v)=abs(T.diff().det()).simplify_full()
Line 517: Line 505:
    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())
    pretty_print(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()))
    show(graphics_array([uvplot,xyplot]))
Line 534: Line 522:
dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):
dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=[1..12]):
Line 539: Line 527:
    html('$f(x)\;=\;%s$'%latex(f))
    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
    pretty_print(html(r'$f(x)\;=\;%s$'%latex(f)))
    pretty_print(html(r'$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1)))
Line 553: Line 541:
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.")
@interact
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1)), a=input_box(default=1), L = input_box(default=1), delta=input_box(label="δ",default=0.1), epsilon=input_box(label=",default=0.1), xm=input_box(label="<i>x</i><sub>min</sub>",default=-1), xM=input_box(label="<i>x</i><sub>max</sub>",default=4)):
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."))
@interact
def delta_epsilon(f = input_box(default=(x^2-x)/(x-1), label="$f$"), a=input_box(default=1, label="$a$"), L = input_box(default=1, label="$L$"), delta=input_box(label=r"$\delta$",default=0.1), epsilon=input_box(label=r"$\varepsilon$",default=0.1), xm=input_box(label=r"$x_{min}$",default=-1), xM=input_box(label=r"$x_{max}$",default=4)):
Line 583: Line 571:
    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(r'<h3>A graphical illustration of $\lim_{x -> 0} \sin(x)/x =1$</h3>'))
    pretty_print(html(r'Below is the unit circle, so the length of the <font color=red>red line</font> is |sin(x)|'))
    pretty_print(html(r'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(r'From the picture, we see that |sin(x)| $\le$ |x| $\le$ |tan(x)|.'))
    pretty_print(html(r'It follows easily from this that cos(x) $\le$ sin(x)/x $\le$ 1 when x is near 0.'))
    pretty_print(html(r'As $\lim_{x ->0} \cos(x) =1$, we conclude that $\lim_{x -> 0} \sin(x)/x =1$.'))
Line 609: Line 597:
def quads(q = selector(quadrics.keys()), a = slider(0,5,1/2,default = 1)): def quads(q = selector(list(quadrics)), a = slider(0,5,1/2,default = 1)):
Line 611: Line 599:
    if a==0 or q=='Cone': html('<center>$'+latex(f)+' \ $'+ '(degenerate)</center>')
    else: html('<center>$'+latex(f)+'$ </center>')
    if a==0 or q=='Cone': pretty_print(latex(f), "   (degenerate)")
    else: pretty_print(latex(f))
Line 640: Line 628:
html("<h1>The midpoint rule for a function of two variables</h1>") pretty_print(html(r"<h1>The midpoint rule for a function of two variables</h1>"))
Line 655: Line 643:
    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(r"$\int_{"+str(R16(y_start))+r"}^{"+str(R16(y_end))+r"} "+ r"\int_{"+str(R16(x_start))+r"}^{"+str(R16(x_end))+r"} "+latex(SR(func))+r"\ dx \ dy$"))
    pretty_print(html(r'<p style="text-align: center;">Numerical approximation: ' + str(num_approx)+r'</p>'))
Line 672: Line 660:
from numpy import linspace from numpy import linspace, asanyarray, diff
Line 724: Line 712:
    y_val = map(scaled_ff,x_val)     y_val = [*map(scaled_ff,x_val)]
Line 727: Line 715:
    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(r"$$\sum_{i=1}^{i=%s}w_i\left(%s\right)= %s\approx %s =\int_{-1}^{1}%s \,dx$$"%(n,
        latex(f), approximation, integral, latex(scaled_func))))
Line 730: Line 718:
    print "Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s"%tuple(error_data)     print("Trapezoid: %s, Simpson: %s, \nMethod: %s, Real: %s" % tuple(error_data))
Line 768: Line 756:
path = parametric_plot( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot( position.list(), (t, start, stop), color = "black" )
Line 772: Line 760:
velocity = derivative(position(t), t)
acceleration = derivative(velocity(t), t)
velocity = derivative(position, t)
acceleration = derivative(velocity, t)
Line 777: Line 765:
dT = derivative(tangent(t), t) dT = derivative(tangent, t)
Line 798: Line 786:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 802: Line 790:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )
    speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 810: Line 798:
    tan = arrow(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))
    tan = arrow(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    acc = arrow(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 837: Line 825:
    print "Position vector defined as r(t)=", position(t)
    print "Speed is ", N(speed(t0
))
    print "Curvature is ", N(curvature)
    print("Position vector defined as r(t)={}".format(position))
    print("Speed is {}".format(N(speed(t=t0))))
    print(
"Curvature is {}".format(N(curvature)))
Line 848: Line 836:
== Vector Calculus, 3-D Motion (FIXME) == == Vector Calculus, 3-D Motion ==
Line 869: Line 857:
assume(t, 'real')
Line 886: Line 875:
path = parametric_plot3d( position(t).list(), (t, start, stop), color = "black" ) path = parametric_plot3d( position.list(), (t, start, stop), color = "black" )
Line 890: Line 879:
velocity = derivative( position(t), t)
acceleration = derivative(velocity(t), t)
velocity = derivative( position, t)
acceleration = derivative(velocity, t)
Line 895: Line 884:
dT = derivative(tangent(t), t) dT = derivative(tangent, t)
Line 898: Line 887:
## dB = derivative(binormal(t), t) ## dB = derivative(binormal, t)
Line 919: Line 908:
    pos_tzero = position(t0)     pos_tzero = position(t=t0)
Line 923: Line 912:
    speed_component = speed(t0)
    tangent_component = speed_deriv(t0)
    normal_component = sqrt( acceleration(t0).norm()^2 - tangent_component^2 )
    speed_component = speed(t=t0)
    tangent_component = speed_deriv(t=t0)
    normal_component = sqrt( acceleration(t=t0).norm()^2 - tangent_component^2 )
Line 932: Line 921:
    tan = arrow3d(pos_tzero, pos_tzero + tangent(t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t0), rgbcolor=(1,0,1))
    tan = arrow3d(pos_tzero, pos_tzero + tangent(t=t0), rgbcolor=(0,1,0) )
    vel = arrow3d(pos_tzero, pos_tzero + velocity(t=t0), rgbcolor=(0,0.5,0))
    nor = arrow3d(pos_tzero, pos_tzero + normal(t=t0), rgbcolor=(0.5,0,0))
    bin = arrow3d(pos_tzero, pos_tzero + binormal(t=t0), rgbcolor=(0,0,0.5))
    acc = arrow3d(pos_tzero, pos_tzero + acceleration(t=t0), rgbcolor=(1,0,1))
    tancomp = arrow3d(pos_tzero, pos_tzero + tangent_component*tangent(t=t0), rgbcolor=(1,0,1) )
    norcomp = arrow3d(pos_tzero, pos_tzero + normal_component*normal(t=t0), rgbcolor=(1,0,1))
Line 962: Line 951:
    print "Position vector: r(t)=", position(t)
    print
"Speed is ", N(speed(t0))
    print
"Curvature is ", N(curvature)
    ## print "Torsion is ", N(torsion)
    print
    print
"Right-click on graphic to zoom to 400%"
    print
"Drag graphic to rotate"
    print("Position vector: r(t)=", position)
    print(
"Speed is ", N(speed(t=t0)))
    print(
"Curvature is ", N(curvature))
    ## print("Torsion is ", N(torsion))
    print()
    print(
"Right-click on graphic to zoom to 400%")
    print(
"Drag graphic to rotate")
Line 1021: Line 1010:
    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.')
    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.'))
Line 1042: Line 1031:
        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))
        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(r'<center><font color="red">$\lim_{(x,?)\rightarrow(x_0,y_0)} f(x,y) =%s$</font>'%str(limit_x)+r' and <font color="red">$\lim_{(?,y)\rightarrow(x_0,y_0)} f(x,y) =%s$</font></center>'%str(limit_y)))
Line 1055: Line 1044:
Line 1081: Line 1069:
    html('Enter $(x_0 ,y_0 )$ above and see what happens as the number of contour levels $\\rightarrow \infty $.')
    html('A surface will have a limit in the center of this graph provided there is not a sudden change in color there.')
    pretty_print(html(r'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.'))
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    pretty_print(table([[surface],['hi']]))     show(surface)
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 html(r'Function $ f(x,y)=%s$ '%latex(f(x,y)))  pretty_print(html(r'Function $ f(x,y)=%s$ '%latex(f(x,y))))
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              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())))
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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 \
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and approximation by differential at shifted point are compared.') and approximation by differential at shifted point are compared.'))
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  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)))
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      order=(1..10)):       order=[1..10]):
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    html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$')     pretty_print(html('$F(x,y) = e^{-(x^2+y^2)/2} \\cos(y) \\sin(x^2+y^2)$'))
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Note that this works in Sage cell, but causes a zip file error in Jupyter
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    html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx))     pretty_print(html(r'<font align=center size=+1>Lateral Surface $ \approx $ %s</font>'%str(line_integral_approx)))
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Note that this works in Sage cell, but causes a zip file error in Jupyter.
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    u(t) = u
    v(t) = v
    w(t) = w
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    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)))

Sage Interactions - Calculus

goto interact main page

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

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

newton.png

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

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

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

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

3D_Irregular_Volume.png

Lateral Surface Area

by John Travis

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

Note that this works in Sage cell, but causes a zip file error in Jupyter

Lateral_Surface.png

Parametric surface example

by Marshall Hampton

Note that this works in Sage cell, but causes a zip file error in Jupyter.

parametric_surface.png

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

3D_Line_Integral.png

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