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Comment: fix bug in Numerical integrals with various rules
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| [[http://aleph.sagemath.org/?z=eJytU02P2yAQvedXjCxFC4njJP06bOOqrdqe9hBVbS-rNMIY1ki2sQDvev99B5zYzq62l5aDPcDw5jHvkQsJmbKCu2MlXKFzImNgMWQxiMbS6xngcOaxD_yQkIJMjpJZd5SlZu5IZHLPjGJZKSyht5sDDcmi46Jx8Mk5o7LWia_GaDPiNMzaMFG1E-aelRaBbwmLM3roqz5oX8qXIK_oe89nmHtuIemhUKWAH6YVIzLHNMKWGV0jxMib-dOEIZLMQpj5kIeQ0yFPSWCZJZJT2Pma12CEa00NPB6pTrMlWyDKDjYjAz98ExEbf3xYF6XP5wsk8FI-Vskm-VZcZhmGYsEvVrZ9P2OIJFStdVCwewEMrLpDqgWr7wTSBVeIgTWQuY3nlkbz0OXxDudbJaxpRJ2TsE1nhatKEu2K7YcvukVxYW8EV1bpGr5r7eCbqnNV38FP67-fg4twc7fGExGdfQy4jLtZjiZDl_jbaUs6Civo0GWDmhva372fbWnwXkpWr2Oy2r5LktWW0pMVUauOprKPe0dsN78x6Fcag0Uh6jfmNpo_scullcGpSlzoiqf-8hpOVXtfjxpMAE_1p5LoptFWOdErw1yQBLvcaEy1oOWFRtGIZQv9QJpSu5EE16U26ZUR-RU2qatUnTL_Z1161vOJZU6EjNcrjaZeHLii-X2zDDZr-g7O-8qL6HW1AaAUNRnaNcneI8bLZMe8wj_OffKInIk3gg9VTShdA9kmbxeX-JODN3jQthUpVS3ILeFxsVDYBJKH4EBBagMKF3icU299UbeVpy4meHABuHyOuCrWb2gAwcnST_4Dcj5Fzv8FOXhiD0u4eSb_HxOSmYw%3D|Interact]] {{{ |
{{{#!sagecell |
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| 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))): |
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| time c, intervals = bisect_method(f, a, b, eps) | c, intervals = bisect_method(f, a, b, eps) |
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| print "f(c) = %r"%f(c) | print "f(c) = %r"%f(x=c) |
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| [[http://aleph.sagemath.org/?z=eJx1U01v2zAMvedXEAGKUKvsxCm2QxsPG7Dt1A3BgJ2yNnAcOVZrS4akJG5-_WjZzhdWATYkPvKRfJTWIgMl9k6rZSlcrteYcUg5iMpyKJNaOmHiaDJh9wOgVUMMWbhLjExWhbDIFpMnD2SVkaXw6FoYuUuc3AmsmQedeWvDm7XxTssssW6ZFTpxy97No0cev_mvn6hTUTn46pyRq60T343R5irBwzWTh5tuEicsWRdpV7g2IEEqqE2iNgK7ntmRL0vJe4PpsUaZQbKymKUMZo1O92CE2xrVyNYn6H2b2BQCIhm39ZzzdL5hUlVCrXvkHbIBiY6jesT8IXdlgcNZHn3-prc0CZgbkUortYLfWjv4IdVaqg38sc3_lx_wyMJPP-LZmOKGbPBFqiZD6gZrugVLzKjY-nlK5U65r9zLjpPwI_MXIsbgjmMQfQrDIGJk8_G7pIhPjr1u5E4E0eQZade1TAIoB8MWurHDmzbs5OCagR34-ZjeuZuXjKZpOR5yOFyYM5K0yWQoE9bxoQtSZCuEwj7LJVdrJSGtZ1Qt6PW-CpgTT1Vo11SFNT8EvRxUx22_J5XMZpXqQpt4tCq2YtRF5xQ9D9_ouiF7gJfuJBV2-CNZ7LbEShMV4p6jCqKgVUwyNlYf8nNu7JGx4pNwSt8dwT7YyoOgJ8zgFv4eH4lfhVQCF8RNVPR_YU_8vNgkfR1xl8v0VQlrY6Jl7XPh--bBCLUtvSAnYbraba73OKd8jxxq6ik-16amluMzgf4Bb3xSAg%3D%3D|Interact]] 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 {{{#!sagecell |
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| 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)): |
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| time z, iterates = newton_method(f, c, eps) | z, iterates = newton_method(f, c, eps) |
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| html(iterates) | pretty_print(html(iterates)) |
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| [[http://aleph.sagemath.org/?z=eJx1UsFu4yAQvfcrkHpgSLBrO4pUVUJaqb-wN29qUYITujbYgJN4v34BO-3uoVzm8QYeb2b4obSXlgv_cJQtamAsGWQlhWxHd4RQNFZhX33uH1BaoucDqzGf_NRrTBF-N1rGKIzpljgM0kZ0snyO8Wx8Cu6C7ypp4Q-ZEoPSv1O0yvURuID0KaGp7xexa3KLD8HXxDvlZwZ7Wpd5_nwg5CXJ3uiMGLpwCzhATBLZBkq-QQZO6VDi5kbIW5WBMA7GajPHHXkcyyc3Wg9w25aBQFs0BzqEsbonspjYwrwt8n28k8SF0d5M1rG6oGVePFUFzXNaHpakYMJ0xoaGuTq2baV7J7tgqhYCGlK_7A6oNRY1SOlPveXkazi1Ms3QGQ8tRXCjYSZVGA_MdxRzzWBCgxwri83aH_rl7g7oMr3kgCZv79yyn3aSSznubK7wSlGrTk79kWxPEXeDFL6x3CvDyn-OxUd3x28s8SMfvLrIpB2NhKfWZ5Ge-iYRjnVSQ2LDXaWVV7xrjnLwZ7bWQP7zclHyKi3DnovzbPQ64PhXPMK_NM4_Qg8gVAjOW7oIk78OvNxY|Interact]] http://sagenb.org/home/pub/2823/ {{{ |
https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2823.sagews {{{#!sagecell |
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| {{{ html('<h2>Tangent line grapher</h2>') |
{{{#!sagecell pretty_print(html('<h2>Tangent line grapher</h2>')) |
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| tanf = f(x0i) + df(x0i)*(x-x0i) | tanf = f(x=x0i) + df(x=x0i)*(x-x0i) |
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| 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] |
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| {{{ | {{{#!sagecell |
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| midys = [func(x_val) for x_val in midxs] | midys = [func(x=x_val) for x_val in midxs] |
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| 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('$\int_{a}^{b}{f(x) dx} {\\approx} \sum_i{f(x_i) \Delta x}$')) |
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| {{{ # by Nick Alexander (based on the work of Marshall Hampton) |
{{{#!sagecell |
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| t = sage.calculus.calculus.var('t') | t = var('t') |
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| 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] |
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| 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] |
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| 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]) |
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| 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))) |
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| {{{ | {{{#!sagecell |
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| html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$') | pretty_print(html('$r=' + latex(b+sin(a1*t)^n1 + cos(a2*t)^n2)+'$')) |
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| {{{ | {{{#!sagecell |
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| 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)))) |
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| {{{ | {{{#!sagecell |
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| == Coordinate Transformations == | == Coordinate Transformations (FIXME in Jupyter) == |
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| {{{ | {{{#!sagecell |
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| # 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 |
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| 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), |
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| 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') 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]])) |
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| {{{ | {{{#!sagecell |
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| 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))) |
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| {{{ 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.") |
{{{#!sagecell 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.")) |
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| {{{ | {{{#!sagecell |
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| 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 $\lim_{x -> 0} \sin(x)/x =1$.')) |
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| {{{ | {{{#!sagecell |
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| {{{ | {{{#!sagecell |
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| 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>")) |
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| 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>')) |
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| {{{ from scipy.special.orthogonal import p_roots |
{{{#!sagecell import scipy import numpy from scipy.special.orthogonal import p_roots, t_roots, u_roots |
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| '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))}} |
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| 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) |
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| 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"]): |
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| scaled_ff = fast_float(scaled_func) | scaled_ff = fast_float(scaled_func, 'x') |
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| 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]) |
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| show(graph,xmin=plot_min,xmax=plot_max) | show(graph,xmin=plot_min,xmax=plot_max,aspect_ratio="auto") |
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| 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))) | 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 704: | Line 729: |
| {{{ | {{{#!sagecell |
| Line 735: | Line 760: |
| velocity = derivative( position(t) ) acceleration = derivative(velocity(t)) |
velocity = derivative(position(t), t) acceleration = derivative(velocity(t), t) |
| Line 738: | Line 763: |
| speed_deriv = derivative(speed) | speed_deriv = derivative(speed, t) |
| Line 740: | Line 765: |
| dT = derivative(tangent(t)) | dT = derivative(tangent(t), t) |
| Line 811: | Line 836: |
| == Vector Calculus, 3-D Motion == | == Vector Calculus, 3-D Motion (FIXME) == |
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| {{{ | {{{#!sagecell |
| Line 853: | Line 878: |
| velocity = derivative( position(t) ) acceleration = derivative(velocity(t)) |
velocity = derivative( position(t), t) acceleration = derivative(velocity(t), t) |
| Line 856: | Line 881: |
| speed_deriv = derivative(speed) | speed_deriv = derivative(speed, t) |
| Line 858: | Line 883: |
| dT = derivative(tangent(t)) | dT = derivative(tangent(t), t) |
| Line 861: | Line 886: |
| ## dB = derivative(binormal(t)) | ## dB = derivative(binormal(t), t) |
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| http://www.sagenb.org/home/pub/2828/ {{{ |
http://sagenb.mc.edu/home/pub/97/ {{{#!sagecell |
| Line 952: | Line 977: |
| ## An updated version of this worksheet may be available at http://sagenb.mc.edu | |
| Line 957: | Line 981: |
| var('x,y,z') Rmin=1/10 |
|
| Line 959: | Line 984: |
| @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 968: | Line 1004: |
| Line 970: | Line 1006: |
| 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 972: | Line 1008: |
| 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 979: | Line 1042: |
| {{{ | {{{#!sagecell |
| Line 996: | Line 1060: |
| 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 1006: | Line 1076: |
| 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') |
|
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| show(limit_point+surface)}}} | # show(limit_point+surface) pretty_print(table([[surface],['hi']])) }}} |
| Line 1016: | Line 1092: |
| {{{ | {{{#!sagecell |
| Line 1069: | Line 1145: |
| {{{ %hide %auto |
{{{#!sagecell |
| Line 1086: | Line 1160: |
| 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 1100: | Line 1174: |
| 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 1130: | Line 1204: |
| {{{ | {{{#!sagecell |
| Line 1134: | Line 1208: |
| 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 1136: | Line 1210: |
| and approximation by differential at shifted point are compared.') | and approximation by differential at shifted point are compared.')) |
| Line 1154: | Line 1228: |
| 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 1168: | Line 1242: |
| {{{ | {{{#!sagecell |
| Line 1196: | Line 1270: |
| 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 1206: | Line 1280: |
| http://www.sagenb.org/home/pub/2829/ {{{ |
https://cloud.sagemath.com/projects/19575ea0-317e-402b-be57-368d04c113db/files/pub/2801-2901/2829.sagews {{{#!sagecell |
| Line 1309: | Line 1383: |
| == Lateral Surface Area == | == Lateral Surface Area (FIXME in Jupyter) == |
| Line 1313: | Line 1387: |
| http://www.sagenb.org/home/pub/2826/ {{{ |
http://sagenb.mc.edu/home/pub/89/ {{{#!sagecell |
| Line 1320: | Line 1394: |
| ## | |
| Line 1322: | Line 1397: |
| @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 1331: | Line 1408: |
| smoother=checkbox(default=false)): | in_3d = checkbox(default=true,label='3D'), smoother=checkbox(default=false), auto_update=true): |
| Line 1333: | Line 1412: |
| ds = sqrt(derivative(u(t),t)^2+derivative(v(t),t)^2) | ds = sqrt(derivative(u,t)^2+derivative(v,t)^2) |
| Line 1337: | Line 1416: |
| 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 1342: | Line 1421: |
| 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 1344: | Line 1427: |
| 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 1354: | Line 1435: |
| 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 1364: | Line 1445: |
| 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 1370: | Line 1455: |
| == Parametric surface example == | == Parametric surface example (FIXME in Jupyter) == |
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| {{{ | {{{#!sagecell |
| Line 1391: | Line 1476: |
| 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 {{{#!sagecell |
| Line 1427: | Line 1512: |
| 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
Contents
-
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
Root Finding Using Bisection
by William Stein
Newton's Method
Note that there is a more complicated Newton's method below.
by William Stein
A contour map and 3d plot of two inverse distance functions
by William Stein
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 (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).
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
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
