︠70d42de2-a80d-45a8-ad60-2186b302540bi︠
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<h1 style="text-align: center;">Calculus in Sage</h1>
<h2 style="text-align: center;"><span style="color: #333333;">William Stein</span></h2>
<h2 style="text-align: center;"><span style="color: #333333;">2009-05-16 at Sage Days 16</span></h2>
<h2 style="text-align: center;">
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<p>&nbsp;</p>
<p style="text-align: center;"><img src="stamp_newton.jpg" alt="" width="222" height="134" /></p>
<p>&nbsp;</p>
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<h2 style="text-align: center;"><img src="stamp_leibniz.jpg" alt="" width="122" height="144" /></h2>
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</table>
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︡934334b2-1234-4de3-b30b-fc91f7b61416︡{"html": "<h1 style=\"text-align: center;\"><br /></h1>\n<h1 style=\"text-align: center;\"><br /></h1>\n<h1 style=\"text-align: center;\"><br /></h1>\n<h1 style=\"text-align: center;\"><br /></h1>\n<h1 style=\"text-align: center;\">Calculus in Sage</h1>\n<h2 style=\"text-align: center;\"><span style=\"color: #333333;\">William Stein</span></h2>\n<h2 style=\"text-align: center;\"><span style=\"color: #333333;\">2009-05-16 at Sage Days 16</span></h2>\n<h2 style=\"text-align: center;\">\n<table style=\"height: 141px;\" border=\"0\" width=\"400\" align=\"center\">\n<tbody>\n<tr>\n<td>\n<p>&nbsp;</p>\n<p style=\"text-align: center;\"><img src=\"stamp_newton.jpg\" alt=\"\" width=\"222\" height=\"134\" /></p>\n<p>&nbsp;</p>\n</td>\n<td>\n<h2 style=\"text-align: center;\"><img src=\"stamp_leibniz.jpg\" alt=\"\" width=\"122\" height=\"144\" /></h2>\n</td>\n</tr>\n</tbody>\n</table>\n<br /></h2>"}︡
︠b1e1d81b-392d-4c69-9060-1565684b3501i︠
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<h1>Creating Symbolic Expressions</h1>
<p>Use the <strong>var</strong> command to define some symbolic variables.  You can separate the variables by commas or spaces in the var command.</p>
<p><em>Tip</em>: Press shift-enter to evaluate an input cell (instead of clicking "evaluate").</p>

︡88593d6f-fe94-4162-b0ac-5d95d0799e84︡{"html": "<h1>Creating Symbolic Expressions</h1>\r\n<p>Use the <strong>var</strong> command to define some symbolic variables.  You can separate the variables by commas or spaces in the var command.</p>\r\n<p><em>Tip</em>: Press shift-enter to evaluate an input cell (instead of clicking \"evaluate\").</p>"}︡
︠cc5c9756-4c36-4893-be1f-291a03e6b006︠
var('x y z epsilon')
︡5d0996b8-5a96-4351-9fe7-475b1bc0aec4︡{"stdout": "(x, y, z, epsilon)"}︡
︠330eb9b5-ab03-471b-a000-e18d1e7db4ea︠
cos(x^3) - y^2*z + epsilon
︡593f35be-b644-4a3d-b57e-60498fdeaf6e︡{"stdout": "-y^2*z + cos(x^3) + epsilon"}︡
︠364e4e4d-c732-4eef-91f8-2129b089c774i︠
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<p><strong>Examples:</strong> Create the following expressions: $\sin^5(x)\cos^2(x), \qquad \displaystyle \frac{x^3}{x^3 + 1}, \qquad k\cdot P \cdot \left(1 - \frac{P}{K}\right)$.</p>
<p><em>Note</em>: that you <em>must</em> put in an asterisk (*) for multiplication.</p>

︡0ef1d048-006b-41ac-ac5e-aa7c8c5c23d1︡{"html": "<p><strong>Examples:</strong> Create the following expressions: $\\sin^5(x)\\cos^2(x), \\qquad \\displaystyle \\frac{x^3}{x^3 + 1}, \\qquad k\\cdot P \\cdot \\left(1 - \\frac{P}{K}\\right)$.</p>\n<p><em>Note</em>: that you <em>must</em> put in an asterisk (*) for multiplication.</p>"}︡
︠648bf9fe-9457-484a-bf51-39b6a0c9d529︠
sin(x)^5*cos(x)^2
︡c85b653f-a2b8-4419-8c11-df6fa4f3e8f5︡{"stdout": "cos(x)^2*sin(x)^5"}︡
︠8dd8fce6-2c0a-4014-abc7-c102610c477d︠
x^3/(x^3+1)
︡df2d65cf-68bc-4bc8-b32a-fb9087f04038︡{"stdout": "x^3/(x^3 + 1)"}︡
︠e517ff1c-b9e5-4650-867c-4bff1507b4ca︠
var('k,K,P')
k*P * (1-P/K)
︡fd0e74a8-67d8-4ce5-91f9-98cc90506d92︡{"stdout": "k*P*(1 - P/K)"}︡
︠2d988611-2836-4716-81d8-94ca2970e970i︠
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<p>Most standard functions are defined in Sage.&nbsp; They are named lowercase, much like in Maple. E.g.,</p>
<p><span style="font-family: courier new,courier;">sin, cos, tan, sec, csc, cot, sinh, cosh, tanh, sech, csch, coth, log, exp, etc.</span></p>

︡15076d39-2842-4c68-80b6-4b442322645a︡{"html": "<p>Most standard functions are defined in Sage.&nbsp; They are named lowercase, much like in Maple. E.g.,</p>\r\n<p><span style=\"font-family: courier new,courier;\">sin, cos, tan, sec, csc, cot, sinh, cosh, tanh, sech, csch, coth, log, exp, etc.</span></p>"}︡
︠5485e6d8-89d3-4d6b-ae5e-9a9f753425c4︠
var('x,y')
sin(x) + cos(y) - tan(x/y) + sec(x*csc(y))^3
︡2acff958-a612-4b8b-93a2-ee3c44276ec5︡{"stdout": "sec(x*csc(y))^3 + cos(y) - tan(x/y) + sin(x)"}︡
︠3b319090-4bf0-4a15-8a47-8be7427898a6i︠
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<p><strong>Example: </strong>Construct the symbolic expresion $\sin(x^{\cos(y)} + \theta) + \coth(2x) + \log(3x)\cdot \exp(y^3)$.&nbsp;</p>

︡1e1c19b0-2b2a-4287-9f5c-e484e12fd029︡{"html": "<p><strong>Example: </strong>Construct the symbolic expresion $\\sin(x^{\\cos(y)} + \\theta) + \\coth(2x) + \\log(3x)\\cdot \\exp(y^3)$.&nbsp;</p>"}︡
︠f06662b2-a919-4558-b41f-bbc112037aa5︠
var('x,y,theta')
show(sin(x^cos(y)+theta) + coth(2*x) + log(3*x) * exp(y^3))
︡881024f5-0109-4587-82d5-2df38c3b6c34︡{"html": "<div class=\"math\">{\\log \\left( {3 x} \\right) {e}^{{y}^{3} } } + \\sin \\left( {x}^{\\cos \\left( y \\right)}  + \\theta \\right) + \\coth \\left( {2 x} \\right)</div>"}︡
︠b13aeb3d-11c9-4fa6-a5e1-4652365c9d4ai︠
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<h2>Making substitutions</h2>
<p>Use the <strong>subs</strong> method to replace any variables by other variables.</p>

︡d2ef74bb-df8d-44ee-9210-34fb31118f75︡{"html": "<h2>Making substitutions</h2>\r\n<p>Use the <strong>subs</strong> method to replace any variables by other variables.</p>"}︡
︠5200130b-6ed5-4b2a-871e-5d5d8ca56b9c︠
var('x,y')
f = sin(x) + cos(y) - x^2 + y
︡29ed36d1-5c69-4540-9e5f-6c1263f11441︡︡
︠b2f91c0d-10f8-4841-b9ff-87fb82eb7c95︠
f.subs(x=5)
︡8ed4f81c-55f7-40b2-9f14-df411424bc00︡{"stdout": "cos(y) + y + sin(5) - 25"}︡
︠0bb5949f-c606-4945-8e51-511a936672d1︠
f.subs(x=y, y=x)
︡26da64fa-85ca-4d6d-b717-3becf1b6bd6d︡{"stdout": "sin(y) - y^2 + cos(x) + x"}︡
︠ef95ffc0-f902-4f97-a2cd-754b4293eb77i︠
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<p><strong>Example: </strong>Replace $x$ by $\sin(y)-x$ in the expression $x^3 + x y$.</p>

︡f91e1b84-7a74-4e1f-aa8c-5086706fde72︡{"html": "<p><strong>Example: </strong>Replace $x$ by $\\sin(y)-x$ in the expression $x^3 + x y$.</p>"}︡
︠6650e282-a7ed-4275-8a6d-b187111184a4︠
var('x,y')
f = x^3 + x*y
f.subs(x=sin(y)-x)
︡917d16a7-7650-41eb-9ac6-0e7a2b31f77b︡{"stdout": "(sin(y) - x)^3 + y*(sin(y) - x)"}︡
︠68ada52d-a098-4278-9e0a-8081a5311c2fi︠
%html
<h2>Expanding Expressions</h2>
<p>To expand a symbolic expression with exponents, use the <strong>expand </strong>method.</p>

︡ec6f2f56-2454-431a-88ce-ee0713521ee9︡{"html": "<h2>Expanding Expressions</h2>\r\n<p>To expand a symbolic expression with exponents, use the <strong>expand </strong>method.</p>"}︡
︠6e2225c7-ae93-47fc-b2c3-a974b8562796︠
var('x,y')
f = (x+2*y)^3
f
︡20aaf512-4b31-479a-bd56-606b1400f3cb︡{"stdout": "(2*y + x)^3"}︡
︠160eea6a-d322-4320-b26b-c3aa19b5c803︠
f.expand().show()                  # tip -- using show makes the output nicer
︡9be4068c-15ec-4676-8dda-0df679ed9553︡{"html": "<div class=\"math\">{8 {y}^{3} } + {{12 x} {y}^{2} } + {{6 {x}^{2} } y} + {x}^{3} </div>"}︡
︠6fda4174-19aa-44c4-90d3-c166eabd610ci︠
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<p><strong>Example:</strong> Expand the expression $(2\sin(x)&nbsp; - \cos(y))^5$.</p>

︡a2077412-aea3-429b-8e6c-266c621244f4︡{"html": "<p><strong>Example:</strong> Expand the expression $(2\\sin(x)&nbsp; - \\cos(y))^5$.</p>"}︡
︠4b8ee132-0fa7-4b04-b307-932e68893f1c︠
f = (2*sin(x) - cos(y))^5
f.expand()
︡f273278c-5993-4be4-9309-7335a564d584︡{"stdout": "-cos(y)^5 + 10*sin(x)*cos(y)^4 - 40*sin(x)^2*cos(y)^3 + 80*sin(x)^3*cos(y)^2 - 80*sin(x)^4*cos(y) + 32*sin(x)^5"}︡
︠3f6983ec-3c64-45e7-ac35-8c4f9c4a25e1i︠
%html
<h1>Creating Symbolic Functions</h1>
<p>To create a symbolic function, use the notation <strong>f(x,y) = x^3 + y</strong>.&nbsp; A symbolic function is just like a symbolic expression, except you can call it without having to explicitly use subs or name variables and be sure that the order is what you want.</p>
<p>&nbsp;</p>

︡89f6a86a-461f-40c7-a9b4-8554ffeb1f31︡{"html": "<h1>Creating Symbolic Functions</h1>\r\n<p>To create a symbolic function, use the notation <strong>f(x,y) = x^3 + y</strong>.&nbsp; A symbolic function is just like a symbolic expression, except you can call it without having to explicitly use subs or name variables and be sure that the order is what you want.</p>\r\n<p>&nbsp;</p>"}︡
︠fdd0f99c-cd38-4d06-aec6-dea9dec8d170︠
f(x,y) = x^3 + y
f
︡b320703e-8ae1-438a-bf99-33fcdd92cc95︡{"stdout": "(x, y) |--> y + x^3"}︡
︠77c09fab-c5d3-4911-93f5-c1999c09e4ce︠
f(2,3)
︡2b248622-4306-41b4-8fb6-bd3bc7a02dc6︡{"stdout": "11"}︡
︠85938322-8725-4299-99d0-cfe60a698c63︠
f(pi,e)
︡52c90818-47f7-416c-8da6-05963499dd1c︡{"stdout": "pi^3 + e"}︡
︠1c734c0d-d704-489a-b29a-8dc37bfb0539i︠
%html
<p><strong>Problem</strong>: Create the functions $x\mapsto x^3 + 1, \qquad (x, y) \mapsto \sin(x) - \cos(y)/y, \qquad (a,x,\theta)\mapsto a x + \theta^2$.</p>

︡b79b50d3-1059-4632-933e-2a3445c98456︡{"html": "<p><strong>Problem</strong>: Create the functions $x\\mapsto x^3 + 1, \\qquad (x, y) \\mapsto \\sin(x) - \\cos(y)/y, \\qquad (a,x,\\theta)\\mapsto a x + \\theta^2$.</p>"}︡
︠6ae7426f-aa4f-42a0-b360-cb290e23f1fb︠
f(x) = x^3 + 1
f
︡5c09681a-34a5-44b4-a8db-4ca37b82f836︡{"stdout": "x |--> x^3 + 1"}︡
︠644a8150-560a-4786-b510-08d66be8057c︠
f(x,y) = sin(x)-cos(y)/y
f
︡9cbc7ce2-1b35-4842-8972-d9c5843a9347︡{"stdout": "(x, y) |--> sin(x) - cos(y)/y"}︡
︠05f44685-1429-498c-92fd-fea363652143︠
f(a,x,theta) = a*x+theta^2
show(f)
︡bf53c2a2-b4ca-45d7-9569-130f48999d48︡{"html": "<div class=\"math\">\\left(a, x, \\theta \\right)\\ {\\mapsto}\\ {a x} + {\\theta}^{2} </div>"}︡
︠298bb9c6-31ea-4cac-93c0-0b6817cfa591i︠
%html
<h1>2D Plotting</h1>
<p>Use the <strong>plot</strong> command to plot a function of 1 variable.&nbsp; <em>TIP</em>: Type <strong>plot(&lt;tab key&gt;</strong> to find out much more about the plot command.</p>

︡1b8550c3-f379-4185-aa52-655d70ab99d9︡{"html": "<h1>2D Plotting</h1>\r\n<p>Use the <strong>plot</strong> command to plot a function of 1 variable.&nbsp; <em>TIP</em>: Type <strong>plot(&lt;tab key&gt;</strong> to find out much more about the plot command.</p>"}︡
︠69b59d5a-62ba-44fe-8b03-df4d052e71cf︠
var('x')
plot(sin(x^2), (x,-3,3))
︡5631c9e6-167b-491e-bd41-407a7dc34bad︡︡
︠6acbf78f-fefd-4b15-80b7-aab99ab62defi︠
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<p>Here's a the same plot, but you can adjust many of the parameters to the plot command interactively.</p>

︡70a0a5e7-446a-46e6-bb05-6db7f865abaa︡{"html": "<p>Here's a the same plot, but you can adjust many of the parameters to the plot command interactively.</p>"}︡
︠6f671fb0-976f-4edf-8bd8-e237de54db9d︠
var('x')
@interact
def plot_example(f=sin(x^2),r=range_slider(-5,5,step_size=1/4,default=(-3,3)), 
                 thickness=(3,(1..10)),
                 adaptive_recursion=(5,(0..10)), adaptive_tolerance=(0.01,(0.001,1)),
                 plot_points=(20,(1..100)),
                 linestyle=['-','--','-.',':'],
                 gridlines=False, fill=False,
                 frame=False, axes=True,
                  c=Color('blue')
                 ):
    show(plot(f, (x,r[0],r[1]), color=c, thickness=thickness, 
                 adaptive_recursion=adaptive_recursion,
                 adaptive_tolerance=adaptive_tolerance, plot_points=plot_points,
                 linestyle=linestyle, fill=fill if fill else None), 
                 gridlines=gridlines, frame=frame, axes=axes)
︡230b8f93-d7b5-47b9-b849-ae2d88882067︡︡
︠61432a52-de2a-4fd6-af25-796a774aa1e1i︠
%html
<p><strong>Example: </strong>I made the following using the above interactive plotter with $\sin(x^2)$.</p>
<p><img src="plotsinx2.png" alt="" /></p>

︡5593f43b-d1e0-4bd3-ab41-91fb00043f00︡{"html": "<p><strong>Example: </strong>I made the following using the above interactive plotter with $\\sin(x^2)$.</p>\n<p><img src=\"plotsinx2.png\" alt=\"\" /></p>"}︡
︠6ad2bbca-c5c4-48e0-a714-895e9adb51c5i︠
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<p>You can plot many other things, including polygons, parametric plots, polar plots, implicit plots, etc.:</p>
<p style="text-align: center;"><strong>line, polygon, circle, text, polar_plot, parametric_plot, circle, implicit_plot</strong></p>
<p style="text-align: left;">You superimpose plots using <strong>+</strong>.</p>

︡52eba0ba-6a4a-4c34-841d-b51ef4dc41ef︡{"html": "<p>You can plot many other things, including polygons, parametric plots, polar plots, implicit plots, etc.:</p>\r\n<p style=\"text-align: center;\"><strong>line, polygon, circle, text, polar_plot, parametric_plot, circle, implicit_plot</strong></p>\r\n<p style=\"text-align: left;\">You superimpose plots using <strong>+</strong>.</p>"}︡
︠42b6d259-4d24-4252-9bac-53696d5370eb︠
var('x')
P = circle((0,0),1) + polar_plot(2 + 2*cos(x), (x, 0, 2*pi), rgbcolor='red')+ plot(sin(x^2),(x,0,4))
show(P, aspect_ratio=1)
︡32f4db47-a74a-437c-99cd-4161959af3e8︡︡
︠3663ac83-68a6-4fde-a9b3-6b28036340afi︠
%html
<p><strong>Example: </strong>Draw 3 concentric circles that are red, green and blue.&nbsp; [Hints: Use rgbcolor, and give the aspect_ratio=1 option to the show command, as above.]</p>

︡255917ee-58dd-4f4d-a286-a42e8b744969︡{"html": "<p><strong>Example: </strong>Draw 3 concentric circles that are red, green and blue.&nbsp; [Hints: Use rgbcolor, and give the aspect_ratio=1 option to the show command, as above.]</p>"}︡
︠24a57e41-25b5-4726-a4e3-9a7cd5ff2e44︠
G = circle((0,0),1,rgbcolor='red') + circle((0,0),2,rgbcolor='green')
G = G + circle((0,0),3,rgbcolor='blue')
G.show(aspect_ratio=1)
︡536d795f-02f6-42fd-ab89-7b746b7fb0f7︡{"html": "<font color='black'><img src='cell://sage0.png'></font>"}︡
︠1278886f-d211-48cf-b9e1-cc0973ee8c7ci︠
%html
<h1>3D Plotting</h1>
<p>You can also plot functions of two variables using the <strong>plot3d</strong> command.&nbsp; Also, there are <strong>line3d, sphere, text3d, cube, parametric_plot3d, </strong>etc. commands.&nbsp;&nbsp; Bill Cauchois will demo 3d plotting much more in the next talk.</p>

︡3c146dc3-e393-41f8-b2cc-8a79fa69532d︡{"html": "<h1>3D Plotting</h1>\n<p>You can also plot functions of two variables using the <strong>plot3d</strong> command.&nbsp; Also, there are <strong>line3d, sphere, text3d, cube, parametric_plot3d, </strong>etc. commands.&nbsp;&nbsp; Bill Cauchois will demo 3d plotting much more in the next talk.</p>"}︡
︠83ab90c8-46f1-4f34-8d2e-161e60e2542b︠
var('x,y')
plot3d(sin(x-y)*cos(x-y^2),(x,-2,2),(y,-2,2))
︡67e036ec-93e1-4e15-be77-97093fd9757b︡︡
︠dc291e92-74e2-4cca-a5d5-5a65ceae5668i︠
%html
<p><strong>Example</strong>: Draw a 3d plot of the function $4x e^{-x^2-y^2}$.</p>

︡6bab7c3d-fd5f-4cab-b49f-85a871dfcfc7︡{"html": "<p><strong>Example</strong>: Draw a 3d plot of the function $4x e^{-x^2-y^2}$.</p>"}︡
︠d39ec6b5-6a95-4b0a-95f9-e3ce1c5aa7d3︠
f(x,y) = 4*x*e^(-x^2-y^2)
plot3d(f,(x,-2,2),(y,-2,2))
︡c34dec1c-fe5e-4b49-ba47-1a53b3263c12︡︡
︠3141f003-2d60-4c7c-b5f9-5ca500fe064ai︠
%html
<h1>Computing Integrals and Derivatives</h1>
<p>You can symbolically integrate or differentiate functions, compute limits, Taylor polynomials, etc.</p>

︡b7e20b26-eb40-4c1f-8afa-ba2b4a9223a4︡{"html": "<h1>Computing Integrals and Derivatives</h1>\r\n<p>You can symbolically integrate or differentiate functions, compute limits, Taylor polynomials, etc.</p>"}︡
︠86fe2890-71e0-4fa4-8ee0-4a648bdfaeb0︠
var('x')
integrate(x^2, x)
︡630a1ef6-b441-4e57-9729-8d144c99f8b9︡{"stdout": "x^3/3"}︡
︠25719f9e-cce7-4621-bb62-a0831663500c︠
show(integrate(sin(x)+tan(2*x),x))
︡2c59407d-9a8a-4601-88ef-7da34b23afbc︡{"html": "<div class=\"math\">\\frac{\\log \\left( \\sec \\left( {2 x} \\right) \\right)}{2} - \\cos \\left( x \\right)</div>"}︡
︠b9d6ee5b-80db-4ed8-95ac-67bad7504e24︠
f = sin(x) - cos(y*x) + 1/(x^3+1)
g = f.integrate(x)
show(g)
︡67a78c20-6244-4eb8-a07b-8f1b7b599e6f︡{"html": "<div class=\"math\">\\frac{-\\sin \\left( {x y} \\right)}{y} - \\frac{\\log \\left( {x}^{2}  - x + 1 \\right)}{6} + \\frac{\\tan^{-1} \\left( \\frac{{2 x} - 1}{\\sqrt{ 3 }} \\right)}{\\sqrt{ 3 }} + \\frac{\\log \\left( x + 1 \\right)}{3} - \\cos \\left( x \\right)</div>"}︡
︠c45c014a-3981-4ab4-9f5d-a6605bb0da8b︠
bool(g.differentiate(x) == f)
︡cd146c39-a2b6-4ece-bfd1-c337caa22b6b︡{"stdout": "True"}︡
︠a1c5f79f-8873-4bbf-8e35-29c76a1146b9︠
h = sin(x)*cos(x)
show(h.taylor(x, 1, 2))
︡dfc489e8-69c0-4109-92fd-987e10a1c5ac︡{"html": "<div class=\"math\">{\\cos \\left( 1 \\right) \\sin \\left( 1 \\right)} + {\\left( -{\\sin \\left( 1 \\right)}^{2}  + {\\cos \\left( 1 \\right)}^{2}  \\right) \\left( x - 1 \\right)} - {{{2 \\cos \\left( 1 \\right)} \\sin \\left( 1 \\right)} {\\left( x - 1 \\right)}^{2} }</div>"}︡
︠7af462af-5517-4714-8d9a-e43a077b040a︠
@interact
def ex_taylor(n=(1..10), h=sin(x)*cos(x)):
    h = SR(h)
    show(plot(h,-1,4,thickness=2) + plot(h.taylor(x,1,n),-1,4, color='red'), ymin=-1,ymax=1)
︡0a57bf29-4096-4dea-92ee-a010cd501539︡︡
︠2e912469-99c4-481b-ab32-569e245e8d73i︠
%html
<p><strong><br /></strong></p>

︡40868ad6-bc1a-4954-80c4-82d96f20b9dd︡{"html": "<p><strong><br /></strong></p>"}︡