(lp1 S'Worksheet \'Tutorial: Calculus\' (2009-04-02 at 17:52)\n{{{id=50|\nvar(\'x\')\n@interact\ndef plot_example(f=sin(x^2),r=range_slider(-5,5,step_size=1/2,default=(-3,3)), \n thickness=(3,(1..10)), c=Color(\'blue\'), \n adaptive_recursion=(5,(0..10)), adaptive_tolerance=(0.01,(0.001,1)),\n plot_points=(20,(1..100)),\n linestyle=[\'-\',\'--\',\'-.\',\':\'],\n gridlines=False, fill=False\n ):\n show(plot(f, (x,r[0],r[1]), color=c, thickness=thickness, \n adaptive_recursion=adaptive_recursion,\n adaptive_tolerance=adaptive_tolerance, plot_points=plot_points,\n linestyle=linestyle), gridlines=gridlines)\n///\n\n
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f ", line 1, in \n File "/Users/wstein/travel/2009/pnw-maa/talk/nb/worksheets/admin/5/code/369.py", line 8, in \n exec compile(ur\'f.integrate(x)\' + \'\\n\', \'\', \'single\')\n File "/Users/wstein/build/sage-3.4/local/lib/python2.5/site-packages/SQLAlchemy-0.4.6-py2.5.egg/", line 1, in \n \n File "/Users/wstein/build/sage-3.4/local/lib/python2.5/site-packages/sage/calculus/calculus.p...\n}}}' p207 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:21)\n{{{id=78|\nf = sin(x) - cos(y*x) + 1/(x^3+1)\nf.integrate(x)\n///\n\n-sin(x*y)/y - log(x^2 - x + 1)/6 + arctan((2*x - 1)/sqrt(3))/sqrt(3) + log(x + 1)/3 - cos(x)\n}}}" p208 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:21)\n{{{id=78|\nf = sin(x) - cos(y*x) + 1/(x^3+1)\nshow(f.integrate(x))\n///\n\n
\\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)
\n}}}' p209 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:21)\n{{{id=75|\nshow(f.diff\n///\n}}}" p210 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:22)\n{{{id=78|\nf = sin(x) - cos(y*x) + 1/(x^3+1)\ng = f.integrate(x)\nshow(g)\n///\n\n
\\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)
\n}}}' p211 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=75|\nshow(g.differentiate\n///\n}}}" p212 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:22)\n{{{id=75|\nshow(g.differentiate(x))\n///\n\n
-\\cos \\left( {x y} \\right) + \\sin \\left( x \\right) + \\frac{2}{{3 \\left( \\frac{{\\left( {2 x} - 1 \\right)}^{2} }{3} + 1 \\right)}} - \\frac{{2 x} - 1}{{6 \\left( {x}^{2} - x + 1 \\right)}} + \\frac{1}{{3 \\left( x + 1 \\right)}}
\n}}}' p213 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=24|\ng.diff\n///\n}}}" p214 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=24|\ng.differentiate\n///\n}}}" p215 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=24|\ng.differentiate(x) == f\n///\n\n-cos(x*y) + sin(x) + 2/(3*((2*x - 1)^2/3 + 1)) - (2*x - 1)/(6*(x^2 - x + 1)) + 1/(3*(x + 1)) == -cos(x*y) + sin(x) + 1/(x^3 + 1)\n}}}" p216 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=24|\nbool(g.differentiate(x) == f)\n///\n\nTrue\n}}}" p217 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:22)\n{{{id=24|\nbool(g.differentiate(x) == f)\n///\n\nTrue\n}}}" p218 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:22)\n{{{id=80|\nshow(g.differentiate(x).expand())\n///\n\n
-\\cos \\left( {x y} \\right) + \\sin \\left( x \\right) - \\frac{{2 x}}{{6 {x}^{2} } - {6 x} + 6} + \\frac{1}{{6 {x}^{2} } - {6 x} + 6} + \\frac{2}{{4 {x}^{2} } - {4 x} + 4} + \\frac{1}{{3 x} + 3}
\n}}}' p219 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=81|\nvar('x')\nintegrate(x^2, x)\n///\n\nx^3/3\n}}}" p220 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(x^2),x)\n///\n\nsqrt(pi)*(((sqrt(2)*I - sqrt(2))*sin(1/4) + (sqrt(2)*I + sqrt(2))*cos(1/4))*erf(((2*sqrt(2)*I + 2*sqrt(2))*x + sqrt(2)*I + sqrt(2))/4) + ((sqrt(2)*I - sqrt(2))*sin(1/4) + (sqrt(2)*I + sqrt(2))*cos(1/4))*erf(((2*sqrt(2)*I + 2*sqrt(2))*x - sqrt(2)*I - sqrt(2))/4) + ((sqrt(2)*I + sqrt(2))*sin(1/4) + (sqrt(2)*I - sqrt(2))*cos(1/4))*erf(((2*sqrt(2)*I - 2*sqrt(2))*x + sqrt(2)*I - sqrt(2))/4) + ((sqrt(2)*I + sqrt(2))*sin(1/4) + (sqrt(2)*I - sqrt(2))*cos(1/4)...\n}}}" p221 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(x),x)\n///\n\n-cos(x)^2/2\n}}}" p222 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(3*x),x)\n///\n\n-cos(4*x)/8 - cos(2*x)/4\n}}}" p223 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(10*x),x)\n///\n\n-cos(11*x)/22 - cos(9*x)/18\n}}}" p224 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(2*x)*tan(x),x)\n///\n\n-(sin(3*x) - 3*sin(x))/6\n}}}" p225 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(2*x)*tan(3*x),x)\n///\n\n(6*integrate(((sin(3*x) + sin(x))*sin(4*x) + (cos(3*x) + cos(x))*cos(4*x) - sin(2*x)*sin(3*x) + (1 - cos(2*x))*cos(3*x) - sin(x)*sin(2*x) - cos(x)*cos(2*x) + cos(x))/(2*sin(4*x)^2 - 4*sin(2*x)*sin(4*x) + 2*cos(4*x)^2 + (4 - 4*cos(2*x))*cos(4*x) + 2*sin(2*x)^2 + 2*cos(2*x)^2 - 4*cos(2*x) + 2), x) - sin(3*x) - 3*sin(x))/6\n}}}" p226 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:23)\n{{{id=82|\nintegrate(cos(x)*sin(2*x)*tan(2*x),x)\n///\n\n(3*sqrt(2)*log(2*sin(x)^2 + 2*sqrt(2)*sin(x) + 2*cos(x)^2 + 2*sqrt(2)*cos(x) + 2) + 3*sqrt(2)*log(2*sin(x)^2 + 2*sqrt(2)*sin(x) + 2*cos(x)^2 - 2*sqrt(2)*cos(x) + 2) - 3*sqrt(2)*log(2*sin(x)^2 - 2*sqrt(2)*sin(x) + 2*cos(x)^2 + 2*sqrt(2)*cos(x) + 2) - 3*sqrt(2)*log(2*sin(x)^2 - 2*sqrt(2)*sin(x) + 2*cos(x)^2 - 2*sqrt(2)*cos(x) + 2) - 4*sin(3*x) - 12*sin(x))/24\n}}}" p227 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:23)\n{{{id=82|\nshow(integrate(cos(x)*sin(2*x)*tan(2*x),x))\n///\n\n
\\frac{{{3 \\sqrt{ 2 }} \\log \\left( {2 {\\sin \\left( x \\right)}^{2} } + {{2 \\sqrt{ 2 }} \\sin \\left( x \\right)} + {2 {\\cos \\left( x \\right)}^{2} } + {{2 \\sqrt{ 2 }} \\cos \\left( x \\right)} + 2 \\right)} + {{3 \\sqrt{ 2 }} \\log \\left( {2 {\\sin \\left( x \\right)}^{2} } + {{2 \\sqrt{ 2 }} \\sin \\left( x \\right)} + {2 {\\cos \\left( x \\right)}^{2} } - {{2 \\sqrt{ 2 }} \\cos \\left( x \\right)} + 2 \\right)} - {{3 \\sqrt{ 2 }} \\log \\left( {2 {\\sin \\l...\n}}}' p228 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:23)\n{{{id=82|\nshow(integrate(sin(2*x)*tan(2*x),x))\n///\n\n
\\frac{\\frac{\\log \\left( \\sin \\left( {2 x} \\right) + 1 \\right)}{2} - \\frac{\\log \\left( \\sin \\left( {2 x} \\right) - 1 \\right)}{2} - \\sin \\left( {2 x} \\right)}{2}
\n}}}' p229 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=82|\nshow(integrate(sin(2*x)*tan(x),x))\n///\n\n
x - \\frac{\\sin \\left( {2 x} \\right)}{2}
\n}}}' p230 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=82|\nshow(integrate(sin(2*x)*tan(2*x),x))\n///\n\n
\\frac{\\frac{\\log \\left( \\sin \\left( {2 x} \\right) + 1 \\right)}{2} - \\frac{\\log \\left( \\sin \\left( {2 x} \\right) - 1 \\right)}{2} - \\sin \\left( {2 x} \\right)}{2}
\n}}}' p231 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=82|\nshow(integrate(sin(x)+tan(x),x))\n///\n\n
\\log \\left( \\sec \\left( x \\right) \\right) - \\cos \\left( x \\right)
\n}}}' p232 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=82|\nshow(integrate(sin(x)+tan(2*x),x))\n///\n\n
\\frac{\\log \\left( \\sec \\left( {2 x} \\right) \\right)}{2} - \\cos \\left( x \\right)
\n}}}' p233 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=82|\nshow(integrate(sin(x)+tan(2*x),x))\n///\n\n
\\frac{\\log \\left( \\sec \\left( {2 x} \\right) \\right)}{2} - \\cos \\left( x \\right)
\n}}}' p234 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=78|\nf = sin(x) - cos(y*x) + 1/(x^3+1)\ng = f.integrate(x)\nshow(g)\n///\n\n
\\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)
\n}}}' p235 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:24)\n{{{id=75|\nshow(g.differentiate(x))\n///\n\n
-\\cos \\left( {x y} \\right) + \\sin \\left( x \\right) + \\frac{2}{{3 \\left( \\frac{{\\left( {2 x} - 1 \\right)}^{2} }{3} + 1 \\right)}} - \\frac{{2 x} - 1}{{6 \\left( {x}^{2} - x + 1 \\right)}} + \\frac{1}{{3 \\left( x + 1 \\right)}}
\n}}}' p236 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:24)\n{{{id=24|\nbool(g.differentiate(x) == f)\n///\n\nTrue\n}}}" p237 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:24)\n{{{id=24|\nbool(g.differentiate(x) == f)\n///\n\nTrue\n}}}" p238 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:24)\n{{{id=79|\nf.taylor\n///\n}}}" p239 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:24)\n{{{id=79|\nf.taylor\n///\n}}}" p240 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:24)\n{{{id=79|\nf.taylor(x,1,3)\n///\n\n(2*sin(1) - 2*cos(y) + 1)/2 + (4*cos(1) + 4*sin(y)*y - 3)*(x - 1)/4 - (4*sin(1) - 4*cos(y)*y^2 - 3)*(x - 1)^2/8 - (8*cos(1) + 8*sin(y)*y^3 - 15)*(x - 1)^3/48\n}}}" p241 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:25)\n{{{id=79|\nshow(f.taylor(x,1,3))\n///\n\n
\\frac{{2 \\sin \\left( 1 \\right)} - {2 \\cos \\left( y \\right)} + 1}{2} + \\frac{{\\left( {4 \\cos \\left( 1 \\right)} + {{4 \\sin \\left( y \\right)} y} - 3 \\right) \\left( x - 1 \\right)}}{4} - \\frac{{\\left( {4 \\sin \\left( 1 \\right)} - {{4 \\cos \\left( y \\right)} {y}^{2} } - 3 \\right) {\\left( x - 1 \\right)}^{2} }}{8} - \\frac{{\\left( {8 \\cos \\left( 1 \\right)} + {{8 \\sin \\left( y \\right)} {y}^{3} } - 15 \\right) {\\left( x - 1 \\right)}^{3} }}{4...\n}}}' p242 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:25)\n{{{id=79|\nshow(sin(x^2).taylor(x,1,3))\n///\n\n
\\sin \\left( 1 \\right) + {{2 \\cos \\left( 1 \\right)} \\left( x - 1 \\right)} + {\\left( {-2 \\sin \\left( 1 \\right)} + \\cos \\left( 1 \\right) \\right) {\\left( x - 1 \\right)}^{2} } - \\frac{{\\left( {6 \\sin \\left( 1 \\right)} + {4 \\cos \\left( 1 \\right)} \\right) {\\left( x - 1 \\right)}^{3} }}{3}
\n}}}' p243 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:25)\n{{{id=79|\nshow(f.taylor(x,1,3))\n///\n\n
\\frac{{2 \\sin \\left( 1 \\right)} - {2 \\cos \\left( y \\right)} + 1}{2} + \\frac{{\\left( {4 \\cos \\left( 1 \\right)} + {{4 \\sin \\left( y \\right)} y} - 3 \\right) \\left( x - 1 \\right)}}{4} - \\frac{{\\left( {4 \\sin \\left( 1 \\right)} - {{4 \\cos \\left( y \\right)} {y}^{2} } - 3 \\right) {\\left( x - 1 \\right)}^{2} }}{8} - \\frac{{\\left( {8 \\cos \\left( 1 \\right)} + {{8 \\sin \\left( y \\right)} {y}^{3} } - 15 \\right) {\\left( x - 1 \\right)}^{3} }}{4...\n}}}' p244 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:25)\n{{{id=83|\nshow(f.taylor(y,1,3))\n///\n\n
\\frac{{\\left( {x}^{3} + 1 \\right) \\sin \\left( x \\right)} - {\\cos \\left( x \\right) {x}^{3} } - \\cos \\left( x \\right) + 1}{{x}^{3} + 1} + {{x \\sin \\left( x \\right)} \\left( y - 1 \\right)} + \\frac{{{\\cos \\left( x \\right) {x}^{2} } {\\left( y - 1 \\right)}^{2} }}{2} - \\frac{{{{x}^{3} \\sin \\left( x \\right)} {\\left( y - 1 \\right)}^{3} }}{6}
\n}}}' p245 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:25)\n{{{id=79|\nshow(f.taylor(sin(x)*cos(x), x, 1, 10)\n///\n line 4\n show(f.taylor(sin(x)*cos(x), x, _sage_const_1 , _sage_const_10 )\n \n^\nSyntaxError: invalid syntax\n}}}" p246 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:25)\n{{{id=79|\nshow(f.taylor(sin(x)*cos(x), x, 1, 10))\n///\n\nTraceback (most recent call last):\n File "", line 1, in \n File "/Users/wstein/travel/2009/pnw-maa/talk/nb/worksheets/admin/5/code/409.py", line 7, in \n exec compile(ur\'show(f.taylor(sin(x)*cos(x), x, _sage_const_1 , _sage_const_10 ))\' + \'\\n\', \'\', \'single\')\n File "/Users/wstein/build/sage-3.4/local/lib/python2.5/site-packages/SQLAlchemy-0.4.6-py2.5.egg/", line 1, in \n \nTypeError: taylor() takes exactly 4 argument...\n}}}' p247 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:26)\n{{{id=79|\nh = sin(x)*cos(x)\nshow(h.taylor(x, 1, 3))\n///\n\n
{\\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} } + \\frac{{\\left( {2 {\\sin \\left( 1 \\right)}^{2} } - {2 {\\cos \\left( 1 \\right)}^{2} } \\right) {\\left( x - 1 \\right)}^{3} }}{3}
\n}}}' p248 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:26)\n{{{id=84|\nplot(h,-1,4) + plot(h.taylor(x,1,3),-1,4)\n///\n}}}" p249 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:26)\n{{{id=84|\nplot(h,-1,4) + plot(h.taylor(x,1,3),-1,4, color='red')\n///\n}}}" p250 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:26)\n{{{id=79|\nh = sin(x)*cos(x)\nshow(h.taylor(x, 1, 2))\n///\n\n
{\\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} }
\n}}}' p251 aS"Worksheet 'Tutorial: Symbolic Calculus' (2009-04-02 at 18:26)\n{{{id=84|\nplot(h,-1,4) + plot(h.taylor(x,1,2),-1,4, color='red')\n///\n}}}" p252 aS'Worksheet \'Tutorial: Symbolic Calculus\' (2009-04-02 at 18:26)\n{{{id=84|\n@interact\ndef ex_taylor(n=[0..5]):\n plot(h,-1,4) + plot(h.taylor(x,1,n),-1,4, color=\'red\')\n///\n\n
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