︠9633dd3a-f656-47c6-a72e-673c1e0f1205s︠
%typeset_mode True
︡f6c08ab6-483e-4fe1-a607-4cd20f569750︡{"done":true}
︠3b77367b-7761-448d-b537-633e216b5ee6s︠
diff(exp(-x^2),x)
︡6f6cd354-c6c4-47b3-a3d7-791ef34949e2︡{"html":"
$\\displaystyle -2 \\, x e^{\\left(-x^{2}\\right)}$
"}︡{"done":true}
︠1cb0ea7e-b434-492a-b365-e459afd9c3acs︠
'diff(exp(-x^2),x)'
︡8bb1ea4c-da7a-4a06-859f-c024b77b8559︡{"html":"diff(exp(-x^2),x)
"}︡{"done":true}
︠253888be-563b-46b1-b2de-21f501180fe8s︠
diff(exp(-x^2),x,2)
︡ac3bce2b-4d0f-435d-9fbe-2df695e4f449︡{"html":"$\\displaystyle 4 \\, x^{2} e^{\\left(-x^{2}\\right)} - 2 \\, e^{\\left(-x^{2}\\right)}$
"}︡{"done":true}
︠f8553814-5ada-4f98-9326-99cc81dceb79s︠
diff(log(x/(x^2+1)),x)
︡96c485f1-970e-4e3f-95c3-b1710a47df82︡{"html":"$\\displaystyle -\\frac{{\\left(x^{2} + 1\\right)} {\\left(\\frac{2 \\, x^{2}}{{\\left(x^{2} + 1\\right)}^{2}} - \\frac{1}{x^{2} + 1}\\right)}}{x}$
"}︡{"done":true}
︠de339f13-877d-41d6-99a1-eebe88ab48a0s︠
diff(log(x/(x^2+1)),x).simplify_full()
︡a560eae5-9234-4b97-ade5-051ed769b99a︡{"html":"$\\displaystyle -\\frac{x^{2} - 1}{x^{3} + x}$
"}︡{"done":true}
︠8e50c7d5-ed8d-4bc0-9ce3-8caef212cd26s︠
︡3924ea89-3cec-4a5f-866b-95847eda6ec9︡{"done":true}
︠76288eae-2e32-4568-9d3d-f47af3786679s︠
diff(x^(x^x),x)
︡98878b93-15f3-40a6-b326-58ffd17e56ac︡{"html":"$\\displaystyle {\\left(x^{x} {\\left(\\log\\left(x\\right) + 1\\right)} \\log\\left(x\\right) + \\frac{x^{x}}{x}\\right)} x^{\\left(x^{x}\\right)}$
"}︡{"done":true}
︠4f1de1f9-fa9c-493d-87ae-502b908cbfccs︠
diff(x^(x^x),x).collect(log(x))
︡2304e0f2-a4f7-4796-b9f1-383f8d40b9f5︡{"html":"$\\displaystyle x^{x} x^{\\left(x^{x}\\right)} \\log\\left(x\\right)^{2} + x^{x} x^{\\left(x^{x}\\right)} \\log\\left(x\\right) + \\frac{x^{x} x^{\\left(x^{x}\\right)}}{x}$
"}︡{"done":true}
︠945e322e-2630-412d-a765-2d83605a316c︠
%html
Derivace funkce dané implicitně
︡0f201bb2-bfe3-4296-9939-bfd81a0066a6︡{"done":true,"html":"Derivace funkce dané implicitně"}
︠a09298e3-c623-478a-a7c9-8f7ad0435748s︠
reset()
︡885ad6e3-2907-4e4d-9ae2-3201dc007ab9︡{"done":true}
︠4e114bb8-8e76-4161-83dd-4bc449c79205s︠
var('x')
︡cb0bef12-d466-4562-aefd-259ffb5f3bb0︡{"html":"$\\displaystyle x$
"}︡{"done":true}
︠df36cd67-12e4-4858-9f31-cf592ef6b3f7s︠
y = function('y')(x);y
︡44144fc2-95f4-47bd-81ea-bb92d975cc70︡{"html":"$\\displaystyle y\\left(x\\right)$
"}︡{"done":true}
︠43595d3e-ffee-4a07-8e05-016633b0c181s︠
dy=y.diff(x);dy
︡b8802223-5df2-4ebf-904c-3718c102f0c5︡{"html":"$\\displaystyle \\frac{\\partial}{\\partial x}y\\left(x\\right)$
"}︡{"done":true}
︠e9c1d2a6-2a15-443a-ba7f-335f7bfb8576s︠
var('c')
︡c6e472d0-4425-4973-92bf-86b96bd921e3︡{"html":"$\\displaystyle c$
"}︡{"done":true}
︠67e0f20c-09b7-4bad-98d5-68559917259fs︠
eq=x^2+y^2==c;eq
︡3c3fbf75-5f39-4ad4-a612-716329af9bee︡{"html":"$\\displaystyle x^{2} + y\\left(x\\right)^{2} = c$
"}︡{"done":true}
︠4605e1c9-19fb-43f6-b205-4bd68b84e92bs︠
dc=diff(eq,x);dc
︡e8777ef4-196f-4f98-916f-b1b371acc764︡{"html":"$\\displaystyle 2 \\, y\\left(x\\right) \\frac{\\partial}{\\partial x}y\\left(x\\right) + 2 \\, x = 0$
"}︡{"done":true}
︠ba1aedc9-6411-4ac3-b46b-8fdadbf86f0bs︠
yx=solve(dc,dy);yx
︡7d1d38ad-de2a-4e27-b636-7689d9ebdd29︡{"html":"[$\\displaystyle \\frac{\\partial}{\\partial x}y\\left(x\\right) = -\\frac{x}{y\\left(x\\right)}$]
"}︡{"done":true}
︠3a575699-e8c8-4c59-8b7e-1e25e613f66bs︠
dcc=diff(eq,x,2);dcc
︡1e069e6e-f086-45dd-bc37-fd94c4cefb3e︡{"html":"$\\displaystyle 2 \\, \\frac{\\partial}{\\partial x}y\\left(x\\right)^{2} + 2 \\, y\\left(x\\right) \\frac{\\partial^{2}}{(\\partial x)^{2}}y\\left(x\\right) + 2 = 0$
"}︡{"done":true}
︠07349ffd-84a6-4626-9596-e6de54a8cf49s︠
dyy=y.diff(x,2);dyy
︡93fab7cb-95a2-4e36-b708-ac7077ab0498︡{"html":"$\\displaystyle \\frac{\\partial^{2}}{(\\partial x)^{2}}y\\left(x\\right)$
"}︡{"done":true}
︠8517a944-4ae4-4f91-a599-979da3ca6a5fs︠
yxx=solve(dcc,dyy);yxx
︡65277b13-4c4f-4bb7-a9bc-a0f7e5deb7e5︡{"html":"[$\\displaystyle \\frac{\\partial^{2}}{(\\partial x)^{2}}y\\left(x\\right) = -\\frac{\\frac{\\partial}{\\partial x}y\\left(x\\right)^{2} + 1}{y\\left(x\\right)}$]
"}︡{"done":true}
︠aac3f48d-6e78-44ed-b6c6-9e0dda5af9e7s︠
yxx[0].rhs().subs(yx).simplify_full()
︡6aa2214f-bc49-4822-94eb-63b47ad6b6ce︡{"html":"$\\displaystyle -\\frac{x^{2} + y\\left(x\\right)^{2}}{y\\left(x\\right)^{3}}$
"}︡{"done":true}
︠a375d90a-53b8-43e3-9749-8b28add31c55︠
%html
Parciální derivace
︡f4b6673c-0127-4101-809f-695efffae2ce︡{"done":true,"html":"Parciální derivace"}
︠670886bc-f264-43e6-bca2-6a4fda12f92fs︠
var('x,y,a')
︡43cd424d-11c5-41dd-8e25-14b074d9a11d︡{"html":"($\\displaystyle x$, $\\displaystyle y$, $\\displaystyle a$)
"}︡{"done":true}
︠31fece86-7d60-496c-91ec-0c49e35b07b3s︠
diff(exp(a*x*y^2),x).diff(y,2)
︡fefc48e6-eb99-41c0-ab99-aa4cbbbcd71f︡{"html":"$\\displaystyle 4 \\, a^{3} x^{2} y^{4} e^{\\left(a x y^{2}\\right)} + 10 \\, a^{2} x y^{2} e^{\\left(a x y^{2}\\right)} + 2 \\, a e^{\\left(a x y^{2}\\right)}$
"}︡{"done":true}
︠c61b9470-a7ce-4dcf-badb-cf1e2f82cfa4s︠
diff(exp(a*x*y^2),x,1,y,2)
︡833aa966-9856-438a-bb30-d75284853f47︡{"html":"$\\displaystyle 4 \\, a^{3} x^{2} y^{4} e^{\\left(a x y^{2}\\right)} + 10 \\, a^{2} x y^{2} e^{\\left(a x y^{2}\\right)} + 2 \\, a e^{\\left(a x y^{2}\\right)}$
"}︡{"done":true}
︠420ab13a-9281-4c06-8dff-05d078430f68s︠
diff(sin(x+y)/y^4, x,5,y,2)
︡c413a308-a750-4885-9a18-9bc5de18f647︡{"html":"$\\displaystyle -\\frac{\\cos\\left(x + y\\right)}{y^{4}} + \\frac{8 \\, \\sin\\left(x + y\\right)}{y^{5}} + \\frac{20 \\, \\cos\\left(x + y\\right)}{y^{6}}$
"}︡{"done":true}
︠ed9f1e7f-5772-4e7e-8cad-a46eb8882b69s︠
var('n')
︡6227b431-16db-485e-9561-70306e40d072︡{"html":"$\\displaystyle n$
"}︡{"done":true}
︠94f418cf-e1b6-4412-b045-2a74a5ab10c2s︠
︡0578d3bf-531f-4ae3-8f82-5ee6e0118708︡{"done":true}
︠9be72114-0b43-482a-97af-ada5b705e811︠
%html
Derivace funkce (ve smyslu datové struktury v Sage)
︡364e74db-ddcc-4831-812d-066b0b3b59ca︡{"done":true,"html":"Derivace funkce (ve smyslu datové struktury v Sage)"}
︠6ce8066d-6faa-4991-8be7-f7ccca4ccadas︠
︡dac6c3ea-c30f-4aad-a1eb-43ec218d110a︡{"done":true}︡{"done":true}
︠6078fb3a-d659-4ccf-abb6-e4fd88276911s︠
g(x)=x^n*exp(sin(x));g
︡7af06f47-15a7-4d16-9abf-384e665fe846︡{"html":"$\\displaystyle x \\ {\\mapsto}\\ x^{n} e^{\\sin\\left(x\\right)}$
"}︡{"done":true}
︠b30cb17d-9a3c-4f42-8298-a093fb35ad3ds︠
g.diff()
︡df7f1259-f0da-4e83-a6d0-99380c91990e︡{"html":"$\\displaystyle x \\ {\\mapsto}\\ \\left(n x^{n - 1} e^{\\sin\\left(x\\right)} + x^{n} \\cos\\left(x\\right) e^{\\sin\\left(x\\right)}\\right)$
"}︡{"done":true}
︠edf45cf9-65ee-4da1-a353-5f873ac97c96s︠
g.diff()(pi/6)
︡a669bcf4-d61f-47b3-9be3-499ad719ed76︡{"html":"$\\displaystyle \\left(\\left(\\frac{1}{6} \\, \\pi\\right)^{n - 1} n e^{\\frac{1}{2}} + \\frac{1}{2} \\, \\sqrt{3} \\left(\\frac{1}{6} \\, \\pi\\right)^{n} e^{\\frac{1}{2}}\\right)$
"}︡{"done":true}
︠e77bf27f-1fca-4b0b-8d2c-95ea587a2dbas︠
g.diff().diff()
︡924c9a84-d4b4-4f38-9939-5dd201592a98︡{"html":"$\\displaystyle \\left(\\begin{array}{r}\nx \\ {\\mapsto}\\ {\\left(n - 1\\right)} n x^{n - 2} e^{\\sin\\left(x\\right)} + 2 \\, n x^{n - 1} \\cos\\left(x\\right) e^{\\sin\\left(x\\right)} + x^{n} \\cos\\left(x\\right)^{2} e^{\\sin\\left(x\\right)} - x^{n} e^{\\sin\\left(x\\right)} \\sin\\left(x\\right)\n\\end{array}\\right)$
"}︡{"done":true}
︠d8087fff-9d7b-4385-bcbd-19a55b667fbbs︠
f(x,y)=x^2*y+2*x*y+x
︡b0059eb3-6c49-42fa-8f9f-07e15f07f78c︡{"done":true}
︠5ac4ff24-a942-453f-8680-fc6f6ef30754s︠
f.diff() #vypočte obě parciální derivace
︡7b134f51-f893-4658-8f10-a3dfd8374815︡{"html":"$\\displaystyle \\left( x, y \\right) \\ {\\mapsto} \\ \\left(2 \\, x y + 2 \\, y + 1,\\,x^{2} + 2 \\, x\\right)$
"}︡{"done":true}
︠19defd9b-6131-43b7-9787-9613e25ecb89s︠
f.diff(x); f.diff(y)
︡65a078a2-1f1b-4ea1-a9d9-8415a33422a2︡{"html":"$\\displaystyle \\left( x, y \\right) \\ {\\mapsto} \\ 2 \\, x y + 2 \\, y + 1$
"}︡{"html":"$\\displaystyle \\left( x, y \\right) \\ {\\mapsto} \\ x^{2} + 2 \\, x$
"}︡{"done":true}
︠513db56e-092b-4013-acde-e1dc37640cf7s︠
f.diff(x).diff(y) #smíšená parciální derivace vzhledem k x a y.
︡a4ae05d4-eb7b-42fb-ae01-83a081da8c58︡{"html":"$\\displaystyle \\left( x, y \\right) \\ {\\mapsto} \\ 2 \\, x + 2$
"}︡{"done":true}
︠02f64a3c-a0ca-4e08-9389-e28390a68e85s︠
F=piecewise([[(-infinity,0),sin(x)],[[0,0],arctan(x)],[(0,infinity),arctan(x)]], var=x)
︡a561c2a6-ff14-405e-857b-2cc3616abd8c︡{"done":true}
︠30826e09-f257-4c03-8176-f0d385453526s︠
plot(F(x), (x,-3*pi,3*pi))
︡6258e888-a580-4e7a-9885-39a43b3bef82︡{"file":{"filename":"/home/user/.sage/temp/project-82c7e719-d1a0-46d5-a44d-4937cec4aba0/381/tmp_r7tkq98d.svg","show":true,"text":null,"uuid":"fa4c165a-39e8-4000-b3c2-e2446a904ada"},"once":false}︡{"done":true}
︠16672bc9-4d84-48fa-8cfd-23b66f82d301s︠
F=piecewise([[(-infinity,0),sin(x)],[[0,0],arctan(x)],[(0,infinity),arctan(x)]], var=x)
︡bf538240-f663-4a85-9f5c-3d6ec66120fa︡{"done":true}
︠8bc99731-d160-4b47-9693-a828d4bed5ees︠
F.derivative()
︡63813d5b-bd94-48d9-a3ef-ad4d9147b1ed︡{"stdout":"(0, 0, 0)*D[0]piecewise(x|-->sin(x) on (-oo, 0), x|-->arctan(x) on {0}, x|-->arctan(x) on (0, +oo); x) + D[1]piecewise(x|-->sin(x) on (-oo, 0), x|-->arctan(x) on {0}, x|-->arctan(x) on (0, +oo); x)\n"}︡{"done":true}
︠8d6d21e7-da6f-41f5-97c4-7fd0202b381bs︠
integral(x/(x^3+1), x, hold=True)
︡210b46e6-b543-4533-a5e7-41262e363672︡{"html":"$\\displaystyle \\int \\frac{x}{x^{3} + 1}\\,{d x}$
"}︡{"done":true}
︠348110d3-9530-4bd9-b79d-244b7d7440b4s︠
︡848f2186-8c47-4739-b7b5-11a762686551︡{"done":true}
︠ad7b0508-0186-4f07-b510-e4a9f24e5c7es︠
i=integral(x/(x^3+1), x);i
︡48bb39aa-cbc2-4d32-aa50-d10cd6613504︡{"html":"$\\displaystyle \\frac{1}{3} \\, \\sqrt{3} \\arctan\\left(\\frac{1}{3} \\, \\sqrt{3} {\\left(2 \\, x - 1\\right)}\\right) + \\frac{1}{6} \\, \\log\\left(x^{2} - x + 1\\right) - \\frac{1}{3} \\, \\log\\left(x + 1\\right)$
"}︡{"done":true}
︠18686422-fc21-4ead-9900-8197c54e9b0ds︠
diff(i,x).simplify_full()
︡e7c0058a-7669-4610-b2b1-bdd7d709bdee︡{"html":"$\\displaystyle \\frac{x}{x^{3} + 1}$
"}︡{"done":true}
︠387cd37c-ec28-4094-8b81-af8a4a9d7437s︠
show(integrate(x/(x^3+1), x, hold=True), '=', i)
︡9e71eebc-ea93-4a5f-acca-f77b966f90a4︡{"html":"$\\displaystyle \\int \\frac{x}{x^{3} + 1}\\,{d x}$ = $\\displaystyle \\frac{1}{3} \\, \\sqrt{3} \\arctan\\left(\\frac{1}{3} \\, \\sqrt{3} {\\left(2 \\, x - 1\\right)}\\right) + \\frac{1}{6} \\, \\log\\left(x^{2} - x + 1\\right) - \\frac{1}{3} \\, \\log\\left(x + 1\\right)$
"}︡{"done":true}
︠da8696a0-b67b-4a2b-ba72-3040ca63ac32s︠
i=integral(x/(x^5+1),x);i
︡930fe3f3-86a5-4624-8073-0a03c819b07e︡{"html":"$\\displaystyle -\\frac{2 \\, \\sqrt{5} \\arctan\\left(\\frac{4 \\, x + \\sqrt{5} - 1}{\\sqrt{2 \\, \\sqrt{5} + 10}}\\right)}{5 \\, \\sqrt{2 \\, \\sqrt{5} + 10}} + \\frac{2 \\, \\sqrt{5} \\arctan\\left(\\frac{4 \\, x - \\sqrt{5} - 1}{\\sqrt{-2 \\, \\sqrt{5} + 10}}\\right)}{5 \\, \\sqrt{-2 \\, \\sqrt{5} + 10}} - \\frac{\\log\\left(2 \\, x^{2} - x {\\left(\\sqrt{5} + 1\\right)} + 2\\right)}{5 \\, {\\left(\\sqrt{5} + 1\\right)}} + \\frac{\\log\\left(2 \\, x^{2} + x {\\left(\\sqrt{5} - 1\\right)} + 2\\right)}{5 \\, {\\left(\\sqrt{5} - 1\\right)}} - \\frac{1}{5} \\, \\log\\left(x + 1\\right)$
"}︡{"done":true}
︠7e97a149-7962-487b-9283-2dfad26d06bas︠
diff(i,x).simplify_full()
︡b1e2d7de-5d22-4622-8df4-4626dab10699︡{"html":"$\\displaystyle \\frac{x}{x^{5} + 1}$
"}︡{"done":true}
︠21abaa8a-5d3f-4417-b1b5-6ce25fb8aa22s︠
integral(2*x*(x^2+1)^24, x)
︡c3000da4-619f-49cb-9739-13862022470a︡{"html":"$\\displaystyle \\frac{1}{25} \\, {\\left(x^{2} + 1\\right)}^{25}$
"}︡{"done":true}
︠9a05f130-789a-4e70-87e1-c39710609624i︠
%html
Určitý integrál
︡3be57830-2642-43ed-9d6e-f6974de9f47f︡{"done":true,"html":"Určitý integrál"}
︠50067c0a-5ea6-4ff8-a8b7-f5e1896f3112s︠
i=integral(x/(x^3+1), (x,1,2));i
︡f23332ce-5af3-4246-9d6d-aa70d216e599︡{"html":"$\\displaystyle \\frac{1}{18} \\, \\sqrt{3} \\pi - \\frac{1}{6} \\, \\log\\left(3\\right) + \\frac{1}{3} \\, \\log\\left(2\\right)$
"}︡{"done":true}
︠bbe1787b-667c-47bc-acb2-3a7a9a9de81as︠
i.n()
︡6d6b7031-1900-4aec-b9f8-2de16f1dbe5e︡{"html":"$\\displaystyle 0.350246906114333$
"}︡{"done":true}
︠a48f9a2b-6dd4-4421-8ed4-2cbd54d8fb47s︠
numerical_integral(x/(x^3+1), 1,2)
︡7a49f6c4-8dcb-4fe2-93e7-83b51fc9e280︡{"html":"($\\displaystyle 0.3502469061143331$, $\\displaystyle 3.888521794718582 \\times 10^{-15}$)
"}︡{"done":true}
︠a99ef376-f2a2-48c8-a1f8-6e87cc210c53s︠
integral(1/x^2, (x,-1,1)
︡d62f4b4c-799f-4c2b-8dc2-0a6e743d59fd︡{"stderr":"Error in lines 1-1\nTraceback (most recent call last):\n File \"/cocalc/lib/python3.8/site-packages/smc_sagews/sage_server.py\", line 1231, in execute\n compile(block + '\\n',\n File \"\", line 1\n integral(Integer(1)/x**Integer(2), (x,-Integer(1),Integer(1))\n ^\nSyntaxError: unexpected EOF while parsing\n"}︡{"done":true}
︠14b950d9-de16-4bca-8089-e2360ddf568bs︠
plot(1/x^2, (x,-1,1), ymin=0, ymax=10)
︡22264e77-fe61-48c4-9a9d-105288d7416a︡{"file":{"filename":"/home/user/.sage/temp/project-82c7e719-d1a0-46d5-a44d-4937cec4aba0/381/tmp_w92dh97z.svg","show":true,"text":null,"uuid":"790fd843-4214-498c-b0e1-3b49c44a5265"},"once":false}︡{"done":true}
︠a41ba8b3-0279-4ba3-9d72-576830eaa8d3︠
%html
Nevlastní integrál
︡063b0833-07ce-4f3e-8acf-916f11c0eb1f︡{"done":true,"html":"Nevlastní integrál"}
︠c2a7c6dc-dbab-46f6-87e7-7593fe6efa63s︠
var('t')
︡4bbb9b5a-52ef-4811-aa37-1df8a65653a9︡{"html":"$\\displaystyle t$
"}︡{"done":true}
︠29190013-7ef0-4655-ad34-33a75cb21a73s︠
integral(t^4*log(t)^2/(1+3*t^2)^3, t,0,oo)
︡fc3c89ec-cec0-45e0-bc15-e4a700e05f9d︡{"html":"$\\displaystyle \\frac{1}{576} \\, \\sqrt{3} \\pi^{3} + \\frac{1}{576} \\, \\sqrt{3} \\pi \\log\\left(3\\right)^{2} - \\frac{1}{108} \\, \\sqrt{3} \\pi \\log\\left(3\\right) + \\frac{1}{216} \\, \\sqrt{3} \\pi$
"}︡{"done":true}
︠935d391a-cff3-4a9d-919f-e3d231c7a268s︠
integral(exp(arcsin(x)), x,0,1)
︡188a6ea6-27b2-4b90-894c-21b08067be7f︡{"html":"$\\displaystyle \\frac{1}{2} \\, e^{\\left(\\frac{1}{2} \\, \\pi\\right)} - \\frac{1}{2}$
"}︡{"done":true}
︠0ca62a22-f6f6-419b-b147-e1ee685ad8a2s︠
integral(sqrt(9-x^2), x)
︡54381f45-24e4-4aff-a232-26847827ee2b︡{"html":"$\\displaystyle \\frac{1}{2} \\, \\sqrt{-x^{2} + 9} x + \\frac{9}{2} \\, \\arcsin\\left(\\frac{1}{3} \\, x\\right)$
"}︡{"done":true}
︠fbccce25-2481-4e93-bb52-69a486a0cc3bi︠
%html
Sumace
︡b78a4152-911f-492b-a733-ae3d98153a6a︡{"done":true,"html":"Sumace"}
︠2e086eb5-b5a0-4048-b85b-c79c1281d7fds︠
var('k,n,a,y')
︡669647d4-4c1d-425c-b370-11601f039f55︡{"html":"($\\displaystyle k$, $\\displaystyle n$, $\\displaystyle a$, $\\displaystyle y$)
"}︡{"done":true}
︠4f8b2953-d939-4650-a145-3866d13662d7s︠
sum(k^7, k,1,20)
︡751419f7-3194-4867-abed-673e3fb7a840︡{"html":"$\\displaystyle 3877286700$
"}︡{"done":true}
︠7f05b22d-cbc3-4d5b-a25a-5b0240ade36as︠
sum(k^7, k,1,n)
︡313cbf1f-9f58-4be6-992d-01bad31f320c︡{"html":"$\\displaystyle \\frac{1}{8} \\, n^{8} + \\frac{1}{2} \\, n^{7} + \\frac{7}{12} \\, n^{6} - \\frac{7}{24} \\, n^{4} + \\frac{1}{12} \\, n^{2}$
"}︡{"done":true}
︠c07741f6-b255-4a64-ad10-9ed0e591e12cs︠
sum(1/(k^2-4), k,3,oo)
︡402c4ba9-e2ad-4699-a719-cba070c575a0︡{"html":"$\\displaystyle \\frac{25}{48}$
"}︡{"done":true}
︠113d425b-b0b0-43b9-8cfb-a5796d540824i︠
%html
Taylorova řada
︡c24131b2-e06d-4072-9b17-86a9b0f38f43︡{"done":true,"html":"Taylorova řada"}
︠12a077ec-590f-4491-afc6-7665f3a5aee0s︠
︡37103149-0fb1-4cc0-b7d8-fffa5fcb203a︡{"done":true}
︠4f63e6cf-45dd-44d3-9201-c780e969de74s︠
t=taylor(sin(tan(x))-tan(sin(x)), x,0,25);t
︡125a6636-de7e-4f48-af8c-bef727124ec1︡{"html":"$\\displaystyle \\frac{48074332710505411}{4616431560515174400000} \\, x^{25} - \\frac{374694625074883}{6690480522485760000} \\, x^{23} - \\frac{2097555460001}{7602818775552000} \\, x^{21} - \\frac{1664108363}{1905468364800} \\, x^{19} - \\frac{10193207}{4358914560} \\, x^{17} - \\frac{311148869}{54486432000} \\, x^{15} - \\frac{95}{7392} \\, x^{13} - \\frac{1913}{75600} \\, x^{11} - \\frac{29}{756} \\, x^{9} - \\frac{1}{30} \\, x^{7}$
"}︡{"done":true}
︠5df86433-2124-4e19-9c94-2a9ffc509575s︠
type(t)
︡83f3a9ba-8fb9-4fdb-8e20-b1f9c969b5f1︡{"stdout":"\n"}︡{"done":true}
︠8c77ddf5-8713-4a99-a68e-ae5b0c1cc738s︠
f=sin(tan(x))-tan(sin(x))
︡c6b9ec60-8916-48fc-a7cd-4b756d397f1a︡{"done":true}
︠76b62aef-4272-4420-8ec5-4e4fd3cf3ea9s︠
t1=f.series(x,25);t1
︡e3aa5d66-a8e0-4044-a73f-104dbf9185c4︡{"html":"$\\displaystyle {(-\\frac{1}{30})} x^{7} + {(-\\frac{29}{756})} x^{9} + {(-\\frac{1913}{75600})} x^{11} + {(-\\frac{95}{7392})} x^{13} + {(-\\frac{311148869}{54486432000})} x^{15} + {(-\\frac{10193207}{4358914560})} x^{17} + {(-\\frac{1664108363}{1905468364800})} x^{19} + {(-\\frac{2097555460001}{7602818775552000})} x^{21} + {(-\\frac{374694625074883}{6690480522485760000})} x^{23} + \\mathcal{O}\\left(x^{25}\\right)$
"}︡{"done":true}
︠f70622f1-44f5-44a2-9abd-fcd5b23ba4ces︠
type(t1)
︡eda98f6e-acd6-4359-8cfb-ae15c850c233︡{"stdout":"\n"}︡{"done":true}
︠6db5d6eb-6f9a-4f8a-80c3-8dd7fb78abb0s︠
︡938adbbd-c99f-44ef-9da7-e15af43cb0b3︡{"done":true}
︠bba5ab26-9de9-4487-b13d-c8f7a607812bs︠
t1.coefficients()
︡6f685191-198f-45f8-8660-006d627a7586︡{"html":"[[$\\displaystyle 0$, $\\displaystyle 0$], [$\\displaystyle 0$, $\\displaystyle 1$], [$\\displaystyle 0$, $\\displaystyle 2$], [$\\displaystyle 0$, $\\displaystyle 3$], [$\\displaystyle 0$, $\\displaystyle 4$], [$\\displaystyle 0$, $\\displaystyle 5$], [$\\displaystyle 0$, $\\displaystyle 6$], [$\\displaystyle -\\frac{1}{30}$, $\\displaystyle 7$], [$\\displaystyle 0$, $\\displaystyle 8$], [$\\displaystyle -\\frac{29}{756}$, $\\displaystyle 9$], [$\\displaystyle 0$, $\\displaystyle 10$], [$\\displaystyle -\\frac{1913}{75600}$, $\\displaystyle 11$], [$\\displaystyle 0$, $\\displaystyle 12$], [$\\displaystyle -\\frac{95}{7392}$, $\\displaystyle 13$], [$\\displaystyle 0$, $\\displaystyle 14$], [$\\displaystyle -\\frac{311148869}{54486432000}$, $\\displaystyle 15$], [$\\displaystyle 0$, $\\displaystyle 16$], [$\\displaystyle -\\frac{10193207}{4358914560}$, $\\displaystyle 17$], [$\\displaystyle 0$, $\\displaystyle 18$], [$\\displaystyle -\\frac{1664108363}{1905468364800}$, $\\displaystyle 19$], [$\\displaystyle 0$, $\\displaystyle 20$], [$\\displaystyle -\\frac{2097555460001}{7602818775552000}$, $\\displaystyle 21$], [$\\displaystyle 0$, $\\displaystyle 22$], [$\\displaystyle -\\frac{374694625074883}{6690480522485760000}$, $\\displaystyle 23$], [$\\displaystyle 0$, $\\displaystyle 24$]]
"}︡{"done":true}
︠96858184-9ca0-472c-9aee-8dd75743d771s︠
t2=t1.truncate();t2
︡d2520205-7069-4a1d-bb7e-c5183232ad39︡{"html":"$\\displaystyle -\\frac{374694625074883}{6690480522485760000} \\, x^{23} - \\frac{2097555460001}{7602818775552000} \\, x^{21} - \\frac{1664108363}{1905468364800} \\, x^{19} - \\frac{10193207}{4358914560} \\, x^{17} - \\frac{311148869}{54486432000} \\, x^{15} - \\frac{95}{7392} \\, x^{13} - \\frac{1913}{75600} \\, x^{11} - \\frac{29}{756} \\, x^{9} - \\frac{1}{30} \\, x^{7}$
"}︡{"done":true}
︠dfe68d50-a4a5-485d-8877-a6aab979edd1s︠
type(t2)
︡a1050b9c-2a3f-4196-89c6-0a439cbe9f90︡{"stdout":"\n"}︡{"done":true}
︠d99a7728-f940-40f7-9b6a-32ab36e4289es︠
taylor(x*y^3,(x,1),(y,-1),4)
︡3325f448-511f-4502-8408-f23ad97f1ded︡{"html":"$\\displaystyle {\\left(x - 1\\right)} {\\left(y + 1\\right)}^{3} - 3 \\, {\\left(x - 1\\right)} {\\left(y + 1\\right)}^{2} + {\\left(y + 1\\right)}^{3} + 3 \\, {\\left(x - 1\\right)} {\\left(y + 1\\right)} - 3 \\, {\\left(y + 1\\right)}^{2} - x + 3 \\, y + 3$
"}︡{"done":true}
︠adb31ec1-d801-4557-b981-441ce2da2b47︠
%html
Výpočty limit
︡9d4d7169-d179-45e9-9888-f9f9fa1a655b︡{"done":true,"html":"Výpočty limit"}
︠8538e7d1-3c9f-4a06-8d85-e4f9bd39c47fs︠
limit( cos(x)^(1/x^3), x=0)
︡dcd6e9f2-7a8a-4198-9071-20f6abb33746︡{"html":"$\\displaystyle \\mathit{und}$
"}︡{"done":true}
︠3bfc5574-44a2-42d9-a40c-a4d83966b8f7s︠
limit( cos(x)^(1/x^3), x=0, dir='plus')
︡f3990382-f6a2-4feb-9170-20b74efd39e0︡{"html":"$\\displaystyle 0$
"}︡{"done":true}
︠fde82e47-42ec-4dde-96f0-7a999fbc0da7s︠
limit( cos(x)^(1/x^3), x=0, dir='minus')
︡2b389a2b-e2b0-4d1e-b65f-8cee21a6c810︡{"html":"$\\displaystyle +\\infty$
"}︡{"done":true}
︠500b052e-0508-4658-9adc-f267f5ab8c86s︠
var('a,b')
︡4425b757-b43f-4f9b-8ade-83a4eb3d0cda︡{"html":"($\\displaystyle a$, $\\displaystyle b$)
"}︡{"done":true}
︠e3ade875-6c41-452f-b94c-4f7ae6a1847cs︠
y=exp(a*x)*cos(b*x);y
︡f05c42c6-76a8-47ef-b3fb-e94be520e886︡{"html":"$\\displaystyle \\cos\\left(b x\\right) e^{\\left(a x\\right)}$
"}︡{"done":true}
︠8fed8640-afca-4672-9597-9d792478c2b5s︠
assume(a>0, b>0)
︡fa3eafbf-d935-4be0-af61-0fe0c412f412︡{"done":true}
︠503e6900-9080-4b7a-a935-2ef5097f8073s︠
︡5921b618-1899-4f2b-a0ae-c23b894ebcc7︡{"done":true}
︠6aa2edbe-c300-4352-86f5-94a65afb46f4s︠
limit(y, x=-oo)
︡cd2c3ad2-351d-4523-8e2f-e1be5ce1d0ed︡{"html":"$\\displaystyle 0$
"}︡{"done":true}
︠fd036cd6-dce8-4cf2-95cd-6f7e31393c0f︠
%html
Diferenciální rovnice
︡814734d3-7b83-4060-9829-d473795e3437︡{"done":true,"html":"Diferenciální rovnice"}
︠cba6b480-9159-4d27-9c7a-6ed863f5c281s︠
y = function('y')(x)
︡ec54442d-3d11-4c52-9f4d-f734c84ed38c︡{"done":true}
︠e8f87acb-72d0-4f9e-83d9-6ddabe709721ss︠
ode=x*diff(y,x)==y;ode
︡a1258596-20b6-4c46-b2d7-8185f54fa68b︡{"html":"$\\displaystyle x \\frac{\\partial}{\\partial x}y\\left(x\\right) = y\\left(x\\right)$
"}︡{"done":true}
︠01c6630d-10d1-4501-94fe-1ec4802e3577s︠
sol=desolve(ode,y);expand(sol)
︡e04b69a7-7828-4374-b784-16cb354f0808︡{"html":"$\\displaystyle C x$
"}︡{"done":true}
︠4e30e257-21f5-4b60-8a5e-506e24a87755s︠
desolve(ode, y, ics=[1,1])
︡fe824752-85c3-43a2-93ea-ab9e247d9461︡{"html":"$\\displaystyle x$
"}︡{"done":true}
︠29b81c07-c63b-48d1-8db7-76d88a2f990fs︠
︡4f67ca7a-7143-4af6-a6d6-da1978940162︡{"done":true}
︠e093cb8b-383e-4719-8a37-d6e3b866dca9s︠
︡4df31106-1aed-4758-ac0c-22a168605274︡{"done":true}