<Text-field style="Heading 1" layout="Heading 1">1. T<Font encoding="UTF-8">\303\266rt\303\251</Font>net</Text-field>

restart;

<Text-field style="Heading 2" layout="Heading 2">E 1.1. P<Font encoding="UTF-8">\303\251lda.</Font> </Text-field>

33!/2^31+41^41;

Il5vOzBELTRfNSpHQC06RyQqKilmRFlMJzRDKlI4KCkqPnJLMWp4M0wi

43!/(2^43-1);

IyJWKysrKz9ea3NdKCpSXm8/OGJ0ak5RUGpJRTovJyIuMkEtJDQneik=

483952545774574373476/122354323571234 mod 1000003;

IicjeSgpKQ==

10*(8+6*I)^(-1/2);

LCQqJiIjNSIiIiksJiIiKUYlKiYiIidGJV4jRiVGJUYlI0YlIiIjISIiRiU=

evalc(%);

LCYiIiQiIiJeI0YkISIi

sqrt(15523/3-98/2);

LCQqJiMiJEMiIiIkIiIiKUYmI0YnIiIjRidGJw==

a:=sin(Pi/3)*exp(2+ln(33));

LCQqKCMiIiIiIiNGJSkiIiRGJEYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJkYmRiUtSSNsbkdGKzYjIiNMRiVGJUYl

simplify(a);

LCQqKCMiI0wiIiMiIiIpIiIkI0YnRiZGJy1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjRiZGJ0Yn

evalf(a);

JCIrJ1IxPDYjISIo

evalf(a,60);

JCJnbmY9dHJfYGUlUUt3PmAkSCYqXHAsZE08PWEmW2lSMTw2IyEjZA==

n:=19380287199092196525608598055990942841820;

Iko/PSVHJTQqZjApZjNjXyc+IzQqPihHIVE+

isprime(n);

SSZmYWxzZUclKnByb3RlY3RlZEc=

ifactor(n);

Ki4pLUkhRzYiNiMiIiNGKCIiIiktRiU2IyIiJEYoRiktRiU2IyIiJkYpKS1GJTYjIiM+Ri1GKSktRiU2IyIkLCIiIiVGKSktRiU2IyIvPk9fWD9HN0YoRik=

nextprime(n);

IkpWPyVHJTQqZjApZjNjXyc+IzQqPihHIVE+

igcd(15990335972848346968323925788771404985,15163659044370489780);

IjQ6M2EocGAjUU9FIg==

a:=(x+y)^12-(x-y)^12;

LCYqJCksJkkieEc2IiIiIkkieUdGJ0YoIiM3RihGKCokKSwmRiZGKEYpISIiRipGKEYu

expand(a);

LC4qKCIjQyIiIkkieUc2IkYlKUkieEdGJyIjNkYlRiUqKCIkUyVGJSlGJiIiJEYlKUYpIiIqRiVGJSooIiUlZSJGJSlGJiIiJkYlKUYpIiIoRiVGJSooRjJGJSlGJkY2RiUpRilGNEYlRiUqKEYsRiUpRiZGMEYlKUYpRi5GJUYlKihGJEYlKUYmRipGJUYpRiVGJQ==

quo(x^3*y-x^3*z+2*x^2*y^2-2*x^2*z^2+x*y^3+x*y^2*z-x*z^3,x+y+z,x);

LCgqJiwmSSJ5RzYiIiIiSSJ6R0YmISIiRicpSSJ4R0YmIiIjRidGJyomLCYqJClGJUYsRidGJyokKUYoRixGJ0YpRidGK0YnRicqJkYlRidGMkYnRic=

gcd(x^3*y-x^3*z+2*x^2*y^2-2*x^2*z^2+x*y^3+x*y^2*z-x*z^3,x+y+z);

IiIi

b:=(x^4-y^4)/(x^3+y^3)-(x^5+y^5)/(x^4-y^4);

LCYqJiwmKiQpSSJ4RzYiIiIlIiIiRioqJClJInlHRihGKUYqISIiRiosJiokKUYnIiIkRipGKiokKUYtRjJGKkYqRi5GKiomLCYqJClGJyIiJkYqRioqJClGLUY5RipGKkYqRiRGLkYu

normal(b);

LCQqKilJInhHNiIiIiQiIiIpSSJ5R0YmRidGKCwqKiRGJEYoRigqJilGJSIiI0YoRipGKCEiIiomRiVGKClGKkYvRihGKCokRilGKEYwRjAsKCokRi5GKEYoKiZGJUYoRipGKEYwKiRGMkYoRihGMEYw

f:=(x+y)*(x-y)^6; g:=(x^2-y^2)*(x-y)^3; f/g;

KiYsJkkieEc2IiIiIkkieUdGJUYmRiYpLCZGJEYmRichIiIiIidGJg==

KiYsJiokKUkieEc2IiIiIyIiIkYpKiQpSSJ5R0YnRihGKSEiIkYpKSwmRiZGKUYsRi0iIiRGKQ==

KigsJkkieEc2IiIiIkkieUdGJUYmRiYpLCZGJEYmRichIiIiIiRGJiwmKiQpRiQiIiNGJkYmKiQpRidGL0YmRipGKg==

normal(f/g);

KiQpLCZJInhHNiIiIiJJInlHRiYhIiIiIiNGJw==

factor(x^6-x^5+x^2+1);

LCoqJClJInhHNiIiIiciIiJGKCokKUYlIiImRighIiIqJClGJSIiI0YoRihGKEYo

factor(5*x^4-4*x^3-48*x^2+44*x+3);

KigsJkkieEc2IiIiIkYmISIiRiYsJkYkRiYiIiRGJ0YmLCgqJiIiJkYmKUYkIiIjRiZGJiomIiM7RiZGJEYmRiZGJkYmRiY=

Factor(x^6-x^5+x^2+1) mod 13;

KiYsKiokKUkieEc2IiIiJCIiIkYpKiYiIzVGKSlGJiIiI0YpRikqJiIiKUYpRiZGKUYpIiM2RilGKSwqRiRGKSomRi1GKUYsRilGKSomRjBGKUYmRilGKSIiJ0YpRik=

factor(x^12-y^12);

Ki4sJkkieEc2IiIiIkkieUdGJSEiIkYmLCgqJClGJyIiI0YmRiYqJClGJEYsRiZGJiomRiRGJkYnRiZGJkYmLCZGJEYmRidGJkYmLChGLUYmRi9GKEYqRiZGJiwmRi1GJkYqRiZGJiwoKiQpRiQiIiVGJkYmKiZGLkYmRitGJkYoKiQpRidGNkYmRiZGJg==

restart;

alias(a=RootOf(x^4-2));

SSJhRzYi

factor(x^12-2*x^8+4*x^4-8,a);

KiwsKCokKUkieEc2IiIiJSIiIkYpKiYiIiNGKSlGJkYrRikhIiJGK0YpRiksKEYkRilGKkYpRitGKUYpLCYqJEYsRilGKSokKUkiYUdGJ0YrRilGKUYpLCZGJkYpRjNGKUYpLCZGJkYpRjNGLUYp

Factor(x^6-2*x^4+4*x^2-8,a) mod 5;

Ki4sJkkieEc2IiIiIiIiJEYmRiYsJkYkRiYiIiNGJkYmLCZGJEYmRiZGJkYmLCZGJEYmIiIlRiZGJiwmRiRGJiokKS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYqJClJI19aR0YyRixGJkYmRidGJkYpRiZGJkYmLCZGJEYmKiZGLEYmRi9GJkYmRiY=

V:=vandermonde([x,y,z]);

LUksdmFuZGVybW9uZGVHNiI2IzclSSJ4R0YkSSJ5R0YkSSJ6R0Yk

with(linalg);

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

V:=vandermonde([x,y,z]);

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRiciIiNJInpHRiM2JEYqRipJInlHRiM2JEYmRiZGJjYkRipGJyokKUYtRipGJjYkRidGJyokKUYrRipGJjYkRipGJkYmNiRGJkYnKiQpSSJ4R0YjRipGJjYkRiZGKkY5NiRGJ0YmRiY=

inverse(V);

PTYiNiQ7IiIiIiIkRiVFXFtsKjYkRiciIiMqJCwqKiZJInhHRiNGJkkiekdGI0YmRiYqJkYuRiZJInlHRiNGJiEiIiokKUYxRipGJkYmKiZGL0YmRjFGJkYyRjI2JEYqRiosJComLCZGLkYmRi9GJkYmRixGMkYyNiRGJkYmKihGL0YmRjFGJiwqRjVGJkYtRjIqJClGLkYqRiZGJkYwRjJGMjYkRipGJyomLCZGLkYmRjFGJkYmLCpGLUYmRjBGMiokKUYvRipGJkYyRjVGJkYyNiRGJ0YnLCQqJEZCRjJGMjYkRipGJiwkKiYsJkYvRiZGMUYmRiZGPEYyRjI2JEYmRicsJCooRi5GJkYxRiZGQkYyRjI2JEYmRioqKEYuRiZGL0YmRixGMjYkRidGJiokRjxGMg==

det(V);

LC4qJkkieUc2IiIiIilJInpHRiUiIiNGJkYmKiYpRiRGKUYmRihGJiEiIiomSSJ4R0YlRiZGJ0YmRiwqJilGLkYpRiZGKEYmRiYqJkYuRiZGK0YmRiYqJkYwRiZGJEYmRiw=

factor(%);

KigsJkkiekc2IiIiIkkieUdGJSEiIkYmLCZGJ0YoSSJ4R0YlRiZGJiwmRiRGKEYqRiZGJg==

e1:=(1-eps)*x+2*y-4*z-1=0;

LywqKiYsJiIiIkYmSSRlcHNHNiIhIiJGJkkieEdGKEYmRiYqJiIiI0YmSSJ5R0YoRiZGJiomIiIlRiZJInpHRihGJkYpRiZGKSIiIQ==

e2:=(3/2-eps)*x+3*y-5*z-2=0;

LywqKiYsJiMiIiQiIiMiIiJJJGVwc0c2IiEiIkYpSSJ4R0YrRilGKSomRidGKUkieUdGK0YpRikqJiIiJkYpSSJ6R0YrRilGLEYoRiwiIiE=

e3:=(5/2+eps)*x+5*y-7*z-3=0;

LywqKiYsJiMiIiYiIiMiIiJJJGVwc0c2IkYpRilJInhHRitGKUYpKiZGJ0YpSSJ5R0YrRilGKSomIiIoRilJInpHRitGKSEiIiIiJEYyIiIh

sols:=solve([e1,e2,e3],[x,y,z]);

NyM3JS9JInhHNiIsJComIyIiIiIiI0YqSSRlcHNHRiYhIiJGLS9JInlHRiYsJCooI0YqIiIlRiosJkYqRioqJiIiKEYqRixGKkYqRipGLEYtRiovSSJ6R0YmIyIiJEYz

subs(eps=10^(-20),sols);

NyM3JS9JInhHNiIhNSsrKysrKysrK10vSSJ5R0YmIyI2MisrKysrKysrKyIiIiUvSSJ6R0YmIyIiJEYs

f:=x^2*y*(1-x-y)^3;

KigpSSJ4RzYiIiIjIiIiSSJ5R0YlRicpLChGJ0YnRiQhIiJGKEYrIiIkRic=

e1:=diff(f,x); e2:=diff(f,y);

LCYqKiIiIyIiIkkieEc2IkYlSSJ5R0YnRiUpLChGJUYlRiYhIiJGKEYrIiIkRiVGJSoqRixGJSlGJkYkRiVGKEYlKUYqRiRGJUYr

LCYqJilJInhHNiIiIiMiIiIpLChGKEYoRiUhIiJJInlHRiZGKyIiJEYoRigqKkYtRihGJEYoRixGKClGKkYnRihGKw==

solve([e1,e2],[x,y]);

NyY3JC9JInhHNiIiIiEvSSJ5R0YmRik3JC9GJSMiIiIiIiQvRikjRi0iIic3JC9GJSwmRi1GLUYpISIiRihGMg==

limit(tan(x)/x,x=0);

IiIi

diff(ln(sec(x)),x);

LUkkdGFuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInhHRic=

series(tan(sinh(x))-sinh(tan(x)),x=0,15);

Ky1JInhHNiIjIiIiIiMhKiIiKCMiIzgiJGMoIiIqIyIlXjkiJitjKCIjNiMiJVZnIidTRUxGKi1JIk9HJSpwcm90ZWN0ZWRHNiNGJiIjOg==

series(BesselJ(0,x)/BesselJ(1,x),x,12);

KzFJInhHNiIiIiMhIiIjRiYiIiUiIiIjRiYiIycqIiIkI0YmIiVPOiIiJiNGJiImU0kjIiIoIyEjOCIoIW9CVyIiKi1JIk9HJSpwcm90ZWN0ZWRHNiNGKSIjNQ==

int(((3*x^2-7*x+15)*exp(x)+3*x^2-14)/(x-exp(x))^2,x);

KiYsKCIjOSIiIiomIiIkRiUpSSJ4RzYiIiIjRiVGJS1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKjYjRikhIiJGJSwmRilGJUYsRjJGMg==

int((3*x^3-x+14)/(x^2+4*x-4),x);

LCoqJiMiIiQiIiMiIiIpSSJ4RzYiRiZGJ0YnKiYiIzdGJ0YpRichIiIqJiMiI2ZGJkYnLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRio2IywoKiRGKEYnRicqJiIiJUYnRilGJ0YnRjpGLUYnRicqKCIjUUYnKUYmI0YnRiZGJy1JKGFyY3RhbmhHRjM2IywkKigjRiciIilGJywmKiZGJkYnRilGJ0YnRjpGJ0YnRj1GJ0YnRidGJw==

int(x*exp(x^3),x);

LCQqKCMiIiIiIiRGJSkhIiJGJEYlLCYqKilJInhHNiIiIiNGJSlGKCNGLkYmRiUtSSZHQU1NQUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGLTYjRjBGJSksJCokKUYsRiZGJUYoRjBGKEYlKipGK0YlRi9GJUY3RigtRjI2JEYwRjhGJUYoRiVGKA==

diff_eqn:=diff(y(x),x$2)+t*diff(y(x),x)-2*t^2*y(x)=0;

LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ4R0YsRi5GLiIiIiomSSJ0R0YsRi9GKEYvRi8qKCIiI0YvKUYxRjNGL0YqRi8hIiIiIiE=

init_conds:=y(0)=t,D(y)(0)=2*t^2;

NiQvLUkieUc2IjYjIiIhSSJ0R0YmLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnLCQqJiIiIyIiIilGKUY0RjVGNQ==

dsolve({diff_eqn,init_conds},y(x));

Ly1JInlHNiI2I0kieEdGJSwmKigjIiIlIiIkIiIiSSJ0R0YlRi0tSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyomRi5GLUYnRi1GLUYtKigjRi1GLEYtRi5GLS1GMDYjLCQqKCIiI0YtRi5GLUYnRi0hIiJGLUY9

Cheby:=proc(n,x) local T,k; T[0]:=1; T[1]:=x; for k from 2 to n do T[k]:=expand(2*x*T[k-1]-T[k-2]); od; T[n]; end;

Zio2JEkibkc2IkkieEdGJTYkSSJUR0YlSSJrR0YlRiVGJUMmPiZGKDYjIiIhIiIiPiZGKDYjRi9GJj8oRikiIiNGL0YkSSV0cnVlRyUqcHJvdGVjdGVkRz4mRig2I0YpLUknZXhwYW5kR0Y2NiMsJiooRjRGL0YmRi8mRig2IywmRilGL0YvISIiRi9GLyZGKDYjLCZGKUYvRjRGQkZCJkYoNiNGJEYlRiVGJQ==

Cheby(7,x);

LCoqJiIjayIiIilJInhHNiIiIihGJUYlKiYiJDciRiUpRiciIiZGJSEiIiomIiNjRiUpRiciIiRGJUYlKiZGKUYlRidGJUYu

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 2" layout="Heading 2">Ex1.3. Feladat.</Text-field>

int(x/(1+exp(x)),x); int(exp(x^2),x); int(sqrt((x^2-1)*(x^2-4)),x); int(sqrt((x-1)*(x-4)),x); int(sqrt((1+x)/(1-x)),x); int(log(x^2-5*x+4),x); int(log(x)/(1+x),x); int(1/log(x),x);

LCgqJiMiIiIiIiNGJSlJInhHNiJGJkYlRiUqJkYoRiUtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGKTYjLCZGJUYlLUkkZXhwR0YtNiNGKEYlRiUhIiItSShwb2x5bG9nR0YtNiRGJiwkRjJGNUY1

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