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Problemas 3, 3* y 4
sepM
(examen de septiembre de matemáticas)
Problema 1a
var('n')
an=n*exp(cos(n))*sin(1/n^2);limit(an,n=infinity)
0
Problema 1b
g(x)=log(abs(x+1/2))+x;dg=diff(g(x),x);dg,g(0),g(1)
(1/2*(2*x + 1)/abs(x + 1/2)^2 + 1, log(1/2), log(3/2) + 1)
plot(g,-2,1,ymin=-3,ymax=1.5,thickness=2,figsize=[1.8,2])
Problema 2 y 8
f(x)=(x+1)/sqrt(2^2x^2);df=factor(diff(f(x),x));sf=factor(diff(df,x))
df,sf,solve(sf,x),find_root(sf,-1,0)
(-(x + 4)/((x - 2)*(x + 2)*sqrt(-x^2 + 4)), 2*(x^2 + 6*x + 2)/((x 2)^2*(x + 2)^2*sqrt(-x^2 + 4)), [x == -sqrt(7) - 3, x == sqrt(7) 3], -0.35424868893540945)
plot(f,-1.8,1.8,gridlines=true,figsize=[2,2])
g(x)=(2*x-1)*exp(-x);dg=factor(diff(g(x),x));sg=factor(diff(dg,x))
G=factor(integrate(g(x),x))
dg,sg,G
(-(2*x - 3)*e^(-x), (2*x - 5)*e^(-x), -(2*x + 1)*e^(-x))
g(1),g(3/2),g(2),g(5/2); N(g(1)),N(g(3/2)),N(g(2)),N(g(5/2))
(e^(-1), 2*e^(-3/2), 3*e^(-2), 4*e^(-5/2))
(0.367879441171442, 0.446260320296860, 0.406005849709838,
0.328339994495595)
-integrate(g,x,0,1/2),integrate(g,x,1,infinity)
(2*e^(-1/2) - 1, 3*e^(-1))
d1=plot(g,x,0,0.5,ymin=-1.5,ymax=0.5,thickness=2,fill=true)
d2=plot(g,x,1,5,ymin=-1.5,ymax=0.5,thickness=2,fill=true)
d3=plot(g,x,-0.2,5,ymin=-1.5,ymax=0.5,thickness=2)
plot(d1+d2+d3,figsize=[5,2])
Problema 5
limit((sin(x)*cos(x)-arctan(x))/log(1+x^3),x=0)
-1/3
Problema 6a
var('n');bn=2^n*n^(100)/factorial(n-1)
N(sum(bn,n,1,10)),N(sum(bn,n,1,56)),N(sum(bn,n,1,57))
(2.82220678648798e97, 2.24380306779007e127, 2.24380306857409e127)
var('n');cn=(7-sin(n))/sqrt(n^3+n)
N(sum(cn,n,1,10)),N(sum(cn,n,1,100))
(10.7428126401201, 13.6800758130844)
(con 1000 ya tarda mucho)
Problema 6b
integrate(f(x),x),integrate(f,x,0,1)
(-sqrt(-x^2 + 4) + arcsin(1/2*x), 1/6*pi - sqrt(3) + 2)
.
sin(3*pi/2),integral(arctan(x)/(1+x^4),x,-1,1)
(-1, 0)