CALCULO DIFERENCIAL E INTEGRAL

Calcular los siguientes límites y esbozar la grafica de algunas de las funciones en algún intervalo apropiado que contenga el punto donde se esta calculando el límite:

a)

>> syms x;>> f=(exp(x)-1)/log10 (1+x);>> limit(f) ans = log(2)+log(5)

>> ezplot(f,[-5,5]);>> ylabel('f');>> grid

b)

>> syms x;>> f=(x+sin(pi*x))/(x-sin(pi*x));>> limit(f) ans = (-1-pi)/(-1+pi) >> syms x;>> f=(x+sin(3.14159*x))/(x-sin(3.14159*x));>> limit(f) ans =

-4663015795180343/2411215981495095>> -4663015795180343/2411215981495095

ans = -1.9339>> (-1-pi)/(-1+pi)

ans = -1.9339

>> ezplot(f,[-5,5]);>> ylabel('f');>> grid

>> ezpolar('(x+sin(pi*x))/(x-sin(pi*x))')

c)

>> syms x;>> f=((exp(x)+x))^(1/x);>> limit(f) ans = exp(2) >> ezplot(f,[-5,5]);>> ylabel('f');>> grid

d)

>> syms x;>> f=(sin (3*x))/(1-2*cos (x));>> limit(f,x,(pi/3)) ans = -3^(1/2) >> limit(f,x,(3.14159/3)) ans =-sin(14148463553350875/4503599627370496)/(1+2*cos(4716154517783625/4503599627370496))

>> ezpolar('(sin (3*x))/(1-2*cos (x))')

e)

>> syms x;>> f=((x^4)+1)^(1/(log (x)));>> limit(f,x,Inf) ans = exp(4) >> ezplot(f,[-10,10]);>> ylabel('f');>> grid

f)

>> syms a x;>> f=((x*exp(a*x))-x)/(1-cos (a*x));>> limit(f) ans = 2/a >> ezplot(f,[-5,5]);>> ylabel('f');>> grid

>> ezpolar('((x*exp(2*x))-x)/(1-cos (2*x))')

UNIVERSIDAD NACIONAL DE SAN AGUSTINUNIVERSIDAD NACIONAL DE SAN AGUSTIN

FACULTAD DE PRODUCCION Y SERVICIOSFACULTAD DE PRODUCCION Y SERVICIOS

ESCUELA PROFESIONAL DE INGENIERIA DE SISTEMASESCUELA PROFESIONAL DE INGENIERIA DE SISTEMAS

METODOS NUMERICOSMETODOS NUMERICOS

CALCULO DIFERENCIAL E INTEGRALCALCULO DIFERENCIAL E INTEGRAL

PRESENTADO POR:PRESENTADO POR:

APAZA APAZA, KAREN GIANNELLAAPAZA APAZA, KAREN GIANNELLA

AREQUIPA – PERUAREQUIPA – PERU

20122012

Ejercicio 2: Calcular las derivadas primeras y segundas de las siguientes funciones en el punto x=2

>> syms x;>> f=(((x^4)/2)-(x/3))*((7-5*(x^6))^2);>> diff(f) ans = (2*x^3-1/3)*(7-5*x^6)^2-60*(1/2*x^4-1/3*x)*(7-5*x^6)*x^5 Hemos hallado la derivada y ahora la calculamos en x=2 :

>> x=2;>> y= (2*x^3-1/3)*(7-5*x^6)^2-60*(1/2*x^4-1/3*x)*(7-5*x^6)*x^5

y = 5.9419e+006

Si volvemos a derivar la ecuación que nos da nos resulta lo siguiente, y de ahí lo calculamos en x=2:

>> syms x;>> f=(2*x^3-1/3)*(7-5*x^6)^2-60*(1/2*x^4-1/3*x)*(7-5*x^6)*x^5;>> diff(f) ans = 6*x^2*(7-5*x^6)^2-60*(2*x^3-1/3)*(7-5*x^6)*x^5-(120*x^3-20)*(7-5*x^6)*x^5+30*(30*x^4-20*x)*x^10-5*(30*x^4-20*x)*(7-5*x^6)*x^4 >> x=2;>> y=6*x^2*(7-5*x^6)^2-60*(2*x^3-1/3)*(7-5*x^6)*x^5-(120*x^3-20)*(7-5*x^6)*x^5+30*(30*x^4-20*x)*x^10-5*(30*x^4-20*x)*(7-5*x^6)*x^4

y = 45715736

Si aplicamos de frente derivada de oren superior 2.

Nos sale el mismo resultado:>> syms x;>> f=(((x^4)/2)-(x/3))*((7-5*(x^6))^2);>> diff(f,2) ans = 6*x^2*(7-5*x^6)^2-120*(2*x^3-1/3)*(7-5*x^6)*x^5+1800*(1/2*x^4-1/3*x)*x^10-300*(1/2*x^4-1/3*x)*(7-5*x^6)*x^4

>> x=2;>> y=6*x^2*(7-5*x^6)^2-120*(2*x^3-1/3)*(7-5*x^6)*x^5+1800*(1/2*x^4-1/3*x)*x^10-300*(1/2*x^4-1/3*x)*(7-5*x^6)*x^4

y = 45715736

>> syms x;>> f=((7*x-5)^2)*((4*(x^3)+3*x)/(x-1)^2);>> diff(f) ans = 14*(7*x-5)*(4*x^3+3*x)/(x-1)^2+(7*x-5)^2*(12*x^2+3)/(x-1)^2-2*(7*x-5)^2*(4*x^3+3*x)/(x-1)^3 Calculamos la dereivad segunda : >> diff(f,2)

ans =98*(4*x^3+3*x)/(x-1)^2+28*(7*x-5)*(12*x^2+3)/(x-1)^2-56*(7*x-5)*(4*x^3+3*x)/(x-1)^3+24*(7*x-5)^2*x/(x-1)^2-4*(7*x-5)^2*(12*x^2+3)/(x-1)^3+6*(7*x-5)^2*(4*x^3+3*x)/(x-1)^4

>> x=2;>> y=14*(7*x-5)*(4*x^3+3*x)/(x-1)^2+(7*x-5)^2*(12*x^2+3)/(x-1)^2-2*(7*x-5)^2*(4*x^3+3*x)/(x-1)^3

y = 2763

>> z=98*(4*x^3+3*x)/(x-1)^2+28*(7*x-5)*(12*x^2+3)/(x-1)^2-56*(7*x-5)*(4*x^3+3*x)/(x-1)^3+24*(7*x-5)^2*x/(x-1)^2-4*(7*x-5)^2*(12*x^2+3)/(x-1)^3+6*(7*x-5)^2*(4*x^3+3*x)/(x-1)^4

z = 3256

>> syms a b c d e f x;>> f=(d*(x^2)+e*x+f)/((x^2)/a)+(x/b)+(1/c);>> diff(f) ans = (2*d*x+e)/x^2*a-2*(d*x^2+e*x+f)/x^3*a+1/b >> diff(f,2) ans = 2*d/x^2*a-4*(2*d*x+e)/x^3*a+6*(d*x^2+e*x+f)/x^4*a >> x=2;>> y=(2*d*x+e)/x^2*a-2*(d*x^2+e*x+f)/x^3*a+1/b y = (d+1/4*e)*a-(d+1/2*e+1/4*(d*x^2+e*x+f)/x^2*a+1/4*x/b+1/4/c)*a+1/b

>> z=2*d/x^2*a-4*(2*d*x+e)/x^3*a+6*(d*x^2+e*x+f)/x^4*a z = 1/2*d*a-(2*d+1/2*e)*a+(3/2*d+3/4*e+3/8*(d*x^2+e*x+f)/x^2*a+3/8*x/b+3/8/c)*a

>> syms a b c x;>> f=(x-a)*(x-b)*(x-c);>> diff(f) ans =(x-b)*(x-c)+(x-a)*(x-c)+(x-a)*(x-b) >> diff(f,2)ans =6*x-2*c-2*b-2*a >> x=2;>> y=(x-b)*(x-c)+(x-a)*(x-c)+(x-a)*(x-b) y =(2-b)*(2-c)+(2-a)*(2-c)+(2-a)*(2-b) >> z=6*x-2*c-2*b-2*a z =12-2*c-2*b-2*a

>> syms a x;>> f=((x+a)^(1/3))*((x-a)^(-1/3));>> diff(f) ans =1/3/(x+a)^(2/3)/(x-a)^(1/3)-1/3*(x+a)^(1/3)/(x-a)^(4/3) >> diff(f,2)

ans = -2/9/(x+a)^(5/3)/(x-a)^(1/3)-2/9/(x+a)^(2/3)/(x-a)^(4/3)+4/9*(x+a)^(1/3)/(x-a)^(7/3) >> x=2;>> y=1/3/(x+a)^(2/3)/(x-a)^(1/3)-1/3*(x+a)^(1/3)/(x-a)^(4/3) y =

1/3/(2+a)^(2/3)/(2-a)^(1/3)-1/3*(2+a)^(1/3)/(2-a)^(4/3) >> z=-2/9/(x+a)^(5/3)/(x-a)^(1/3)-2/9/(x+a)^(2/3)/(x-a)^(4/3)+4/9*(x+a)^(1/3)/(x-a)^(7/3) z = -2/9/(2+a)^(5/3)/(2-a)^(1/3)-2/9/(2+a)^(2/3)/(2-a)^(4/3)+4/9*(2+a)^(1/3)/(2-a)^(7/3)

>> syms x;>> f=(1+(((x^2)-7)^(1/2)))^(1/2);>> diff(f) ans = 1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(1/2)*x >> diff(f,2) ans = -1/4/(1+(x^2-7)^(1/2))^(3/2)/(x^2-7)*x^2-1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(3/2)*x^2+1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(1/2) >> x=2;>> y=1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(1/2)*x

y = -0.2041 - 0.3536i

>> z=-1/4/(1+(x^2-7)^(1/2))^(3/2)/(x^2-7)*x^2-1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(3/2)*x^2+1/2/(1+(x^2-7)^(1/2))^(1/2)/(x^2-7)^(1/2)

z = -0.2381 - 0.5303i

Ejercicio 3 Calcular las siguientes integrales indefinidas:

>> syms x;>> f=(log10 (x))/(sqrt(x));>> int(f) ans = 2/log(10)*log(x)*x^(1/2)-4/log(10)*x^(1/2)

>> syms x;>> f=atan(sqrt(x));>> int(f) ans = x*atan(x^(1/2))-x^(1/2)+atan(x^(1/2))

>> syms a b x;>> f=((a)/((x^2)+(b^2)));>> int(f) ans =a/b*atan(x/b)

>> syms x;>> f=((x+1)/((x^2)+4*x+5));>> int(f) ans =1/2*log(x^2+4*x+5)-atan(x+2)

>> syms x;>> f=((x^2)/((x^2)-x+1));>> int(f) ans =x+1/2*log(x^2-x+1)-1/3*3^(1/2)*atan(1/3*(2*x-1)*3^(1/2))

>> syms x;>> f=((x^2)/((x^2)+x+1));>> int(f) ans = x-1/2*log(x^2+x+1)-1/3*3^(1/2)*atan(1/3*(2*x+1)*3^(1/2))

Ejercicio 4 Calcular las integrales que se calcularon en el ejercicio anterior, pero haciendolas todas ellas definidas en el intervalo [10;12], ofreciendo el resultado en forma algebraica y en su forma numérica, con 6 decimales.

>> syms x;>> f=(log10 (x))/(sqrt(x));>> int(f,10,12) ans =2*(-10^(1/2)*log(2)-10^(1/2)*log(5)+2*10^(1/2)+4*3^(1/2)*log(2)+2*3^(1/2)*log(3)-4*3^(1/2))/(log(2)+log(5)) >> y=2*(-10^(1/2)*log(2)-10^(1/2)*log(5)+2*10^(1/2)+4*3^(1/2)*log(2)+2*3^(1/2)*log(3)-4*3^(1/2))/(log(2)+log(5))

y = 0.627910

>> syms x;>> f=atan(sqrt(x));>> int(f,10,12) ans = -11*atan(10^(1/2))+10^(1/2)+13*atan(2*3^(1/2))-2*3^(1/2) >> y=-11*atan(10^(1/2))+10^(1/2)+13*atan(2*3^(1/2))-2*3^(1/2)

y = 2.555366

>> syms a b x;>> f=((a)/((x^2)+(b^2)));>> int(f,10,12) ans =a*(-atan(10/b)+atan(12/b))/b >> y=a*(-atan(10/b)+atan(12/b))/b y =

a*(-atan(10/b)+atan(12/b))/b

>> syms x;>> f=((x+1)/((x^2)+4*x+5));>> int(f,10,12) ans = -1/2*log(5)-1/2*log(29)+atan(12)+1/2*log(197)-atan(14) >> y=-1/2*log(5)-1/2*log(29)+atan(12)+1/2*log(197)-atan(14)

y = 0.141401

>> syms x;>> f=((x^2)/((x^2)-x+1));>> int(f,10,12) ans =1/3*3^(1/2)*atan(19/3*3^(1/2))+2-1/2*log(13)-1/3*3^(1/2)*atan(23/3*3^(1/2))+1/2*log(19) >> y=1/3*3^(1/2)*atan(19/3*3^(1/2))+2-1/2*log(13)-1/3*3^(1/2)*atan(23/3*3^(1/2))+1/2*log(19)

y = 2.180655

>> syms x;>> f=((x^2)/((x^2)+x+1));>> int(f,10,12) ans =1/3*3^(1/2)*atan(7*3^(1/2))+2+1/2*log(3)+1/2*log(37)-1/3*3^(1/2)*atan(25/3*3^(1/2))-1/2*log(157) >> y=1/3*3^(1/2)*atan(7*3^(1/2))+2+1/2*log(3)+1/2*log(37)-1/3*3^(1/2)*atan(25/3*3^(1/2))-1/2*log(157)

y = 1.819067