129.sagews

6.1. $\left\{ 1+\frac{1}{2^k} \right\}$.

k=var('k'); limit(1+1/2^k,k=10)

1025/1024

RR(_)

1.00097656250000

6.2. $f(x)=\left( 1+\frac{1}{x} \right)^x$, x \to \infty$,$x \to 5$, x \to 1,2$.

f=(1+1/x)^x

f.limit(x=oo)

e

f.limit(x=5)

7776/3125

RR(_)

2.48832000000000

f.limit(x=1.2)

2.069615754672029

6.3. $\;y=\frac{\sqrt{7+x}-1}{x^2-4}\;$ $\;x=2\;$.

limit((sqrt(7+x)-1)/(x^2-4),x=2,dir='plus')

+Infinity

limit((sqrt(7+x)-1)/(x^2-4),x=2,dir='minus')

-Infinity

6.4. $\sum_{k=1}^{50}\frac{1}{k(k+1)}$.

sum([1/(k*(k+1)) for k in [1..50]])

50/51

6.5. $\prod_{k=3}^{10}\frac{k^3-4}{k^2-1}$.

prod([(k^3-4)/(k^2-1) for k in [3..10]])

463185169513/254016

RR(prod([(k^3-4)/(k^2-1) for k in [3..10]]))

1.82344879658368e6

6.6. $y=2e^{x}\ln^{2}{(x+1)}$.

diff(2*exp(x)*(ln(x+1))^2)

2*e^x*log(x + 1)^2 + 4*e^x*log(x + 1)/(x + 1)

6.7. $y=\cos^2{(x^2)}$.

diff(cos(x^2)**2)

-4*x*cos(x^2)*sin(x^2)

()

diff(diff(cos(x^2)**2))

8*x^2*sin(x^2)^2 - 4*cos(x^2)*sin(x^2) - 8*x^2*cos(x^2)^2

()

diff(cos(x^2)**2,2)

8*x^2*sin(x^2)^2 - 4*cos(x^2)*sin(x^2) - 8*x^2*cos(x^2)^2

6.8. $f(x)=-\frac{1}{3}x^3+\frac{3}{2}x^2-2x+1$ $x=3$.

df(x)=diff(-1/3*x^3+3/2*x^2-2*x+1); df(x)

-x^2 + 3*x - 2

df(3)

-2

6.9. $f(x,y)=\frac{\sin{x}}{\cos{2y}}$.

diff(sin(x)/cos(2*y),x)

cos(x)/cos(2*y)

diff(sin(x)/cos(2*y),y)

2*sin(x)*sin(2*y)/cos(2*y)^2

6.10. $y=\frac{\arccos^2{(7x)}}{\sqrt{1-49x^2}}$.

integral((acos(7*x)^2)/sqrt(1-49*x^2))

-arccos(7*x)^3/21

6.11. \int_0^{\pi/2} \sin{x} \cos^2{x} \,dx$.

integral(sin(x)*cos(x)^2,x,0,pi/2)

1/3

6.12. \int_{-1}^0 \int_{-y/2}^{-y} {2xy}-{x^2} \,dxdy$.

integral(integral(2*x*y-x^2,x,-y/2,-y),y,-1,0)

-25/96

6.13. x=1$\LaTeX$y=e^{−x^2}$

taylor(exp(-x^2),x,1,5)

e^-1 - 2*e^-1*(x - 1) + e^-1*(x - 1)^2 + 2*(x - 1)^3/(3*e) - 5*(x - 1)^4/(6*e) + (x - 1)^5/(15*e)

6.14. $y=\sin{(\sin{x})}$.

taylor(sin(sin(x)),x,0,7)

x - x^3/3 + x^5/10 - 8*x^7/315