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2 c sum() product(), limit, , , , Maple.
Maple sum(f,k=m..n), f , k, m n . , , .. sum() .
,
>sum(2^n*(n+1)/n!, n=0..20)
>evalf(%)
22.16716830
> sum(n/(n+1)!, n=0..infinity)
cx.
>sum(n*(n+2)*x^n, n=1..infinity)
product(), , sum(). product(f, k=m..n) : f , k k- , m n k.
,
>product(1-1/n^2, n=2..infinity)
f (x)
series(f(x), x=a, n), , , n .
taylor(f(x), x=a, n) f(x) x=a n -1 .
series taylor , series. , , convert(%,polynom).
f (x 1,, xn) (x 1,, xn) (a 1,, an) n mtaylor(f(x), [x1,,xn], n). , readlib(mtaylor).
. 0=0, 5 .
> f(x)=series(exp(-x)*sqrt(x+1), x=0, 5);
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(0, 0) 6- .
> readlib(mtaylor):
> f=mtaylor(sin(x^2+y^2), [x=0,y=0], 7);
Maple : , . : , . (, , ..) . . .
:
1) limit(expr,x=a,par), expr , , a , , par (left , right ) (real , complex ).
2) Limit(expr,x=a,par), , . :
> Limit(sin(2*x)/x,x=0);
> limit(sin(2*x)/x,x=0);
, :
> Limit(x*(Pi/2+arctan(x)),x=-infinity)= limit(x*(Pi/2+arctan(x)), x=-infinity);
: left righ . :
> Limit(1/(1+exp(1/x)),x=0,left)= limit(1/(1+exp(1/x)),x=0,left);
> Limit(1/(1+exp(1/x)),x=0,right)=limit(1/(1+exp(1/x)), x=0,right);
Maple :
1) diff(f,x), f , , x , .
2) Diff(f,x), , . . , . simplifyfactor expand, , .
:
> Diff(sin(x^2),x)=diff(sin(x^2),x);
x$n, n ; :
> Diff(cos(2*x)^2,x$4)=diff(cos(2*x)^2,x$4);
:
> simplify(%);
> combine(%);
Maple , , .
f (x 1,, xm) diff. : diff(f,x1$n1,x2$n2,, xm$nm), x1,, xm , , $ . , : diff(f,x,y).
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2- .
> restart; f:=(x-y)/(x+y):
> Diff(f,x$2)=simplify(diff(f,x$2));
> Diff(f,y$2)=simplify(diff(f,y$2));
> Diff(f,x,y)=diff(f,x,y);
.
, , , . (. II). , , , , .
f (x) [ x 1, x 2] iscont(f,x=x1..x2). f , true (); f , false (). , x=-infinity..+infinity, f . , true, , . . :
1) discont(f,x), f , , x . .
2) singular(f,x), f , x . , .
. set. , , set convert .
1.
> iscont(exp(1/(x+3)),x=-infinity..+infinity);
false
, . :
> discont(exp(1/(x+3)),x);
{-3}
:
x =-3.
2.
> iscont(tan(x/(2-x)),x=-infinity..infinity);
false
> singular(tan(x/(2-x)),x);
{ x =2},{ x =2 }
_ N . :
: x =2 x =2p(2 n +1)/(p(2 n +1)-2).
Maple extrema(f,{cond},x,s), f - , . {cond} , , , s , . {}, . :
> extrema(arctan(x)-ln(1+x^2)/2,{},x,x0);x0;
{{ x =1}}
, .
, , , . f (x) maximize(f,x,x=x1..x2), f (x) minimize(f, x, x=x1..x2). infinity x=-infinity..+infinity, maximize minimize , , , . , . :
> maximize(exp(-x^2),x);
, location. () (). :
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>minimize(x^4-x^2, x, location);
, { , }
.
, , maximize(f,{x1,,xn},range), minimize(f,{x1,,xn}, range), , , , .
2- :
1) int(f, x), f , x ;
2) Int(f, x) , int. Int .
int Int , ,
> Int((1+cos(x))^2, x=0..Pi)=int((1+cos(x))^2, x=0..Pi);
continuous: int(f, x, continuous), Maple . . , int , , x=0..+infinity.
, , - . , , , >0 <0. , :
>Int(exp(-a*x),x=0..+infinity)=int(exp(-a*x),x=0..+infinity);
- , . assume(expr1), expr1 . additionally(expr2), expr2 , .
, :
>assume(a>0);
>Int(exp(-a*x),x=0..+infinity)=int(exp(-a*x),x=0..+infinity);