4.
IV. : .
1. .
2. .
3. .
4. .
1.
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);
1.
1. 
. :
> Limit((1-x)*tan(Pi*x/2),x=1)=
limit((1-x)*tan(Pi*x/2),x=1);
2. 
. :
> Limit(arctan(1/(1-x)),x=1,left)=
limit(arctan(1/(1-x)), x=1, left);
> Limit(arctan(1/(1-x)),x=1,right)=
limit(arctan(1/(1-x)),x=1, right);
.
Maple :
1) diff(f,x), f , , x , .
2) Diff(f,x), , . 
. , . simplify factor expand, , .
:
> Diff(sin(x^2),x)=diff(sin(x^2),x);
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x$n, n ; :
> Diff(cos(2*x)^2,x$4)=diff(cos(2*x)^2,x$4);
:
> simplify(%);
> combine(%);
.
D(f) f -. :
> D(sin);
cos
:
> D(sin)(Pi):eval(%);
-1
> f:=x-> ln(x^2)+exp(3*x):
> D(f);
2.
1. 
> Diff(sin(2*x)^3-cos(2*x)^3,x)=
diff(sin(2*x)^3-cos(2*x)^3,x);
2. 
. :
> Diff(exp(x)*(x^2-1),x$24)=
diff(exp(x)*(x^2-1),x$24):
> collect(%,exp(x));
3. 
x =p/2, x =p.
> y:=sin(x)^2/(2+sin(x)): d2:=diff(y,x$2):
> x:=Pi; d2y(x)=d2;
x:=p d2y(p)=1
> x:=Pi/2;d2y(x)=d2;
:= 
3.
Maple , , .
.
f (x 1,, xm) diff. : diff(f,x1$n1,x2$n2,, xm$nm), x1,, xm , , $ . , 
: diff(f,x,y).
3.
1. 
.
> f:=arctan(x/y):
>D iff(f,x)=simplify(diff(f,x));
> Diff(f,y)=simplify(diff(f,y));
.
2. 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);
.
4.
, , , . (. 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 . .
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2) singular(f,x), f , x . , .
readlib(name), name .
. set. , , set convert .
4.1.
1. 
> readlib(iscont): readlib(discont):
> iscont(exp(1/(x+3)),x=-infinity..+infinity);
false
, . :
> discont(exp(1/(x+3)),x);
{-3}
:
x =-3.
2. 
> readlib(singular):
> 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 , . {}, . set. :
> 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);
, , .
Maple 6 maximize minimize . , location. () (). :
> minimize(x^4-x^2, x, location);
, { 
, 
}
.
4.2.
1. max min 
.
> y:=(x^2-1/2)*arcsin(x)/2+x*sqrt(1-x^2)/4-
Pi*x^2/12:
> extrema(y,{},x,'s');s;
. x . , (0,0) (1/2, p/24+ 
). , , . maximize minimize.
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> ymin:= minimize(y,x=0..1/2);
> ymax:= maximize(y,x=0..1/2);
:
: 
, 
.
. Maple, Enter. , 
sqrt(3).
.
:
, : miny(x)=y(1/2)=;
: -Pi/24+sqrt(3)/16
Enter;
.
2. 
. :
> f:=x^2*ln(x):
maximize(f,x=1..2);
> minimize(f,x=1..2);
:
: 
, min f(x)=0
3. 
. :
> restart:y:=x^3/(4-x^2): readlib(extrema):
readlib(maximize): readlib(minimize):
> extrema(y,{},x,'s');s;
{ 
}
{{ x =0},{ 
},{ 
}}
. :
> d2:=diff(y,x$2): x:=0: d2y(x):=d2;
d2y(0):=0
> x:=2*sqrt(3):d2y(x):=d2;
> x:=-2*sqrt(3):d2y(x):=d2;
, x =0 ; 
, max; 
, min. :
(
), (
).
.
extrema(f,{cond},{x,y,},'s'), cond , . , f, s , . , .
, extrema , , . , , subs.
, , maximize(f,{x1,,xn},range), minimize(f,{x1,,xn}, range), , , , .
, ( ) , , -. simplex, maximize ( minimize), range . simplex . maximize minimize . , . NONNEGATIVE
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- ,
- ,
- .
, (, , ).
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