Задание 4
Вариант 3
3.
>>syms a x
>>limit(sqrt(x.^2+1)-sqrt(x.2-1),x,inf)
Ans=0
Вариант 5
. (1+3tg2x)ctg2x
>> syms a x
>> limit((1+3*tan(x)^2)*cot(x)^2,x,0)
ans =
Inf
Вариант6
6.
Решение:
>> syms a x
>>limit((exp(x)-exp(-x)/sin(x),x,0)
Ans =2
Вариант 7
>>syms x
>>limit((2*x-asin(x))/(2*x+atan(x)),x,0)
Ans=1/3
Вариант 11
11.
>> syms x
>> limit((2*tan(x)-x^4+x^2)/atan(3*x),x,0)
ans =
2/3
Задание 5
Вариант 3
3. f(x,y) = e-xlny
>>syms x y
>>f=exp(-x*log(y));
>>x1=diff(f,x)
X1=-log(y)*exp(-x*log(y))
y1=-x/y*exp(-x*log(y))
>>subs([x1 y1],[x y],[1 2])
>>xy=diff(x1,y)
>>[m]=simple(xy)
>>[m]=simple(yx)
Вариант 4
f(x,y) = cos(e-xlny)
>> syms x y;
>> f = cos(exp((-x)*log(y)))
f =
cos(exp(-x*log(y)))
>> diff(f,x)
ans =
sin(exp(-x*log(y)))*log(y)*exp(-x*log(y))
>> diff(f,y)
ans =
sin(exp(-x*log(y)))*x/y*exp(-x*log(y))
>> diff(f,x,2)
ans =
-cos(exp(-x*log(y)))*log(y)^2*exp(-x*log(y))^2-sin(exp(-x*log(y)))*log(y)^2*exp(-x*log(y))
>> diff(f,y,2)
ans =
-cos(exp(-x*log(y)))*x^2/y^2*exp(-x*log(y))^2-sin(exp(-x*log(y)))*x/y^2*exp(-x*log(y))-sin(exp(-x*log(y)))*x^2/y^2*exp(-x*log(y))
Вариант 5
. f(x,y) = ln(lnxy)
>> syms x y
>> f=log(log(x*y));
>> x1=diff(f,x)
x1 =
1/(x*log(x*y))
>> pretty(x1)
1
----------
x log(x y)
>> y1=diff(f,y)
y1 =
1/(y*log(x*y))
>> pretty(y1)
1
----------
y log(x y)
>> subs([x1 y1], [x y], [1 2])
ans =
1.4427 0.7213
>> xy=diff(x1,y)
xy =
-1/(x*y*log(x*y)^2)
>> [m]=simple(xy)
m =
-1/(x*y*log(x*y)^2)
>> pretty(m)
1
- -------------
2
x y log(x y)
>> pretty(y1)
1
----------
y log(x y)
>> pretty(xy)
1
- -------------
2
x y log(x y)
>> yx=diff(y1,x)
yx =
-1/(x*y*log(x*y)^2)
>> [m]=simple(yx)
m =
-1/(x*y*log(x*y)^2)
>> pretty(m)
1
- -------------
2
x y log(x y)
>> pretty(yx)
1
- -------------
2
x y log(x y)
Вариант 8
f(x,y) = arccos(sin(x-y))
>> f = acos(sin(x-y));
>> diff(f,x)
ans =
-cos(x-y)/(1-sin(x-y)^2)^(1/2)
>> diff(f,y)
ans =
cos(x-y)/(1-sin(x-y)^2)^(1/2)
>> diff(f,x,2)
ans =
sin(x-y)/(1-sin(x-y)^2)^(1/2)-cos(x-y)^2/(1-sin(x-y)^2)^(3/2)*sin(x-y)
>> diff(f,y,2)
ans =
sin(x-y)/(1-sin(x-y)^2)^(1/2)-cos(x-y)^2/(1-sin(x-y)^2)^(3/2)*sin(x-y)
Вариант 11
11. f(x,y) = arcctg(ex-ey)
>> syms x y
>> f=acot(exp(x)-exp(y))
f =
acot(exp(x)-exp(y))
>> x1=diff(f,x)
x1 =
-exp(x)/(1+(exp(x)-exp(y))^2)
>> pretty(x1)
exp(x)
- ----------------------
2
1 + (exp(x) - exp(y))
>> y1=diff(f,y)
y1 =
exp(y)/(1+(exp(x)-exp(y))^2)
>> pretty(y1)
exp(y)
----------------------
2
1 + (exp(x) - exp(y))
>> subs([x1 y1],[x y],[1 2])
ans =
-0.1191 0.3239
>> xy=diff(x1,y)
xy =
-2*exp(x)/(1+(exp(x)-exp(y))^2)^2*(exp(x)-exp(y))*exp(y)
>> [m]=simple(xy)
m =
-2/(1+(exp(x)-exp(y))^2)^2*(exp(x)-exp(y))*exp(x+y)
>> pretty(m)
(exp(x) - exp(y)) exp(x + y)
-2 ----------------------------
2 2
(1 + (exp(x) - exp(y)) )
>> yx=diff(y1,x)
yx =
-2*exp(x)/(1+(exp(x)-exp(y))^2)^2*(exp(x)-exp(y))*exp(y)
> [m]=simple(yx)
m =
-2/(1+(exp(x)-exp(y))^2)^2*(exp(x)-exp(y))*exp(x+y)
>> pretty(m)
(exp(x) - exp(y)) exp(x + y)
-2 ----------------------------
2 2
(1 + (exp(x) - exp(y)))