помогите решить

[kexhtsavor]

помогите пожалуйста

[Monopole]

Sin1/x в аргументе арктангенса или просто умножается на него ?

[Monopole]

Странно, вроде бы решение в случае, когда arctg(x)*sin(1/x) тривиальное, но Matematica 5.0 не считает его.....
Короче, когда x ->0, аргумент синуса стремитсья к бесконечности, но синус ограниченная функция, а арктангенс в нуле дает ноль. А произведение ограниченной функции на ноль тоже ноль, поэтому в предле то, что у тебя под корнем дает ноль и ты получаешь вответе Ln2.

[Monopole]

Во втормо случае, когда у тебя arctg(x*sin(1/x)) овтет тот же. Но теперь у тебя x*sin(1/x)) в пределе дает произведеие ограниченной функции на ноль, что есть ноль, а арктангенс нуля опять ноль.

[Monopole]

С Matematica 5.0, проблема решена, я ошибся в написании. Так что мое решение верно. :)

[kexhtsavor]

большое спосибо очень памог :)

[Monopole]

Да не за что....обращайся, если что

[Meme]

Ребята может кто нибудь знает как решить энто?

1. (i) Suppose f : R^n -> R and g : R^n ->R are two functions such that both are homogenous of degree r: Let h : R^n ->R be a function such that for each x E R^n; h(x) = f(x) + g(x): Show whether the function h is also homogenous of degree r:

[Monopole]

Ребята может кто нибудь знает как решить энто?

1. (i) Suppose f : R^n -> R and g : R^n ->R are two functions such that both are homogenous of degree r: Let h : R^n ->R be a function such that for each x E R^n; h(x) = f(x) + g(x): Show whether the function h is also homogenous of degree r:

Lets take arbitrary point x from R^n. Then h(x)=f(x)+g(x). Using homogenousy of functions f and g one has f(a*x)=a^r*f(x) and g(a*x)=a^r*g(x), therefore h(a*x)=f(a*x)+g(a*x)=a^r*f(x)+a^r*g(x)=a^r*(f(x)+g (x))=a^r*h(x).

[Meme]

Lets take arbitrary point x from R^n. Then h(x)=f(x)+g(x). Using homogenousy of functions f and g one has f(a*x)=a^r*f(x) and g(a*x)=a^r*g(x), therefore h(a*x)=f(a*x)+g(a*x)=a^r*f(x)+a^r*g(x)=a^r*(f(x)+g (x))=a^r*h(x).

What concerned me in this problem is that h(x) is given as a function of a sum of two other functions of one variable. According to homogenity rule the sum of the exponents of the polynomal should be homogenous of some degree.
I 'coined' a function for f(x) and a function of g(x) then differentiated and messed up everything.

So say we had a product of two functions h(x)=f(x)*g(x), would we again choose an arbitrary point p from the real line and follow the same logic multiplying the two functions?

[Monopole]

What concerned me in this problem is that h(x) is given as a function of a sum of two other functions of one variable. According to homogenity rule the sum of the exponents of the polynomal should be homogenous of some degree.
I 'coined' a function for f(x) and a function of g(x) then differentiated and messed up everything.

So say we had a product of two functions h(x)=f(x)*g(x), would we again choose an arbitrary point p from the real line and follow the same logic multiplying the two functions?

Well, we have not a "function of a sum" as you said, we have a function which is given as a sum of two other functions. But, these functionas are both homogeneous functions of the same dergee r. This fact provide the h(x) function to be hmogeneous again with the same degree r.
In case of produc h(x)=f(x)*g(x) we get homogeneous function again, but the dergee will be different.
Suppose f is homogeneous function with degree r, which means f(a*x)=a^r*f(x), and g is homogeneious function with degree q, i.e. g(a*x)=a^q*g(x), then
h(a*x)=f(a*x)g(a*x)=a^rf(x)a^qg(x)=a^(r+q)f(x)g(x) =a^(r+q)h(x).
Therefore, h is homogeneous function with degree r+q.

[Meme]

Well, we have not a "function of a sum" as you said, we have a function which is given as a sum of two other functions. But, these functionas are both homogeneous functions of the same dergee r. This fact provide the h(x) function to be hmogeneous again with the same degree r.
In case of produc h(x)=f(x)*g(x) we get homogeneous function again, but the dergee will be different.
Suppose f is homogeneous function with degree r, which means f(a*x)=a^r*f(x), and g is homogeneious function with degree q, i.e. g(a*x)=a^q*g(x), then
h(a*x)=f(a*x)g(a*x)=a^rf(x)a^qg(x)=a^(r+q)f(x)g(x) =a^(r+q)h(x).
Therefore, h is homogeneous function with degree r+q.

Got it sunshine, ta!:)