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Operation ResearchQuestion and Answers: Page 4

Question Number 1987 Answers: 0 Comments: 1

lets a<b and f:[a,b]→R integable into [a,b] and continuous lets I a closed subset of [a,b] proof that (or give a conter example) ∫_I fdx=0 ∀I⊂[a,b]⇒f=0

$lets a < b and f : [a, b] \to R integable into [a, b] and continuous$ $lets I a closed subset of [a, b]$ $proof that (or give a conter example)$ $\int_{I} f d x = 0 \forall I \subset [a, b] \Rightarrow f = 0$

Question Number 1858 Answers: 1 Comments: 0

[((x(ρ,θ,ξ))),((y(ρ,θ,ξ))),((z(ρ,θ,ξ))) ]= [((ρe^ξ cos θ)),((ρe^ξ sin θ)),(ξ) ] { ((ρ∈[0,+∞))),((θ∈[0,2π))),((ξ∈R)) :} r(ρ,θ,ξ)=x(ρ,θ,ξ)i+y(ρ,θ,ξ)j+z(ρ,θ,ξ)k a_ρ =(∂r/∂ρ) a_θ =(∂r/∂θ) a_ξ =(∂r/∂ξ) a_ρ ∙a_ρ +a_θ ∙a_θ +a_ξ ∙a_ξ =??? a_ρ ∙a_θ +a_ρ ∙a_ξ +a_θ ∙a_ξ =??? a_ρ ×a_θ +a_ρ ×a_ξ +a_θ ×a_ξ =???

$[\begin{matrix} x (ρ, θ, ξ) \\ y (ρ, θ, ξ) \\ z (ρ, θ, ξ) \end{matrix}] = [\begin{matrix} ρ e^{ξ} \cos θ \\ ρ e^{ξ} \sin θ \\ ξ \end{matrix}] {\begin{cases} ρ \in [0, + \infty) \\ θ \in [0, 2 π) \\ ξ \in R \end{cases}$ $r (ρ, θ, ξ) = x (ρ, θ, ξ) i + y (ρ, θ, ξ) j + z (ρ, θ, ξ) k$ $a_{ρ} = \frac{\partial r}{\partial ρ}$ $a_{θ} = \frac{\partial r}{\partial θ}$ $a_{ξ} = \frac{\partial r}{\partial ξ}$ $a_{ρ} \cdot a_{ρ} + a_{θ} \cdot a_{θ} + a_{ξ} \cdot a_{ξ} = ? ? ?$ $a_{ρ} \cdot a_{θ} + a_{ρ} \cdot a_{ξ} + a_{θ} \cdot a_{ξ} = ? ? ?$ $a_{ρ} \times a_{θ} + a_{ρ} \times a_{ξ} + a_{θ} \times a_{ξ} = ? ? ?$

Question Number 1448 Answers: 1 Comments: 2

x=(x_1 ,x_2 ),y=(y_1 ,y_2 ) η:[0,1)^4 →[0,1] η(x,y):=med[(((1−x_1 )^y_1 +(1−y_1 )^x_1 )/2),(((1−x_2 )^y_2 +(1−y_2 )^x_2 )/2)] med(x,y):=((min(x,y)+max(x,y))/2) η(x,y)=^? η(y,x) η(x,y)=0⇔^? x=y η(x,z)≤^? η(x,y)+η(y,z)

$x = (x_{1}, x_{2}), y = (y_{1}, y_{2})$ $η : {[0, 1)}^{4} \to [0, 1]$ $η (x, y) := med [\frac{{(1 - x_{1})}^{y_{1}} + {(1 - y_{1})}^{x_{1}}}{2}, \frac{{(1 - x_{2})}^{y_{2}} + {(1 - y_{2})}^{x_{2}}}{2}]$ $med (x, y) := \frac{\min (x, y) + \max (x, y)}{2}$ $η (x, y) \overset{?}{=} η (y, x)$ $η (x, y) = 0 \overset{?}{\Leftrightarrow} x = y$ $η (x, z) \overset{?}{⩽} η (x, y) + η (y, z)$

Question Number 1351 Answers: 0 Comments: 2

W{f(x)}(t)=∫_0 ^(1/t) f(x)ln(xt)dx,t>0 W{f(x)+g(x)}(t)=^? W{f(x)}(t)+W{g(x)}(t) W{cf(x)}(t)=^? cW{f(x)}(t) W{1}(t)=? W{x}(t)=? W{x^n }(t)=?,n∈N W{f′(x)}(t)=?

$W {f (x)} (t) = \underset{0}{\int^{1 / t}} f (x) \ln (x t) d x, t > 0$ $W {f (x) + g (x)} (t) \overset{?}{=} W {f (x)} (t) + W {g (x)} (t)$ $W {c f (x)} (t) \overset{?}{=} c W {f (x)} (t)$ $W {1} (t) = ?$ $W {x} (t) = ?$ $W {x^{n}} (t) = ?, n \in N$ $W {f^{'} (x)} (t) = ?$

Question Number 1055 Answers: 0 Comments: 0

f:[0,1]→R g:[0,1]×N→R g_n (x)=f[g_(n−1) (x)] g_0 (x)=x A{f}(n)=∫_0 ^1 f(t)g_n (t)dt A{f+g}=^? A{f}+A{g} A{kf}=^? kA{f}

$f : [0, 1] \to R$ $g : [0, 1] \times N \to R$ $g_{n} (x) = f [g_{n - 1} (x)]$ $g_{0} (x) = x$ $A {f} (n) = \underset{0}{\int^{1}} f (t) g_{n} (t) d t$ $A {f + g} \overset{?}{=} A {f} + A {g}$ $A {k f} \overset{?}{=} k A {f}$

Question Number 1034 Answers: 1 Comments: 0

x=i+f,i∈N,f∈[0,1) x^2 =(i+f)^2 =i^2 +2if+f^2 x∈R_+ i+2f=2⇒x^2 =2i+f^2 2if=0⇒x^2 =i^2 +f^2

$x = i + f, i \in N, f \in [0, 1)$ $x^{2} = {(i + f)}^{2} = i^{2} + 2 i f + f^{2}$ $x \in R_{+}$ $i + 2 f = 2 \Rightarrow x^{2} = 2 i + f^{2}$ $2 i f = 0 \Rightarrow x^{2} = i^{2} + f^{2}$

Question Number 746 Answers: 1 Comments: 1

lets ⊞:(R^+ )^2 →R^+ defined by x⊞y=(√(⌊x⌋⌈x⌉))+y 1. x⊞y=^? y⊞x 2.x⊞(y⊞z)=^? (x⊞y)⊞z 3.∃e,∀x∈R^+ ,x⊞e=x ? 4.∃e,∀x∈R^+ ,e⊞x=x ?

$l e t s ⊞ : {(R^{+})}^{2} \to R^{+}$ $d e f i n e d b y x ⊞ y = \sqrt{⌊ x ⌋ ⌈ x ⌉} + y$ $1 . x ⊞ y \overset{?}{=} y ⊞ x$ $2 . x ⊞ (y ⊞ z) \overset{?}{=} (x ⊞ y) ⊞ z$ $3 . \exists e, \forall x \in R^{+}, x ⊞ e = x ?$ $4 . \exists e, \forall x \in R^{+}, e ⊞ x = x ?$

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