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Let A ∈ R^(N×N) be a symmetric positive definite matrix and b ∈ R^N a vector. If x ∈ R^N , evaluate the integral Z(A,b) = ∫e^(−(1/2)x^T Ax + b^T x) dx as a function of A and b.

$${Let}\:{A}\:\in\:{R}^{{N}×{N}} \:{be}\:{a}\:{symmetric}\:{positive} \\ $$$${definite}\:{matrix}\:{and}\:{b}\:\in\:{R}^{{N}} \:{a}\:{vector}. \\ $$$${If}\:{x}\:\in\:{R}^{{N}} ,\:{evaluate}\:{the}\:{integral} \\ $$$${Z}\left({A},{b}\right)\:=\:\int{e}^{−\frac{\mathrm{1}}{\mathrm{2}}{x}^{{T}} {Ax}\:+\:{b}^{{T}} {x}} {dx}\:{as}\:{a}\:{function} \\ $$$${of}\:{A}\:{and}\:{b}. \\ $$

Question Number 203898 Answers: 0 Comments: 2

Let f(W) be a function of vector W ∈ R^N , i.e. f(W) = (1/(1 + e^(−W^T x) )) Determine the first derivative and matrix of second derivatives of f with respect to W

$${Let}\:{f}\left({W}\right)\:{be}\:{a}\:{function}\:{of}\:{vector}\:{W}\:\in\: {R}^{{N}} , \\ $$$${i}.{e}.\:{f}\left({W}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}\:+\:{e}^{−{W}^{{T}} {x}} } \\ $$$${Determine}\:{the}\:{first}\:{derivative}\:{and} \\ $$$${matrix}\:{of}\:{second}\:{derivatives}\:{of}\:{f}\:{with} \\ $$$${respect}\:{to}\:{W} \\ $$$$ \\ $$

Question Number 203897 Answers: 3 Comments: 0

Question Number 203891 Answers: 1 Comments: 0

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Question Number 203884 Answers: 1 Comments: 0

find ∫_0 ^∞ (x^3 /((1+x)^4 (x+2)^5 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{3}} }{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} }{dx} \\ $$

Question Number 203881 Answers: 2 Comments: 1

Question Number 203879 Answers: 1 Comments: 0

Complete and balance the following chemical reaction equation (a)Ca + H_2 O → (b)BaCl + H_2 SO_4 → (c)CuSO_4 .5H_2 O →^(Heat strongly) (d)ZnCO_3 +HCl → 31/1/2024

$$\mathrm{Complete}\:\mathrm{and}\:\mathrm{balance}\:\mathrm{the}\:\mathrm{following}\:\mathrm{chemical}\:\mathrm{reaction}\: \\ $$$$\mathrm{equation} \\ $$$$\left(\mathrm{a}\right)\mathrm{Ca}\:+\:\mathrm{H}_{\mathrm{2}} \mathrm{O}\:\:\rightarrow\:\: \\ $$$$\left(\mathrm{b}\right)\mathrm{BaC}{l}\:+\:\mathrm{H}_{\mathrm{2}} \mathrm{SO}_{\mathrm{4}} \:\:\rightarrow \\ $$$$\left(\mathrm{c}\right)\mathrm{CuSO}_{\mathrm{4}} .\mathrm{5H}_{\mathrm{2}} \mathrm{O}\:\:\:\overset{\mathrm{Heat}\:\mathrm{strongly}} {\rightarrow} \\ $$$$\left(\mathrm{d}\right)\mathrm{ZnCO}_{\mathrm{3}} \:+\mathrm{HC}{l}\:\:\rightarrow \\ $$$$\mathrm{31}/\mathrm{1}/\mathrm{2024} \\ $$

Question Number 203878 Answers: 1 Comments: 0

A compound M is composed of 52.2% carbon ,13% hydrogenand the rest is oxygen.if the molecuar mass of M is 138 (a)The empirical formular (b)The molecular formular 31/1/2024

$$\mathrm{A}\:\mathrm{compound}\:\mathrm{M}\:\mathrm{is}\:\mathrm{composed}\:\mathrm{of}\:\mathrm{52}.\mathrm{2\%}\: \\ $$$$\mathrm{carbon}\:,\mathrm{13\%}\:\mathrm{hydrogenand}\:\mathrm{the}\:\mathrm{rest}\: \\ $$$$\mathrm{is}\:\mathrm{oxygen}.\mathrm{if}\:\mathrm{the}\:\mathrm{molecuar}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{M}\:\mathrm{is}\:\mathrm{138} \\ $$$$\left(\mathrm{a}\right)\mathrm{The}\:\mathrm{empirical}\:\mathrm{formular} \\ $$$$\left(\mathrm{b}\right)\mathrm{The}\:\mathrm{molecular}\:\mathrm{formular} \\ $$$$\mathrm{31}/\mathrm{1}/\mathrm{2024} \\ $$

Question Number 203877 Answers: 0 Comments: 0

Write a balanced Ionic chemical equation for the following reaction (a)HCl+CaCO_3 →CaCl_2 +CO_(2(g)) +H_2 O_((l)) (b)NH_4 OH_( (aq)) +HCl_((aq)) →NH_4 Cl_((aq)) +H_2 O_((l)) 31/1/2024

$$\mathrm{Write}\:\mathrm{a}\:\mathrm{balanced}\:\mathrm{Ionic}\:\mathrm{chemical}\:\mathrm{equation} \\ $$$$\:\mathrm{for}\:\mathrm{the}\:\mathrm{following}\:\mathrm{reaction} \\ $$$$\left(\mathrm{a}\right){HCl}+{C}\mathrm{aCO}_{\mathrm{3}} \rightarrow\mathrm{Ca}{Cl}_{\mathrm{2}} +{C}\mathrm{O}_{\mathrm{2}\left(\mathrm{g}\right)} +\mathrm{H}_{\mathrm{2}} \mathrm{O}_{\left({l}\right)} \\ $$$$\left({b}\right){NH}_{\mathrm{4}} {OH}_{\:\left({aq}\right)} +{HCl}_{\left({aq}\right)} \:\rightarrow{NH}_{\mathrm{4}} \mathrm{C}{l}_{\left({aq}\right)} +{H}_{\mathrm{2}} \mathrm{O}_{\left({l}\right)} \\ $$$$\mathrm{31}/\mathrm{1}/\mathrm{2024} \\ $$

Question Number 203875 Answers: 1 Comments: 0

Hmmm..... I have one Question. f(t)∈C^∞ , {C_ ^𝛂 mean can derivate 𝛂 times.} where t∈R , Can f(t) integrable when S∈R\{Q}?? Ex. integral ∫_1 ^( e) ln(z)dz S∈[1,e] But Except Q in set S like.. S^′ =S\{Q} than Can integrable In S′

$$\mathrm{Hmmm}.....\:\mathrm{I}\:\mathrm{have}\:\mathrm{one}\:\mathrm{Question}. \\ $$$${f}\left({t}\right)\in{C}^{\infty} \:,\:\left\{{C}_{\:} ^{\boldsymbol{\alpha}} \:\mathrm{mean}\:\mathrm{can}\:\mathrm{derivate}\:\boldsymbol{\alpha}\:\mathrm{times}.\right\} \\ $$$$\mathrm{where}\:{t}\in\mathbb{R}\:,\:\mathrm{Can}\:{f}\left({t}\right)\:\:\mathrm{integrable}\:\mathrm{when}\:{S}\in\mathbb{R}\backslash\left\{\mathbb{Q}\right\}?? \\ $$$$\mathrm{Ex}.\:\mathrm{integral}\:\int_{\mathrm{1}} ^{\:{e}} \:\mathrm{ln}\left({z}\right)\mathrm{d}{z}\:{S}\in\left[\mathrm{1},{e}\right]\: \\ $$$$\mathrm{But}\:\mathrm{Except}\:\mathbb{Q}\:\mathrm{in}\:\mathrm{set}\:{S}\:\mathrm{like}..\:{S}^{'} ={S}\backslash\left\{\mathbb{Q}\right\}\: \\ $$$$\mathrm{than}\:\mathrm{Can}\:\mathrm{integrable}\:\mathrm{In}\:{S}' \\ $$

Question Number 203867 Answers: 1 Comments: 0

find ∫(√((1−x^3 )/(1+x^3 )))dx

$${find}\:\int\sqrt{\frac{\mathrm{1}−{x}^{\mathrm{3}} }{\mathrm{1}+{x}^{\mathrm{3}} }}{dx} \\ $$

Question Number 203859 Answers: 0 Comments: 3

Show that the surface z = xy has neither a maximum nor a minimum point

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{surface}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{xy} \\ $$$$\mathrm{has}\:\mathrm{neither}\:\mathrm{a}\:\mathrm{maximum}\:\mathrm{nor}\:\mathrm{a}\:\mathrm{minimum}\:\mathrm{point} \\ $$

Question Number 203858 Answers: 0 Comments: 1

Classsify the critical points of the function f(x,y) = x^2 y + (1/3)y^3 − x^2 − y^2 + 2 Thank you in advance!

$$\mathrm{Classsify}\:\mathrm{the}\:\mathrm{critical}\:\mathrm{points}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\mathrm{y}^{\mathrm{3}} \:−\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{y}^{\mathrm{2}} \:+\:\mathrm{2} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{in}\:\mathrm{advance}! \\ $$

Question Number 203846 Answers: 1 Comments: 0

lim_(n→∞) Π_(i=2) ^n (((i^2 −1)/i^2 ))