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Question Number 92798    Answers: 0   Comments: 0

Find the value of 𝛉 in the Mean Value Theorem f(x+h) = f(x) +h f^( β€²) (x+ΞΈh) if f(x)= (1/x) .

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\theta}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Mean}\:\mathrm{Value} \\ $$$$\:\:\mathrm{Theorem}\: \\ $$$$\:\:\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)\:+\mathrm{h}\:\mathrm{f}^{\:'} \left(\mathrm{x}+\theta\mathrm{h}\right)\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\:\frac{\mathrm{1}}{\mathrm{x}}\:. \\ $$

Question Number 92795    Answers: 0   Comments: 0

Define Clairautβ€²s equation and solve y= px +(√(a^2 p^2 +b^2 ))

$$\boldsymbol{\mathrm{Define}}\:\boldsymbol{\mathrm{Clairaut}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{equation}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{solve}} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{y}}=\:\boldsymbol{\mathrm{px}}\:+\sqrt{\boldsymbol{\mathrm{a}}^{\mathrm{2}} \boldsymbol{\mathrm{p}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} } \\ $$

Question Number 92790    Answers: 0   Comments: 0

∫_0 ^1 ((Ο€/4)βˆ’tan^(βˆ’1) (x))(dx/(1βˆ’x^2 ))

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\pi}{\mathrm{4}}βˆ’{tan}^{βˆ’\mathrm{1}} \left({x}\right)\right)\frac{{dx}}{\mathrm{1}βˆ’{x}^{\mathrm{2}} } \\ $$

Question Number 92788    Answers: 0   Comments: 0

Question Number 92785    Answers: 0   Comments: 2

a) Find E(x^x ) then E(x^x^x ) for x∈]0,1[ b) find lim_(xβ†’0) E(x^x^x )

$$\left.\mathrm{a}\left.\right)\:\mathrm{Find}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}} \right)\:\mathrm{then}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \right)\:\mathrm{for}\:\mathrm{x}\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{E}\left(\mathrm{x}^{\mathrm{x}^{\mathrm{x}} } \right) \\ $$

Question Number 92782    Answers: 0   Comments: 0

$$ \\ $$

Question Number 92781    Answers: 1   Comments: 1

tan^(βˆ’1) (x) = sin^(βˆ’1) (((2x)/(x^2 +1)))

$$\mathrm{tan}^{βˆ’\mathrm{1}} \left({x}\right)\:=\:\mathrm{sin}^{βˆ’\mathrm{1}} \left(\frac{\mathrm{2}{x}}{{x}^{\mathrm{2}} +\mathrm{1}}\right)\: \\ $$

Question Number 92778    Answers: 0   Comments: 1

Question Number 92772    Answers: 0   Comments: 1

Question Number 92771    Answers: 0   Comments: 0

Question Number 92769    Answers: 0   Comments: 0

calculate Ξ£_(k=0) ^(nβˆ’1) C_n ^k cos^2 (((kΟ€)/n)) (nβ‰₯2)

$${calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}βˆ’\mathrm{1}} \:{C}_{{n}} ^{{k}} \:{cos}^{\mathrm{2}} \left(\frac{{k}\pi}{{n}}\right)\:\:\:\:\:\left({n}\geqslant\mathrm{2}\right) \\ $$

Question Number 92768    Answers: 0   Comments: 0

calculate A_n = Ξ£_(k=0) ^n C_n ^k cos^3 (kx) and Ξ£_(k=0) ^n C_n ^k sin^3 (kx)

$${calculate}\:{A}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}^{\mathrm{3}} \left({kx}\right)\:\:{and}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{sin}^{\mathrm{3}} \left({kx}\right) \\ $$

Question Number 92763    Answers: 0   Comments: 1

study the sequence u_(n+1) =(√(u_n ^2 +(1/n))) and u_1 =1

$${study}\:{the}\:{sequence}\:{u}_{{n}+\mathrm{1}} =\sqrt{{u}_{{n}} ^{\mathrm{2}} +\frac{\mathrm{1}}{{n}}} \\ $$$${and}\:{u}_{\mathrm{1}} =\mathrm{1} \\ $$

Question Number 92767    Answers: 0   Comments: 0

let U_n =∫_0 ^1 e^(βˆ’nx) cosx dx 1) determine lim_(nβ†’+∞) n^2 U_n 2)calculate Ξ£_(n=0) ^∞ U_n

$${let}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{βˆ’{nx}} {cosx}\:{dx} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{lim}_{{n}\rightarrow+\infty} {n}^{\mathrm{2}} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 92741    Answers: 0   Comments: 9

lim_(x→0) ((2/(sin^2 (x))) + (1/(ln(cos (x))))) =

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{2}}{\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:+\:\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{cos}\:\left(\mathrm{x}\right)\right)}\right)\:= \\ $$

Question Number 92743    Answers: 0   Comments: 2

y βˆ’ y_1 = m ( x βˆ’ x_1 ) β‡’ y βˆ’ 1 = 5 ( x βˆ’ 2 ) y βˆ’ 1 = 5x βˆ’ 10 y = 5x βˆ’ 10 + 1 ∴ y = 5x βˆ’ 9

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:\:βˆ’\:\:{y}_{\mathrm{1}} \:\:=\:\:{m}\:\left(\:{x}\:βˆ’\:{x}_{\mathrm{1}} \:\right) \\ $$$$\:\:\:\Rightarrow \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:βˆ’\:\mathrm{1}\:=\:\:\mathrm{5}\:\left(\:{x}\:βˆ’\:\mathrm{2}\:\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:βˆ’\:\mathrm{1}\:=\:\:\mathrm{5}{x}\:βˆ’\:\mathrm{10} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:=\:\:\mathrm{5}{x}\:βˆ’\:\mathrm{10}\:+\:\mathrm{1} \\ $$$$\:\:\:\:\therefore\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{y}\:=\:\:\mathrm{5}{x}\:βˆ’\:\mathrm{9} \\ $$$$\: \\ $$

Question Number 92740    Answers: 0   Comments: 0

∫ ((ln(x))/(cos x)) dx ?

$$\int\:\frac{\mathrm{ln}\left({x}\right)}{\mathrm{cos}\:{x}}\:{dx}\:? \\ $$

Question Number 92729    Answers: 1   Comments: 0

if p_n is the product of the terms in the nth row of the pascalβ€²s triangle find lim_(nβ†’βˆž) ((p_(nβˆ’1) p_(n+1) )/((p_n )^2 ))

$${if}\:{p}_{{n}} \:{is}\:{the}\:{product}\:{of}\:{the}\:{terms}\:{in} \\ $$$${the}\:{nth}\:{row}\:{of}\:{the}\:{pascal}'{s}\:{triangle} \\ $$$${find} \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\frac{{p}_{{n}βˆ’\mathrm{1}} {p}_{{n}+\mathrm{1}} }{\left({p}_{{n}} \right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 92727    Answers: 1   Comments: 0

Solve: x^y = y^x ....... (i) 3^x = 15^y ...... (ii) x β‰  y, x, y ∈ R

$$\mathrm{Solve}:\:\:\:\:\:\mathrm{x}^{\mathrm{y}} \:\:=\:\:\mathrm{y}^{\mathrm{x}} \:\:\:\:\:.......\:\:\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}^{\mathrm{x}} \:\:=\:\:\mathrm{15}^{\mathrm{y}} \:\:\:\:......\:\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\mathrm{x}\:\:\neq\:\:\mathrm{y},\:\:\:\:\:\:\:\mathrm{x},\:\:\mathrm{y}\:\in\:\mathbb{R} \\ $$

Question Number 92723    Answers: 1   Comments: 0

Question Number 92717    Answers: 2   Comments: 5

Question Number 92708    Answers: 1   Comments: 0

solve the differential equations. (a) (x + 3y^2 )(d^2 y/dx^2 ) + 6y ((dy/dx))^2 + 2(dy/dx) + 2 = 0 (b) (2yβˆ’x)(d^2 y/dx^2 ) + 2((dy/dx))^2 βˆ’2 (dy/dx) + 2 = 0

$$\mathrm{solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equations}. \\ $$$$\:\left(\mathrm{a}\right)\:\left({x}\:+\:\mathrm{3}{y}^{\mathrm{2}} \right)\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{6}{y}\:\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \:+\:\mathrm{2}\frac{{dy}}{{dx}}\:+\:\mathrm{2}\:=\:\mathrm{0} \\ $$$$\left(\mathrm{b}\right)\:\left(\mathrm{2}{y}βˆ’{x}\right)\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{2}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \:βˆ’\mathrm{2}\:\frac{{dy}}{{dx}}\:+\:\mathrm{2}\:=\:\mathrm{0} \\ $$

Question Number 92702    Answers: 2   Comments: 0

yβ€²β€²+2yβ€²βˆ’3y=e^x +e^(2x)

$${y}''+\mathrm{2}{y}'βˆ’\mathrm{3}{y}={e}^{{x}} +{e}^{\mathrm{2}{x}} \\ $$

Question Number 92701    Answers: 0   Comments: 3

Question Number 92700    Answers: 0   Comments: 0

∫_(βˆ’(Ο€/2)) ^(Ο€/2) ((tan (x^2 )dx)/x) ?

$$\underset{βˆ’\frac{\pi}{\mathrm{2}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{tan}\:\left({x}^{\mathrm{2}} \right){dx}}{{x}}\:?\: \\ $$

Question Number 92691    Answers: 1   Comments: 3

find x in eq tan^(βˆ’1) (x)= cos^(βˆ’1) (x)

$$\mathrm{find}\:{x}\:{in}\:{eq}\:\mathrm{tan}^{βˆ’\mathrm{1}} \left({x}\right)=\:\mathrm{cos}^{βˆ’\mathrm{1}} \left({x}\right) \\ $$

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