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Question Number 159292    Answers: 1   Comments: 0

Find: 𝛀 =∫_( 0) ^( ∞) ((x arctan(x))/((x + 1)(x^2 + 1))) dx

$$\mathrm{Find}:\:\:\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\frac{\mathrm{x}\:\mathrm{arctan}\left(\mathrm{x}\right)}{\left(\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}\right)}\:\mathrm{dx} \\ $$$$ \\ $$

Question Number 159157    Answers: 1   Comments: 0

Question Number 159184    Answers: 0   Comments: 2

Question Number 159153    Answers: 0   Comments: 0

Question Number 159152    Answers: 1   Comments: 0

Question Number 159146    Answers: 0   Comments: 0

Find the absolute extrema of f(x)= 2 csc x + cot x on the interval ((Ο€/2), ((3Ο€)/2) ]

$${Find}\:{the}\:{absolute}\:{extrema}\:{of} \\ $$$${f}\left({x}\right)=\:\mathrm{2}\:\mathrm{csc}\:{x}\:+\:\mathrm{cot}\:{x}\:{on}\:{the}\: \\ $$$${interval}\:\left(\frac{\pi}{\mathrm{2}},\:\frac{\mathrm{3}\pi}{\mathrm{2}}\:\right] \\ $$

Question Number 159143    Answers: 1   Comments: 0

Question Number 159142    Answers: 2   Comments: 1

lim_(xβ†’βˆ’(Ο€/4)) (((Ο€/( (√8))) βˆ’(√2) x. tan x)/(sin x+cos x)) =?

$$\:\underset{{x}\rightarrowβˆ’\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\frac{\pi}{\:\sqrt{\mathrm{8}}}\:βˆ’\sqrt{\mathrm{2}}\:{x}.\:\mathrm{tan}\:{x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}\:=? \\ $$

Question Number 159137    Answers: 1   Comments: 4

Yesterday,Mr John earned $50 mowing lawns, today Mr. John earned 60percent of what he earned yesterday mowing lawns. how much money did Mr Jacob earned mowing lawns today??

$$\mathrm{Yesterday},\mathrm{Mr}\:\mathrm{John}\:\mathrm{earned}\:\$\mathrm{50} \\ $$$$\mathrm{mowing}\:\mathrm{lawns},\:\mathrm{today}\:\mathrm{Mr}.\:\mathrm{John} \\ $$$$\mathrm{earned}\:\mathrm{60percent}\:\mathrm{of}\:\mathrm{what}\:\mathrm{he}\:\mathrm{earned} \\ $$$$\mathrm{yesterday}\:\mathrm{mowing}\:\mathrm{lawns}.\:\mathrm{how} \\ $$$$\mathrm{much}\:\mathrm{money}\:\mathrm{did}\:\mathrm{Mr}\:\mathrm{Jacob}\:\mathrm{earned} \\ $$$$\mathrm{mowing}\:\mathrm{lawns}\:\mathrm{today}?? \\ $$

Question Number 159136    Answers: 0   Comments: 0

Question Number 159135    Answers: 1   Comments: 4

Question Number 159131    Answers: 1   Comments: 0

Question Number 159130    Answers: 0   Comments: 0

f ∈ C^0 (R,R) , ((f(x))/x)β†’_(xβ†’+∞) 0 Does lim_(xβ†’+∞) f(x+1) βˆ’ f(x) exist?

$${f}\:\in\:\mathcal{C}^{\mathrm{0}} \left(\mathbb{R},\mathbb{R}\right)\:,\:\frac{{f}\left({x}\right)}{{x}}\underset{{x}\rightarrow+\infty} {\rightarrow}\mathrm{0} \\ $$$${Does}\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}+\mathrm{1}\right)\:βˆ’\:{f}\left({x}\right)\:{exist}? \\ $$

Question Number 159123    Answers: 2   Comments: 0

Question Number 159122    Answers: 0   Comments: 0

In order to monitor buses in a travel agency, the manager decides to monitor the number of break downs of the buses using the sequence {x_n } defined by x_(n+1) = 1.05 x_n + 4. Given that x_0 = 40. is the number of break downs by the buses from the 1^(st) of january 2000, and that for every n∈N, we denote x_n the number of breakdowns of the buses as from 1^(st) of january of the year (2000 + n) (a) Calculate x_1 , x_2 , x_3 (b) Consider the sequence {y_n } defined by y_n = x_n + 80 for all n ∈ N (i) express y_(n+1) in terms of y_n and deduce the nature of the sequence {y_n }. (ii) Express y_n in terms of n. deduce x_n in terms of n (iv) find the number of break downs that will be registered by 1^(st) january 2021.

$$\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{monitor}\:\mathrm{buses}\:\mathrm{in}\:\mathrm{a}\:\mathrm{travel} \\ $$$$\mathrm{agency},\:\mathrm{the}\:\mathrm{manager}\:\mathrm{decides}\:\mathrm{to}\:\mathrm{monitor} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}\:\mathrm{defined}\:\mathrm{by} \\ $$$${x}_{{n}+\mathrm{1}} \:=\:\mathrm{1}.\mathrm{05}\:{x}_{{n}} \:+\:\mathrm{4}.\:\mathrm{Given}\:\mathrm{that}\:{x}_{\mathrm{0}} \:=\:\mathrm{40}. \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{by}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{of}\:\mathrm{january}\:\mathrm{2000},\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{every}\:{n}\in\mathbb{N},\:\mathrm{we}\:\mathrm{denote}\:{x}_{{n}} \:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{breakdowns}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses}\:\mathrm{as}\:\mathrm{from}\:\mathrm{1}^{\mathrm{st}} \\ $$$$\mathrm{of}\:\mathrm{january}\:\mathrm{of}\:\mathrm{the}\:\mathrm{year}\:\left(\mathrm{2000}\:+\:{n}\right) \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} \:,\:{x}_{\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}\:\mathrm{defined} \\ $$$$\mathrm{by}\:{y}_{{n}} \:=\:{x}_{{n}} \:+\:\mathrm{80}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\:\mathbb{N} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{express}\:{y}_{{n}+\mathrm{1}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{y}_{{n}} \:\mathrm{and} \\ $$$$\mathrm{deduce}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Express}\:{y}_{{n}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}.\:\mathrm{deduce}\:{x}_{{n}} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\left(\mathrm{iv}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs} \\ $$$$\mathrm{that}\:\mathrm{will}\:\mathrm{be}\:\mathrm{registered}\:\mathrm{by}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{january}\: \\ $$$$\mathrm{2021}. \\ $$

Question Number 159121    Answers: 0   Comments: 0

Consider f(x) = x^3 + 2x βˆ’1. Use the intermidiate value theorem and the Rolle theorem to establish that the equation f(x) = 0 has a unique solution denoted a_0 ∈] 0,1[.

$$\mathrm{Consider} \\ $$$${f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{2}{x}\:βˆ’\mathrm{1}. \\ $$$$\mathrm{Use}\:\mathrm{the}\:\mathrm{intermidiate}\:\mathrm{value}\:\mathrm{theorem}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{Rolle}\:\mathrm{theorem}\:\mathrm{to}\:\mathrm{establish}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{equation}\:{f}\left({x}\right)\:=\:\mathrm{0}\:\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution} \\ $$$$\left.\mathrm{denoted}\:{a}_{\mathrm{0}} \in\right]\:\mathrm{0},\mathrm{1}\left[.\:\right. \\ $$

Question Number 159119    Answers: 0   Comments: 0

Assume x;y;z>0 and x^2 +y^2 +z^2 =12 Prove that: Ξ£_(cycl) (((x/y) + 1 + (y/x))/((1/x) + (1/y))) ≀ 9

$$\mathrm{Assume}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{12} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\underset{\boldsymbol{\mathrm{cycl}}} {\sum}\:\frac{\frac{\mathrm{x}}{\mathrm{y}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{y}}{\mathrm{x}}}{\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}}\:\leqslant\:\mathrm{9} \\ $$

Question Number 159118    Answers: 0   Comments: 0

Let F(r) = { ((mkr, 0 ≀ r < R)),((((mgR^2 )/r^2 ), rβ‰₯ R)) :} find D_F

$$\mathrm{Let}\:{F}\left({r}\right)\:=\:\begin{cases}{{mkr},\:\:\mathrm{0}\:\leqslant\:{r}\:<\:{R}}\\{\frac{{m}\mathrm{g}{R}^{\mathrm{2}} }{{r}^{\mathrm{2}} },\:{r}\geqslant\:{R}}\end{cases} \\ $$$$\mathrm{find}\:{D}_{{F}} \\ $$

Question Number 159111    Answers: 1   Comments: 0

Given A^β†’ =5t^2 i^β†’ +tj^β†’ βˆ’t^3 k^β†’ and B^β†’ =sin(t)i^β†’ βˆ’cos(t)j^β†’ . Calculate ((d(A^β†’ .B^β†’ ))/dx) ; ((d(A^β†’ ∧B^β†’ ))/dx) and ((d(A^β†’ .A^β†’ ))/dx).

$${Given}\:\overset{\rightarrow} {{A}}=\mathrm{5}{t}^{\mathrm{2}} \overset{\rightarrow} {{i}}+{t}\overset{\rightarrow} {{j}}βˆ’{t}^{\mathrm{3}} \overset{\rightarrow} {{k}}\:{and} \\ $$$$\overset{\rightarrow} {{B}}={sin}\left({t}\right)\overset{\rightarrow} {{i}}βˆ’{cos}\left({t}\right)\overset{\rightarrow} {{j}}. \\ $$$${Calculate}\:\frac{{d}\left(\overset{\rightarrow} {{A}}.\overset{\rightarrow} {{B}}\right)}{{dx}}\:;\:\frac{{d}\left(\overset{\rightarrow} {{A}}\wedge\overset{\rightarrow} {{B}}\right)}{{dx}}\:\:{and} \\ $$$$\frac{{d}\left(\overset{\rightarrow} {{A}}.\overset{\rightarrow} {{A}}\right)}{{dx}}. \\ $$

Question Number 159109    Answers: 0   Comments: 0

Show that βˆ€ V_i ^β†’ and V_j ^β†’ : V_i ^β†’ ∧[V_j ^β†’ ∧(V_j ^β†’ ∧V_i ^β†’ )]=βˆ’V_j ^β†’ ∧[V_i ^β†’ ∧(V_i ^β†’ ∧V_j ^β†’ )]

$${Show}\:{that}\:\forall\:\overset{\rightarrow} {{V}}_{{i}} \:{and}\:\overset{\rightarrow} {{V}}_{{j}} : \\ $$$$\overset{\rightarrow} {{V}}_{{i}} \wedge\left[\overset{\rightarrow} {{V}}_{{j}} \wedge\left(\overset{\rightarrow} {{V}}_{{j}} \wedge\overset{\rightarrow} {{V}}_{{i}} \right)\right]=βˆ’\overset{\rightarrow} {{V}}_{{j}} \wedge\left[\overset{\rightarrow} {{V}}_{{i}} \wedge\left(\overset{\rightarrow} {{V}}_{{i}} \wedge\overset{\rightarrow} {{V}}_{{j}} \right)\right] \\ $$

Question Number 160090    Answers: 0   Comments: 2

There are 40 oranges, 20 apples, and 20 lemons in a bag. What is the minimum number of fruits that you have to take out of the bag with your eyes closed before you are sure that one of them is an orange?

$$\mathrm{There}\:\mathrm{are}\:\mathrm{40}\:\mathrm{oranges},\:\mathrm{20}\:\mathrm{apples},\:\mathrm{and}\: \\ $$$$\mathrm{20}\:\mathrm{lemons}\:\mathrm{in}\:\mathrm{a}\:\mathrm{bag}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{minimum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{fruits}\:\mathrm{that}\:\mathrm{you}\: \\ $$$$\mathrm{have}\:\mathrm{to}\:\mathrm{take}\:\mathrm{out}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bag}\:\mathrm{with}\:\mathrm{your}\: \\ $$$$\mathrm{eyes}\:\mathrm{closed}\:\mathrm{before}\:\mathrm{you}\:\mathrm{are}\:\mathrm{sure}\:\mathrm{that}\: \\ $$$$\mathrm{one}\:\mathrm{of}\:\mathrm{them}\:\mathrm{is}\:\mathrm{an}\:\mathrm{orange}? \\ $$

Question Number 159125    Answers: 0   Comments: 3

for a, b, c >0 and a+b+c=2 find min(2ab^2 +b^3 c, 2bc^2 +c^3 a, 2ca^2 +a^3 b) or disprove that such a minimum doesnβ€²t exist.

$${for}\:{a},\:{b},\:{c}\:>\mathrm{0}\:{and}\:{a}+{b}+{c}=\mathrm{2} \\ $$$${find}\:{min}\left(\mathrm{2}{ab}^{\mathrm{2}} +{b}^{\mathrm{3}} {c},\:\mathrm{2}{bc}^{\mathrm{2}} +{c}^{\mathrm{3}} {a},\:\mathrm{2}{ca}^{\mathrm{2}} +{a}^{\mathrm{3}} {b}\right) \\ $$$${or}\:{disprove}\:{that}\:{such}\:{a}\:{minimum} \\ $$$${doesn}'{t}\:{exist}. \\ $$

Question Number 159106    Answers: 0   Comments: 0

∫_0 ^Ο€ ((sin(nz))/(z^4 sin(Ο€z)))dz

$$\int_{\mathrm{0}} ^{\pi} \frac{\mathrm{sin}\left({nz}\right)}{\mathrm{z}^{\mathrm{4}} \mathrm{sin}\left(\pi{z}\right)}{dz} \\ $$

Question Number 159099    Answers: 1   Comments: 0

Question Number 159097    Answers: 0   Comments: 0

f(x)=log_(2 ) ^x (((x^2 βˆ’1)/( (√(x+1)))))+∣x∣^2

$${f}\left({x}\right)={log}_{\mathrm{2}\:} ^{{x}} \left(\frac{{x}^{\mathrm{2}} βˆ’\mathrm{1}}{\:\sqrt{{x}+\mathrm{1}}}\right)+\mid{x}\mid^{\mathrm{2}} \\ $$

Question Number 159092    Answers: 1   Comments: 0

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