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Question Number 140959 Answers: 1 Comments: 0

Let a,b ≥ 0. Prove that (1/4)∙(((2+a)(2+b))/((1+a)(1+b))) ≥ ((4−a−b)/(4+a+b))

$$\mathrm{Let}\:{a},{b}\:\geqslant\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{4}}\centerdot\frac{\left(\mathrm{2}+{a}\right)\left(\mathrm{2}+{b}\right)}{\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{b}\right)}\:\geqslant\:\frac{\mathrm{4}−{a}−{b}}{\mathrm{4}+{a}+{b}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 140958 Answers: 0 Comments: 0

find e^ (((−1 1)),((2 −1)) )

$$\mathrm{find}\:\mathrm{e}^{\begin{pmatrix}{−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\mathrm{1}}\end{pmatrix}} \\ $$

Question Number 140956 Answers: 1 Comments: 0

.....advanced......calculus..... prove that: 𝛗:= ∫_(−∞) ^( ∞) ((sin^4 (x).cos^4 (x))/x^2 )dx=(π/(16)) m.n

$$\:\:\:\:\:\:\:\:\:\:.....{advanced}......{calculus}..... \\ $$$$\:\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\boldsymbol{\phi}:=\:\int_{−\infty} ^{\:\infty} \frac{{sin}^{\mathrm{4}} \left({x}\right).{cos}^{\mathrm{4}} \left({x}\right)}{{x}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{16}} \\ $$$$\:\:{m}.{n} \\ $$

Question Number 140966 Answers: 1 Comments: 0

.......nice......calculus..... if Σ_(n=0) ^∞ (((√(cos (nπ))) )/((2n)!!)) = ω then Re(ω):=??

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.......{nice}......{calculus}..... \\ $$$$\:\:\:\:\:{if}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\sqrt{{cos}\:\left({n}\pi\right)}\:}{\left(\mathrm{2}{n}\right)!!}\:=\:\omega \\ $$$$\:\:\:\:\:\:\:{then}\:\:\:{Re}\left(\omega\right):=?? \\ $$

Question Number 140941 Answers: 1 Comments: 2

Question Number 140939 Answers: 1 Comments: 0

x^(sgn (x^3 −x)) = x^2 − (4/9)

$$\:{x}^{\mathrm{sgn}\:\left({x}^{\mathrm{3}} −{x}\right)} \:=\:{x}^{\mathrm{2}} \:−\:\frac{\mathrm{4}}{\mathrm{9}}\: \\ $$

Question Number 140938 Answers: 0 Comments: 0

Question Number 140937 Answers: 0 Comments: 0

can someone please share maple with me or show me the site to download free?

$${can}\:{someone}\:{please}\:{share}\:{maple}\:{with}\:{me}\:{or}\:{show}\:{me}\:{the}\:{site}\:{to}\:{download}\:{free}? \\ $$

Question Number 140930 Answers: 3 Comments: 0

∫_0 ^π (dx/( (√2) −cos x))

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{2}}\:−\mathrm{cos}\:\mathrm{x}} \\ $$

Question Number 141002 Answers: 1 Comments: 0

If a>0 and one root of ax^2 +bx+c=0 is less than −2 and the other is greater than 2, then (A) 4a+2∣b∣+c<0 (B) 4a+2∣b∣+c>0 (C) 4a+2∣b∣+c=0 (D) a+b=c

$$\mathrm{If}\:{a}>\mathrm{0}\:\mathrm{and}\:\mathrm{one}\:\mathrm{root}\:\mathrm{of}\:{a}\mathrm{x}^{\mathrm{2}} +{b}\mathrm{x}+\mathrm{c}=\mathrm{0}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:−\mathrm{2} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2},\:\mathrm{then} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{4}{a}+\mathrm{2}\mid{b}\mid+{c}<\mathrm{0} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{4}{a}+\mathrm{2}\mid{b}\mid+{c}>\mathrm{0} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{4}{a}+\mathrm{2}\mid{b}\mid+{c}=\mathrm{0} \\ $$$$\left(\mathrm{D}\right)\:{a}+{b}={c} \\ $$

Question Number 140915 Answers: 1 Comments: 2

Question Number 142212 Answers: 0 Comments: 0

Question Number 140916 Answers: 0 Comments: 4

Question Number 140908 Answers: 0 Comments: 0

ab=c let (a−p)(b−q)=0 ⇒ c−(aq+bp)+pq=0 q=((bp−c)/(p−a)) say 4×2=8 (4−3)(2−((6−8)/(−1)))=1×0=0 And if q=b p=((ab−c)/(2b))=0 And if p+q=c (p−a)(c−p)=bp−c p^2 −(a+c−b)p+c(1−a)=0 2p=(a+c−b)±(√((a+c−b)^2 −4c(1−a))) ________________________

$$\:\:{ab}={c} \\ $$$${let}\:\:\:\left({a}−{p}\right)\left({b}−{q}\right)=\mathrm{0} \\ $$$$\Rightarrow\:\:{c}−\left({aq}+{bp}\right)+{pq}=\mathrm{0} \\ $$$${q}=\frac{{bp}−{c}}{{p}−{a}} \\ $$$${say}\:\:\:\mathrm{4}×\mathrm{2}=\mathrm{8} \\ $$$$\:\:\:\:\:\left(\mathrm{4}−\mathrm{3}\right)\left(\mathrm{2}−\frac{\mathrm{6}−\mathrm{8}}{−\mathrm{1}}\right)=\mathrm{1}×\mathrm{0}=\mathrm{0} \\ $$$${And}\:\:{if}\:\:{q}={b} \\ $$$$\:\:{p}=\frac{{ab}−{c}}{\mathrm{2}{b}}=\mathrm{0} \\ $$$${And}\:{if}\:{p}+{q}={c} \\ $$$$\left({p}−{a}\right)\left({c}−{p}\right)={bp}−{c} \\ $$$${p}^{\mathrm{2}} −\left({a}+{c}−{b}\right){p}+{c}\left(\mathrm{1}−{a}\right)=\mathrm{0} \\ $$$$\mathrm{2}{p}=\left({a}+{c}−{b}\right)\pm\sqrt{\left({a}+{c}−{b}\right)^{\mathrm{2}} −\mathrm{4}{c}\left(\mathrm{1}−{a}\right)} \\ $$$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \\ $$

Question Number 140907 Answers: 1 Comments: 0

(√(sin x)) cos x −(√(sin^5 x)) cos x = cos^3 x (√(sin x))

$$\:\sqrt{\mathrm{sin}\:\mathrm{x}}\:\mathrm{cos}\:\mathrm{x}\:−\sqrt{\mathrm{sin}\:^{\mathrm{5}} \mathrm{x}}\:\mathrm{cos}\:\mathrm{x}\:=\:\mathrm{cos}\:^{\mathrm{3}} \mathrm{x}\:\sqrt{\mathrm{sin}\:\mathrm{x}} \\ $$

Question Number 140905 Answers: 2 Comments: 0

lim_(x→0) ((1−cos (1−cos x))/(x (tan x−x))) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{x}\:\left(\mathrm{tan}\:\mathrm{x}−\mathrm{x}\right)}\:=? \\ $$

Question Number 140900 Answers: 1 Comments: 1

{ ((x^3 =xyz+1)),((y^3 =xyz+2)),((z^3 =xyz−3)) :}

$$\:\begin{cases}{\mathrm{x}^{\mathrm{3}} =\mathrm{xyz}+\mathrm{1}}\\{\mathrm{y}^{\mathrm{3}} =\mathrm{xyz}+\mathrm{2}}\\{\mathrm{z}^{\mathrm{3}} =\mathrm{xyz}−\mathrm{3}}\end{cases} \\ $$

Question Number 140899 Answers: 0 Comments: 2

Question Number 140896 Answers: 2 Comments: 0

the function f with variable x satisfies the equation x^2 f ′(x) +2x f(x) = arctan x for 0 < arctan x <(π/2) and f(1)=(π/4). find f(x).

$$\mathrm{the}\:\mathrm{function}\:\mathrm{f}\:\mathrm{with}\:\mathrm{variable}\:\mathrm{x} \\ $$$$\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\mathrm{x}^{\mathrm{2}} \:\mathrm{f}\:'\left(\mathrm{x}\right)\:+\mathrm{2x}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{arctan}\:\mathrm{x}\:\mathrm{for}\: \\ $$$$\mathrm{0}\:<\:\mathrm{arctan}\:\mathrm{x}\:<\frac{\pi}{\mathrm{2}}\:\mathrm{and}\:\mathrm{f}\left(\mathrm{1}\right)=\frac{\pi}{\mathrm{4}}. \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right). \\ $$

Question Number 140890 Answers: 1 Comments: 0

1+2x+3x^2 +4x^3 +...+(n+1)x^n =?

$$\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}^{\mathrm{3}} +...+\left({n}+\mathrm{1}\right){x}^{\boldsymbol{{n}}} =? \\ $$

Question Number 140874 Answers: 2 Comments: 1

Question Number 140869 Answers: 1 Comments: 0

the velocities of air particles above and below the wing of an aircraft speeding down the runway at a given instant are 210m/s and 200m/s respectively.If the density of air is 1.2kg/m^3 ,what is the pressure difference between the upper and lower surface of the wing?

$${the}\:\mathrm{velocities}\:\mathrm{of}\:\mathrm{air}\:\mathrm{particles}\: \\ $$$$\mathrm{above}\:\mathrm{and}\:\mathrm{below}\:\mathrm{the}\:\mathrm{wing}\:\mathrm{of}\:\mathrm{an}\:\mathrm{aircraft} \\ $$$$\:\mathrm{speeding}\:\mathrm{down}\:\mathrm{the}\:\mathrm{runway}\:\mathrm{at}\:\mathrm{a} \\ $$$$\:\mathrm{given}\:\mathrm{instant} \\ $$$$\mathrm{are}\:\mathrm{210m}/\mathrm{s}\:\mathrm{and}\:\mathrm{200m}/\mathrm{s}\: \\ $$$$\mathrm{respectively}.\mathrm{If}\:\mathrm{the}\:\mathrm{density}\:\mathrm{of}\:\mathrm{air}\:\mathrm{is}\: \\ $$$$\mathrm{1}.\mathrm{2kg}/\mathrm{m}^{\mathrm{3}} ,\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{pressure} \\ $$$$\:\mathrm{difference}\:\mathrm{between}\:\mathrm{the}\:\mathrm{upper}\:\mathrm{and} \\ $$$$\mathrm{lower}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{wing}? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 140866 Answers: 1 Comments: 0

∫_0 ^1 ((ln 2−ln (1+x^2 ))/(1−x)) dx =?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{ln}\:\mathrm{2}−\mathrm{ln}\:\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}−\mathrm{x}}\:\mathrm{dx}\:=?\: \\ $$

Question Number 140870 Answers: 1 Comments: 0

Ocean waves are observed to travel along the water surface during a developing storm. A Coast Guard weather station observes that there is a vertical distance from high point to low point of 4.6 meters and horizontal distance of 8.6 meters between adjacent crests.The waves splash into the station once every 6.2 seconds Determine the frequency and the speeed of these waves.

$${O}\mathrm{cean}\:\mathrm{waves}\:\mathrm{are}\:\mathrm{observed}\:\mathrm{to} \\ $$$$\mathrm{travel}\:\mathrm{along}\:\mathrm{the}\:\mathrm{water}\:\mathrm{surface} \\ $$$$\:\mathrm{during}\:\mathrm{a}\:\mathrm{developing}\:\mathrm{storm}. \\ $$$$\mathrm{A}\:\mathrm{Coast}\:\mathrm{Guard}\:\mathrm{weather}\:\mathrm{station} \\ $$$$\:\mathrm{observes}\:\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\mathrm{a}\:\mathrm{vertical}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{high}\:\mathrm{point}\:\mathrm{to}\:\mathrm{low} \\ $$$$\:\mathrm{point}\:\mathrm{of}\:\mathrm{4}.\mathrm{6}\:\mathrm{meters}\:\mathrm{and}\:\mathrm{horizontal} \\ $$$$\mathrm{distance}\:\mathrm{of}\:\mathrm{8}.\mathrm{6}\:\mathrm{meters}\:\mathrm{between}\: \\ $$$$\mathrm{adjacent}\:\mathrm{crests}.\mathrm{The}\:\mathrm{waves}\:\mathrm{splash}\: \\ $$$$\mathrm{into}\:\mathrm{the}\:\mathrm{station}\:\mathrm{once}\:\mathrm{every}\:\mathrm{6}.\mathrm{2}\:\mathrm{seconds} \\ $$$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{frequency}\:\mathrm{and} \\ $$$$\:\mathrm{the}\:\mathrm{speeed}\:\mathrm{of}\:\mathrm{these}\:\mathrm{waves}. \\ $$

Question Number 140863 Answers: 1 Comments: 4

1+(√3^a ) = 2(√2^a )

$$\mathrm{1}+\sqrt{\mathrm{3}^{\boldsymbol{{a}}} }\:=\:\mathrm{2}\sqrt{\mathrm{2}^{\boldsymbol{{a}}} } \\ $$

Question Number 140888 Answers: 1 Comments: 0

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