Question and Answers Forum

AllQuestion and Answers: Page 1697

Question Number 33649 Answers: 1 Comments: 3

Consider f:R^+ →R such that f(3)=1 for a∈R^+ and f(x).f(y) + f((3/x)).f((3/y)) = 2f(xy) ∀ x,y ∈ R^+ . Then find f(x) ?

$${Consider}\:{f}:{R}^{+} \rightarrow{R}\:{such}\:{that} \\ $$$${f}\left(\mathrm{3}\right)=\mathrm{1}\:{for}\:{a}\in{R}^{+} \:{and}\: \\ $$$${f}\left({x}\right).{f}\left({y}\right)\:+\:{f}\left(\frac{\mathrm{3}}{{x}}\right).{f}\left(\frac{\mathrm{3}}{{y}}\right)\:=\:\mathrm{2}{f}\left({xy}\right) \\ $$$$\forall\:{x},{y}\:\in\:{R}^{+} .\:{Then}\:{find}\:{f}\left({x}\right)\:? \\ $$

Question Number 33644 Answers: 1 Comments: 0

∫(sec^2 x)e^(tanx) dx

$$\:\int\left(\boldsymbol{\mathrm{sec}}^{\mathrm{2}} \boldsymbol{{x}}\right)\boldsymbol{{e}}^{\boldsymbol{\mathrm{tan}{x}}} \boldsymbol{{dx}} \\ $$

Question Number 33643 Answers: 1 Comments: 0

∫((2x+3)/(x^2 +4))dx

$$\:\int\frac{\mathrm{2}\boldsymbol{{x}}+\mathrm{3}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}}\boldsymbol{{dx}} \\ $$

Question Number 33629 Answers: 1 Comments: 2

Question Number 33628 Answers: 0 Comments: 4

Question Number 33622 Answers: 0 Comments: 0

Question Number 33619 Answers: 1 Comments: 3

∫x^(5/2) (1−x)^(3/2) dx

$$\int{x}^{\mathrm{5}/\mathrm{2}} \left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} {dx} \\ $$

Question Number 33616 Answers: 1 Comments: 4

please help me please is there any app for practicing calculus for a CBT exam.I mean one that has a timer so I can asses my speed.

$${please}\:{help}\:{me} \\ $$$$ \\ $$$${please}\:{is}\:{there}\:{any}\:{app}\:{for}\:{practicing} \\ $$$${calculus}\:{for}\:\:{a}\:{CBT}\:{exam}.{I}\:{mean} \\ $$$${one}\:{that}\:{has}\:{a}\:{timer}\:{so}\:{I}\:{can}\:{asses} \\ $$$${my}\:{speed}. \\ $$

Question Number 33599 Answers: 1 Comments: 2

calculatef(a)= ∫_(−a) ^a (dx/((t^2 +x^2 )^(3/2) )) with a>0 .

$${calculatef}\left({a}\right)=\:\:\int_{−{a}} ^{{a}} \:\:\:\:\frac{{dx}}{\left({t}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\:\:{with}\:{a}>\mathrm{0}\:. \\ $$

Question Number 33597 Answers: 0 Comments: 0

study and give the graph of f(x) =e^(2/(lnx)) .

$${study}\:{and}\:{give}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:={e}^{\frac{\mathrm{2}}{{lnx}}} \:. \\ $$

Question Number 33596 Answers: 0 Comments: 0

1) prove that ∀(a,b)∈R^2 ∣sinb −sina∣≤∣b−a∣ 2)let give the sequence x_0 =0 and x_(n+1) =a +(1/2)sin(x_n ) prove that for m≥n ∣x_m −x_n ∣ ≤ ((∣a∣)/2^(n−1) ) 3) prove that (x_n ) is convergent and its limit is solution of the equation x = a +(1/2) sinx .

$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\forall\left({a},{b}\right)\in{R}^{\mathrm{2}} \:\:\:\:\mid{sinb}\:−{sina}\mid\leqslant\mid{b}−{a}\mid \\ $$$$\left.\mathrm{2}\right){let}\:{give}\:{the}\:{sequence}\:\:{x}_{\mathrm{0}} =\mathrm{0}\:{and} \\ $$$${x}_{{n}+\mathrm{1}} ={a}\:+\frac{\mathrm{1}}{\mathrm{2}}{sin}\left({x}_{{n}} \right)\:{prove}\:{that}\:{for}\:{m}\geqslant{n} \\ $$$$\mid{x}_{{m}} \:−{x}_{{n}} \mid\:\leqslant\:\:\frac{\mid{a}\mid}{\mathrm{2}^{{n}−\mathrm{1}} } \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\left({x}_{{n}} \right)\:{is}\:{convergent}\:{and}\:{its}\:{limit}\:{is}\:{solution} \\ $$$${of}\:{the}\:{equation}\:\:{x}\:=\:{a}\:+\frac{\mathrm{1}}{\mathrm{2}}\:{sinx}\:. \\ $$

Question Number 33595 Answers: 0 Comments: 0

find lim_(x→0) ((ln(1+sinx) −x(√(1−x)))/(sinx −shx)) .

$${find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\frac{{ln}\left(\mathrm{1}+{sinx}\right)\:−{x}\sqrt{\mathrm{1}−{x}}}{{sinx}\:−{shx}}\:\:. \\ $$

Question Number 33594 Answers: 0 Comments: 1

calculate lim_(x→1^− ) (1/((1−x)^α ))(arcsinx −(π/2)) .

$${calculate}\:\:{lim}_{{x}\rightarrow\mathrm{1}^{−} } \:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\alpha} }\left({arcsinx}\:−\frac{\pi}{\mathrm{2}}\right)\:. \\ $$

Question Number 33593 Answers: 0 Comments: 0

calculate lim_(x→0) ((2(1−cosx)sinx −x^3 (1−x^2 )^(1/4) )/(sin^5 x −x^5 ))

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:\:\:\frac{\mathrm{2}\left(\mathrm{1}−{cosx}\right){sinx}\:−{x}^{\mathrm{3}} \:\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} }{{sin}^{\mathrm{5}} {x}\:−{x}^{\mathrm{5}} } \\ $$

Question Number 33592 Answers: 0 Comments: 0

let f(x) =e^(−x^2 ) 1) prove that f^((n)) (x) = p_n (x).e^(−x^2 ) where p_n is a polynome with deg=n 2) prove that ∀ n≥1 p_(n+1) (x) +α(x)p_n (x) +β(n)p_(n−1) (x) =0 find α and β 3)calculate p_0 ,p_1 ,p_2 ,p_3 4) calculate p_n ^(′′) (x) interms of p^′ (x) and p_n (x).

$${let}\:{f}\left({x}\right)\:={e}^{−{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}^{\left({n}\right)} \left({x}\right)\:=\:{p}_{{n}} \left({x}\right).{e}^{−{x}^{\mathrm{2}} } \:\:\:{where}\:{p}_{{n}} {is}\:{a}\:{polynome} \\ $$$${with}\:{deg}={n} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\geqslant\mathrm{1}\: \\ $$$${p}_{{n}+\mathrm{1}} \left({x}\right)\:+\alpha\left({x}\right){p}_{{n}} \left({x}\right)\:+\beta\left({n}\right){p}_{{n}−\mathrm{1}} \left({x}\right)\:=\mathrm{0}\:\:{find}\:\alpha\:{and}\:\beta \\ $$$$\left.\mathrm{3}\right){calculate}\:{p}_{\mathrm{0}} ,{p}_{\mathrm{1}} ,{p}_{\mathrm{2}} ,{p}_{\mathrm{3}} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{p}_{{n}} ^{''} \:\left({x}\right)\:{interms}\:{of}\:{p}^{'} \left({x}\right)\:{and}\:{p}_{{n}} \left({x}\right). \\ $$

Question Number 33591 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (2/(n^3 +3n^2 +2n)) .

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{2}}{{n}^{\mathrm{3}} \:\:+\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{2}{n}}\:. \\ $$

Question Number 33590 Answers: 0 Comments: 1

let α >1 calculate f(α) = ∫_α ^(+∞) ((x^2 −x+1)/((x−1)^2 (x+1)^2 )) dx .

$${let}\:\alpha\:>\mathrm{1}\:\:{calculate}\:{f}\left(\alpha\right)\:=\:\int_{\alpha} ^{+\infty} \:\:\frac{{x}^{\mathrm{2}} −{x}+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\:. \\ $$

Question Number 33589 Answers: 0 Comments: 1

1) decompose F(x) = (1/((x^2 +4)(x−3)^2 )) 2) calculate ∫_4 ^(+∞) (dx/((x^2 +4)(x−3)^2 )) .

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)\:=\:\:\:\frac{\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{4}} ^{+\infty} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)\left({x}−\mathrm{3}\right)^{\mathrm{2}} }\:. \\ $$

Question Number 33588 Answers: 1 Comments: 1

find the value of Σ_(n=1) ^∞ (1/(n^2 (n+1)(2n+1))) .