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AllQuestion and Answers: Page 1781

Question Number 30749 Answers: 0 Comments: 0

let f(x)=arcsinx with x∈[0,1] 1) prove that (1−x^2 )f^(′′) (x) −xf^′ (x)=0 2)prove that (1−x^2 )f^((n+2)) (x)=(2n+1)x f^((n+1)) (x) +n^2 f^((n)) (x) 3) prove that f^((n)) (x) ≥0 ∀n .

$${let}\:{f}\left({x}\right)={arcsinx}\:{with}\:{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{''} \left({x}\right)\:−{xf}^{'} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)=\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+{n}^{\mathrm{2}} {f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left({n}\right)} \left({x}\right)\:\geqslant\mathrm{0}\:\forall{n}\:. \\ $$

Question Number 30748 Answers: 0 Comments: 0

let a>0 and b>0 find lim_(x→0^+ ) ( ((a^x +b^x )/2))^(1/x) .

$${let}\:{a}>\mathrm{0}\:{and}\:{b}>\mathrm{0}\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\left(\:\:\frac{{a}^{{x}} \:+{b}^{{x}} }{\mathrm{2}}\right)^{\frac{\mathrm{1}}{{x}}} \:\:. \\ $$

Question Number 30747 Answers: 0 Comments: 0

let f(x)=(√(1+x^2 )) 1)find a d.e.wich verify f(x) 2) prove that ∀x∈R ,∀n∈N (1+x^2 )f^((n+2)) (x)+(2n+1)x f^((n+1)) (x) +(n^2 −1)f^((n)) (x)=0 3) prove that f^((2n+1)) (0)=0 ∀n∈ N

$${let}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right){find}\:{a}\:{d}.{e}.{wich}\:{verify}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall{x}\in{R}\:,\forall{n}\in{N} \\ $$$$\left(\mathrm{1}+{x}^{\mathrm{2}} \right){f}^{\left({n}+\mathrm{2}\right)} \left({x}\right)+\left(\mathrm{2}{n}+\mathrm{1}\right){x}\:{f}^{\left({n}+\mathrm{1}\right)} \left({x}\right)\:+\left({n}^{\mathrm{2}} −\mathrm{1}\right){f}^{\left({n}\right)} \left({x}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\:{f}^{\left(\mathrm{2}{n}+\mathrm{1}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:\forall{n}\in\:{N} \\ $$

Question Number 30745 Answers: 0 Comments: 0

let U_n ={z∈C/z^n =1} simlify A_n = Σ_(α∈U_n ) (x+α)^n and B_n =Σ_(α∈ U_n ) (x−α)^n .

$${let}\:\:{U}_{{n}} =\left\{{z}\in{C}/{z}^{{n}} =\mathrm{1}\right\}\:\:{simlify} \\ $$$${A}_{{n}} =\:\sum_{\alpha\in{U}_{{n}} } \:\left({x}+\alpha\right)^{{n}} \:{and}\:{B}_{{n}} =\sum_{\alpha\in\:{U}_{{n}} } \:\:\left({x}−\alpha\right)^{{n}} . \\ $$

Question Number 30744 Answers: 0 Comments: 0

let p(x)= (x−1)^n −x^n +1 with n integr find n in ordre that p(x) have a double root.

$${let}\:{p}\left({x}\right)=\:\left({x}−\mathrm{1}\right)^{{n}} \:−{x}^{{n}} \:+\mathrm{1}\:\:{with}\:{n}\:{integr}\:{find}\:{n} \\ $$$${in}\:{ordre}\:{that}\:{p}\left({x}\right)\:{have}\:{a}\:{double}\:{root}. \\ $$

Question Number 30743 Answers: 0 Comments: 1

decompose inside R[x] p(x)=x^(2n+1) −1 then find Π_(k=1) ^n sin( ((kπ)/(2n+1))) .

$${decompose}\:{inside}\:{R}\left[{x}\right]\:\:{p}\left({x}\right)={x}^{\mathrm{2}{n}+\mathrm{1}} \:−\mathrm{1}\:{then}\:{find} \\ $$$$\prod_{{k}=\mathrm{1}} ^{{n}} \:{sin}\left(\:\frac{{k}\pi}{\mathrm{2}{n}+\mathrm{1}}\right)\:. \\ $$

Question Number 30742 Answers: 0 Comments: 0

prove that ∀p ∈N it exist one polynomial Q_(2p) / sin(2p+1)θ=sin^(2p+1) θ Q_(2p) (cotanθ) and degQ_(2p) =2p 2) prove that Π_(k=1) ^p tan(((kπ)/(2p+1)))=(√(2p+1)) .

$${prove}\:{that}\:\forall{p}\:\in{N}\:\:{it}\:{exist}\:{one}\:{polynomial}\:{Q}_{\mathrm{2}{p}} \:/ \\ $$$${sin}\left(\mathrm{2}{p}+\mathrm{1}\right)\theta={sin}^{\mathrm{2}{p}+\mathrm{1}} \theta\:{Q}_{\mathrm{2}{p}} \:\left({cotan}\theta\right)\:{and}\:{degQ}_{\mathrm{2}{p}} =\mathrm{2}{p} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{p}} \:{tan}\left(\frac{{k}\pi}{\mathrm{2}{p}+\mathrm{1}}\right)=\sqrt{\mathrm{2}{p}+\mathrm{1}}\:. \\ $$$$ \\ $$

Question Number 30741 Answers: 0 Comments: 0

let give D= R_+ ^2 −{(0,0)} and α from R let C_1 ={(x,y)∈ D/0<x^2 +y^2 ≤1 } C_2 ={(x,y) ∈D / x^2 +y^2 ≥1} study the convergence of I= ∫∫_C_1 ((dxdy)/(((√(x^2 +y^2 )) )^α )) and J=∫∫_C_2 ((dxdy)/(((√(x^2 +y^2 )) )^α )) .

$${let}\:{give}\:{D}=\:{R}_{+} ^{\mathrm{2}} \:−\left\{\left(\mathrm{0},\mathrm{0}\right)\right\}\:{and}\:\alpha\:{from}\:{R}\:{let} \\ $$$${C}_{\mathrm{1}} =\left\{\left({x},{y}\right)\in\:{D}/\mathrm{0}<{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \leqslant\mathrm{1}\:\right\} \\ $$$${C}_{\mathrm{2}} \:=\left\{\left({x},{y}\right)\:\in{D}\:/\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \geqslant\mathrm{1}\right\}\:{study}\:{the}\:{convergence}\:{of} \\ $$$${I}=\:\int\int_{{C}_{\mathrm{1}} } \:\:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:\right)^{\alpha} }\:\:{and}\:{J}=\int\int_{{C}_{\mathrm{2}} } \:\:\frac{{dxdy}}{\left(\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:}\:\right)^{\alpha} }\:. \\ $$

Question Number 30738 Answers: 1 Comments: 1

A boy can swim with a speed of 26m/s in still water.He wants to swim across a 150m river from a point A to point B which is directly opposite the other side of the river.The river flows with a speed of 10m/s. i)if he always swim in the direction parallel to AB,find how far he lands downstream of B. ii)In what direction relative to the bank must he swim so as to cross directly from A to B.

$${A}\:{boy}\:{can}\:{swim}\:{with}\:{a}\:{speed}\:{of} \\ $$$$\mathrm{26}{m}/{s}\:{in}\:{still}\:{water}.{He}\:{wants}\:{to} \\ $$$${swim}\:{across}\:{a}\:\mathrm{150}{m}\:{river}\:{from} \\ $$$${a}\:{point}\:{A}\:{to}\:{point}\:{B}\:{which}\:{is}\: \\ $$$${directly}\:{opposite}\:{the}\:{other}\:{side} \\ $$$${of}\:{the}\:{river}.{The}\:{river}\:{flows}\:{with} \\ $$$${a}\:{speed}\:{of}\:\mathrm{10}{m}/{s}. \\ $$$$\left.{i}\right){if}\:{he}\:{always}\:{swim}\:{in}\:{the}\: \\ $$$${direction}\:{parallel}\:{to}\:{AB},{find}\:{how} \\ $$$${far}\:{he}\:{lands}\:{downstream}\:{of}\:{B}. \\ $$$$\left.{ii}\right){In}\:{what}\:{direction}\:{relative}\:{to} \\ $$$${the}\:{bank}\:{must}\:{he}\:{swim}\:{so}\:{as}\:{to} \\ $$$${cross}\:{directly}\:{from}\:{A}\:{to}\:{B}. \\ $$

Question Number 30737 Answers: 0 Comments: 1

∫(1/(x^2 +ln x))dx

$$\int\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{ln}\:{x}}{dx} \\ $$

Question Number 30739 Answers: 0 Comments: 0

let (u_n ) / u_1 =1−i and ∀p∈{2,3,...n} u_p =u_(p−1) j with j=e^(i((2π)/3)) 1)verify that u_1 +u_2 +u_3 =0 2)prove that ∀p∈ {4,5,...,n} u_p =u_(p−3) 3)find the value of S_n =Σ_(i=1) ^n u_i 4)calculate α_n = Σ_(p=0) ^(n−1) cos(−(π/4) +((2pπ)/3)) and β_n = Σ_(p=0) ^(n−1) sin(−(π/4) +((2pπ)/3)).

$${let}\:\left({u}_{{n}} \right)\:/\:{u}_{\mathrm{1}} =\mathrm{1}−{i}\:{and}\:\:\forall{p}\in\left\{\mathrm{2},\mathrm{3},...{n}\right\}\:{u}_{{p}} ={u}_{{p}−\mathrm{1}} {j}\:{with} \\ $$$${j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right){verify}\:{that}\:{u}_{\mathrm{1}} \:+{u}_{\mathrm{2}} \:+{u}_{\mathrm{3}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{p}\in\:\left\{\mathrm{4},\mathrm{5},...,{n}\right\}\:\:{u}_{{p}} ={u}_{{p}−\mathrm{3}} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:{S}_{{n}} \:=\sum_{{i}=\mathrm{1}} ^{{n}} \:{u}_{{i}} \\ $$$$\left.\mathrm{4}\right){calculate}\:\:\alpha_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right)\:{and} \\ $$$$\beta_{{n}} =\:\sum_{{p}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{sin}\left(−\frac{\pi}{\mathrm{4}}\:+\frac{\mathrm{2}{p}\pi}{\mathrm{3}}\right). \\ $$

Question Number 30719 Answers: 1 Comments: 4

Question Number 30711 Answers: 1 Comments: 0

Question Number 33450 Answers: 1 Comments: 1

Question Number 30704 Answers: 0 Comments: 0

Question Number 30665 Answers: 0 Comments: 0

find ∫_0 ^π (dx/(1+cos(2x) +sin(2x))) .

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)\:+{sin}\left(\mathrm{2}{x}\right)}\:. \\ $$

Question Number 30657 Answers: 0 Comments: 2

lim_(x→∞) e^(−(x^2 /2))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \\ $$

Question Number 30646 Answers: 1 Comments: 2

Question Number 30631 Answers: 0 Comments: 0

Question Number 30628 Answers: 1 Comments: 0

Question Number 30627 Answers: 1 Comments: 0

Question Number 30687 Answers: 1 Comments: 0

Given that LCM(A,B,C)=252 LCM(A,B)=36 & LCM(A,C)=63; then: LCM(B,C)=? Pl determine all possible answers.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{LCM}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right)=\mathrm{252} \\ $$$$\mathrm{LCM}\left(\mathrm{A},\mathrm{B}\right)=\mathrm{36}\:\&\:\mathrm{LCM}\left(\mathrm{A},\mathrm{C}\right)=\mathrm{63}; \\ $$$$\mathrm{then}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{LCM}\left(\mathrm{B},\mathrm{C}\right)=? \\ $$$$\mathrm{Pl}\:\mathrm{determine}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{answers}. \\ $$

Question Number 30615 Answers: 1 Comments: 0

Question Number 30614 Answers: 1 Comments: 0

Consider that two cars are accelerating along the same road and if the distance between them was observed to be increasing,what deduction can you make as regards the acceleration? a)it implies that the trailing car has the smaller acceleration b)it implies that the two cars are accelerating at the same rate c)it implies nothing about the acceleration d)it implies that the leading car has the greater acceleration.

$${Consider}\:{that}\:{two}\:{cars}\:{are}\: \\ $$$${accelerating}\:{along}\:{the}\:{same}\:{road} \\ $$$${and}\:{if}\:{the}\:{distance}\:{between}\:{them} \\ $$$${was}\:{observed}\:{to}\:{be}\:{increasing},{what} \\ $$$${deduction}\:{can}\:{you}\:{make}\:{as}\:{regards} \\ $$$${the}\:{acceleration}? \\ $$$$\left.{a}\right){it}\:{implies}\:{that}\:{the}\:{trailing}\:{car} \\ $$$${has}\:{the}\:{smaller}\:{acceleration} \\ $$$$\left.{b}\right){it}\:{implies}\:{that}\:{the}\:{two}\:{cars}\:{are} \\ $$$${accelerating}\:{at}\:{the}\:{same}\:{rate} \\ $$$$\left.{c}\right){it}\:{implies}\:{nothing}\:{about}\:{the} \\ $$$${acceleration} \\ $$$$\left.{d}\right){it}\:{implies}\:{that}\:{the}\:{leading}\:{car}\: \\ $$$${has}\:{the}\:{greater}\:{acceleration}. \\ $$

Question Number 30613 Answers: 1 Comments: 1

A car negotiates a bend of radius 20m with an acceleration of 12m/s^2 .What is the maximum speed the car can attain without skidding?

$${A}\:{car}\:{negotiates}\:{a}\:{bend}\:{of}\:{radius} \\ $$$$\mathrm{20}{m}\:{with}\:{an}\:{acceleration}\:{of}\: \\ $$$$\mathrm{12}{m}/{s}^{\mathrm{2}} .{What}\:{is}\:{the}\:{maximum} \\ $$$${speed}\:{the}\:{car}\:{can}\:{attain}\:{without} \\ $$$${skidding}? \\ $$

Question Number 30612 Answers: 0 Comments: 0

A car negotiates a bend of radius 20m with an acceleration of 12m/s^2 .What is the maximum speed the car can attain without skidding?

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