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Question Number 33701 Answers: 0 Comments: 0

let Σ f_n (x) with f_n (x) = ((sin(nx))/(n^2 (n+1))) and S its sum x∈[−π,π] prove that ∀(x,y)∈[−π,π]^2 x≠y ⇒∣S(x)−S(y)∣<∣x−y∣ .

$${let}\:\Sigma\:{f}_{{n}} \left({x}\right)\:{with}\:{f}_{{n}} \left({x}\right)\:=\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)}\:\:{and}\:{S}\:{its}\:{sum} \\ $$$${x}\in\left[−\pi,\pi\right]\:{prove}\:{that}\:\forall\left({x},{y}\right)\in\left[−\pi,\pi\right]^{\mathrm{2}} \\ $$$${x}\neq{y}\:\Rightarrow\mid{S}\left({x}\right)−{S}\left({y}\right)\mid<\mid{x}−{y}\mid\:. \\ $$

Question Number 33699 Answers: 0 Comments: 3

let S(x)=Σ_(n=0) ^∞ f_n (x) with f_n (x)= (((−1)^n )/(n!(x+n))) x∈]0,+∞[ 1) prove that S id defined .calculate S(1) and prove that ∀x>0 xS(x) −S(x+1) =(1/e) 2) prove that S is C^∞ on R^(+∗) 3) prove that S(x) ∼ (1/x) (x→0^+ ) .

$${let}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:{f}_{{n}} \left({x}\right)\:\:{with}\:{f}_{{n}} \left({x}\right)=\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!\left({x}+{n}\right)} \\ $$$$\left.{x}\in\right]\mathrm{0},+\infty\left[\right. \\ $$$$\left.\mathrm{1}\right)\:\:{prove}\:{that}\:{S}\:{id}\:{defined}\:.{calculate}\:{S}\left(\mathrm{1}\right)\:{and} \\ $$$${prove}\:{that}\:\forall{x}>\mathrm{0}\:\:{xS}\left({x}\right)\:−{S}\left({x}+\mathrm{1}\right)\:=\frac{\mathrm{1}}{{e}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{S}\:{is}\:{C}^{\infty} \:{on}\:{R}^{+\ast} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{S}\left({x}\right)\:\sim\:\frac{\mathrm{1}}{{x}}\:\left({x}\rightarrow\mathrm{0}^{+} \right)\:. \\ $$

Question Number 33698 Answers: 0 Comments: 0

let f_n (x)= n^x e^(−nx) with x>0 1) study the simple and uniform convervence for Σ f_n (x) 2) let S(x)= Σ_(n=1) ^∞ f_n (x).prove that S(x) ∼ (1/x) ( x→0^+ )

$${let}\:\:{f}_{{n}} \left({x}\right)=\:{n}^{{x}} \:{e}^{−{nx}} \:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:{and}\:{uniform}\:{convervence}\:{for} \\ $$$$\Sigma\:\:{f}_{{n}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{f}_{{n}} \left({x}\right).{prove}\:{that} \\ $$$${S}\left({x}\right)\:\sim\:\frac{\mathrm{1}}{{x}}\:\left(\:{x}\rightarrow\mathrm{0}^{+} \right) \\ $$

Question Number 33695 Answers: 0 Comments: 1

find lim_(n→+∞) ∫_0 ^∞ (e^(−(x/n)) /(1+x^2 ))dx.

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{e}^{−\frac{{x}}{{n}}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}. \\ $$

Question Number 33694 Answers: 0 Comments: 1

calculate lim_(n→+∞) ∫_0 ^∞ (dx/(x^n +e^x )) .

$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{{x}^{{n}} \:\:+{e}^{{x}} }\:\:. \\ $$

Question Number 33690 Answers: 1 Comments: 0

∫(dx/((x)^(1/3) +(√x)))

$$\int\frac{\boldsymbol{\mathrm{dx}}}{\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}}+\sqrt{\boldsymbol{\mathrm{x}}}} \\ $$

Question Number 33689 Answers: 2 Comments: 1

∫(x/(x^3 +1))dx

$$\int\frac{{x}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx} \\ $$

Question Number 33688 Answers: 1 Comments: 0

given that f(x)=(1/2)(10^x +10^(−x) ) prove that 2f(x) f(y)=f(x+y)+f(x−y)

Question Number 33681 Answers: 0 Comments: 0

Let a, b, c are positive real numbers such that (1/a) + (1/b) + (1/c) = 3 . Prove that : a + b + c + (4/(1 + (((abc)^2 ))^(1/3) )) ≥ 5

$${Let}\:\:{a},\:{b},\:{c}\:\:\:{are}\:\:{positive}\:{real}\:\:{numbers}\:\:{such}\:\:{that}\:\:\:\frac{\mathrm{1}}{{a}}\:+\:\frac{\mathrm{1}}{{b}}\:+\:\frac{\mathrm{1}}{{c}}\:\:=\:\:\mathrm{3}\:. \\ $$$${Prove}\:{that}\::\:\:\:{a}\:+\:{b}\:+\:{c}\:\:+\:\:\frac{\mathrm{4}}{\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\left({abc}\right)^{\mathrm{2}} }}\:\:\:\geqslant\:\:\mathrm{5} \\ $$

Question Number 33677 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((xlnx)/(x−1))dx .

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{xlnx}}{{x}−\mathrm{1}}{dx}\:. \\ $$

Question Number 33676 Answers: 0 Comments: 0

How fast is the height of a balloon changing when 500m away at an angle of π/4 rad and the angle is increasing by 0.2rad/min

$${How}\:{fast}\:{is}\:{the}\:{height}\:{of}\:{a}\:{balloon} \\ $$$${changing}\:{when}\:\mathrm{500}{m}\:{away}\:{at}\:{an} \\ $$$${angle}\:{of}\:\pi/\mathrm{4}\:{rad}\:{and}\:{the}\:{angle}\:{is} \\ $$$${increasing}\:{by}\:\mathrm{0}.\mathrm{2}{rad}/{min} \\ $$

Question Number 33671 Answers: 0 Comments: 0

Question Number 33663 Answers: 0 Comments: 0

Qn:(a) given A and B are disjoint sets shade in venn diagram (i)B∩C (ii)A−(B∪C) (iii)(B∪C)−A Qn:(b) Given that ∣A∣=19 ∣B∣=23 ∣C∣=24 ∣A∪B∣=30 ∣(A∩B)−C∣=5 ∣(B∩C)∣=10 ∣(A∪B)^′ ∣=20 and ∣C−(A∪B)∣=10 then find (i)∣(A∩C)∣ (ii)∣(B−′C)∣′ (iii)∣𝛍∣ (iv)∣(B′∪C)∩A′∣

Question Number 33660 Answers: 4 Comments: 0

from sinhu=tan𝛝 prove that (i)tanh(u/2)=tan(𝛝/2) (ii)coshu=sec𝛝 (iii)u=log(sec𝛝+tan𝛝) (iv)tanhu=sin𝛝

Question Number 33659 Answers: 1 Comments: 0

solve for x and y coshy−7sinhx=3 and coshy−3sinh^2 x=2

$$\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{y}} \\ $$$$\:\boldsymbol{\mathrm{cosh}{y}}−\mathrm{7}\boldsymbol{\mathrm{sinh}{x}}=\mathrm{3}\:\boldsymbol{\mathrm{and}} \\ $$$$\:\boldsymbol{\mathrm{cosh}{y}}−\mathrm{3}\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \boldsymbol{{x}}=\mathrm{2} \\ $$

Question Number 33658 Answers: 1 Comments: 1

If f(x)= x^3 −3x+1 then find number of different real solutions of f(f(x))=0 ?

$$\boldsymbol{{I}}{f}\:{f}\left({x}\right)=\:{x}^{\mathrm{3}} −\mathrm{3}{x}+\mathrm{1} \\ $$$${then}\:{find}\:{number}\:{of}\:{different}\:\:\:{real} \\ $$$${solutions}\:{of}\:{f}\left({f}\left({x}\right)\right)=\mathrm{0}\:? \\ $$

Question Number 33651 Answers: 1 Comments: 0

Let function f(x) be defined as f(x)= x^2 +bx+c , where b,c∈R . And f(1) − 2f(5) +f(9) =32. Find no. of ordered pairs (b,c) such that ∣f(x)∣≤8 ∀ x∈ [1,9] ?

$${Let}\:{function}\:{f}\left({x}\right)\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)=\:{x}^{\mathrm{2}} +{bx}+{c}\:,\:{where}\:{b},{c}\in{R}\:. \\ $$$${And}\:{f}\left(\mathrm{1}\right)\:−\:\mathrm{2}{f}\left(\mathrm{5}\right)\:+{f}\left(\mathrm{9}\right)\:=\mathrm{32}. \\ $$$${Find}\:{no}.\:{of}\:{ordered}\:{pairs}\:\left({b},{c}\right) \\ $$$${such}\:{that}\:\mid{f}\left({x}\right)\mid\leqslant\mathrm{8}\:\forall\:{x}\in\:\left[\mathrm{1},\mathrm{9}\right]\:? \\ $$

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