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Question Number 208418    Answers: 1   Comments: 0

u_(n+1) = u_n −u_n ^3 ; u_0 ∈ ]0, 1[ . show that u_n ∈ ]0, 1[ . show that u_n converges to 0 v_n = (1/u_(n+1) ^2 ) − (1/u_n ^2 ) . express v_n interms of u_n . show that v_n converges to 2 f(x) = ((2−x)/((1−x)^2 )) . show that f is increasing and deduce that v_n is decreasing . show that v_n ≥ 2

$$\left.{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} −{u}_{{n}} ^{\mathrm{3}} \:;\:{u}_{\mathrm{0}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$\left..\:{show}\:{that}\:{u}_{{n}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$.\:{show}\:{that}\:{u}_{{n}} \:{converges}\:{to}\:\mathrm{0} \\ $$$${v}_{{n}} \:=\:\frac{\mathrm{1}}{{u}_{{n}+\mathrm{1}} ^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{{u}_{{n}} ^{\mathrm{2}} } \\ $$$$.\:{express}\:{v}_{{n}} \:{interms}\:{of}\:{u}_{{n}} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:{converges}\:{to}\:\mathrm{2} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{2}−{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} } \\ $$$$.\:{show}\:{that}\:{f}\:{is}\:{increasing}\:{and}\:{deduce}\:{that}\: \\ $$$${v}_{{n}} \:{is}\:{decreasing} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:\geqslant\:\mathrm{2} \\ $$

Question Number 208412    Answers: 2   Comments: 0

Question Number 208409    Answers: 2   Comments: 0

Find: ∫_0 ^( 2) ∣1 − x∣ dx = ?

$$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\mid\mathrm{1}\:−\:\mathrm{x}\mid\:\mathrm{dx}\:=\:? \\ $$

Question Number 208398    Answers: 3   Comments: 0

write z = (1/( (√3)+i)) in e^(iθ)

$${write}\:{z}\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}+{i}}\:{in}\:{e}^{{i}\theta} \\ $$

Question Number 208395    Answers: 0   Comments: 0

Question Number 208387    Answers: 0   Comments: 2

Find the value of the scalar for which the vector a = 3i + 2j is perpendicular to b = 4i - 3j

Find the value of the scalar for which the vector a = 3i + 2j is perpendicular to b = 4i - 3j

Question Number 208385    Answers: 1   Comments: 3

Question Number 208384    Answers: 2   Comments: 0

$$\:\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 208381    Answers: 1   Comments: 0

g(x) = lnx^2 f(x) = ((x + 25))^(1/3) Find: lim_(x→e) (f(g(x)) = ?

$$\mathrm{g}\left(\mathrm{x}\right)\:=\:\mathrm{lnx}^{\mathrm{2}} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:+\:\mathrm{25}} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\boldsymbol{\mathrm{e}}} {\mathrm{lim}}\:\left(\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\:=\:?\right. \\ $$

Question Number 208377    Answers: 2   Comments: 1

sin x − sin (π/6) > 0 x = ?

$$\mathrm{sin}\:\mathrm{x}\:−\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}}\:>\:\mathrm{0} \\ $$$$\mathrm{x}\:=\:? \\ $$

Question Number 208370    Answers: 2   Comments: 3

if (fof)(x)=f(x)+x and f(1)=1 find fofofofofofofofofof(1)

$${if}\:\:\:\left({fof}\right)\left({x}\right)={f}\left({x}\right)+{x}\:\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{1}\:\:\: \\ $$$${find}\:\:{fofofofofofofofofof}\left(\mathrm{1}\right) \\ $$

Question Number 208367    Answers: 0   Comments: 0

1. Find the length of each of the following (a) {x : −3 < x < 7} (b) {x : 2 ≤ x ≤ 6} ∪ {−3 ≤ x ≤ −1} (c) {x : −2 ≤ x < 5} ∪ {1 < x ≤ 7} 2. Let I=(a, b). Prove that I is measurable and m(I) = L(I).

$$\mathrm{1}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\:\:\:\:\:\left(\mathrm{a}\right)\:\left\{\mathrm{x}\::\:−\mathrm{3}\:<\:\mathrm{x}\:<\:\mathrm{7}\right\} \\ $$$$\:\:\:\:\:\left(\mathrm{b}\right)\:\left\{\mathrm{x}\::\:\mathrm{2}\:\leqslant\:\mathrm{x}\:\leqslant\:\mathrm{6}\right\}\:\cup\:\left\{−\mathrm{3}\:\leqslant\:\mathrm{x}\:\leqslant\:−\mathrm{1}\right\} \\ $$$$\:\:\:\:\:\left(\mathrm{c}\right)\:\left\{\mathrm{x}\::\:−\mathrm{2}\:\leqslant\:\mathrm{x}\:<\:\mathrm{5}\right\}\:\cup\:\left\{\mathrm{1}\:<\:\mathrm{x}\:\leqslant\:\mathrm{7}\right\} \\ $$$$ \\ $$$$\mathrm{2}.\:\mathrm{Let}\:\mathrm{I}=\left(\mathrm{a},\:\mathrm{b}\right).\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{I}\:\mathrm{is}\:\mathrm{measurable} \\ $$$$\mathrm{and}\:\mathrm{m}\left(\mathrm{I}\right)\:=\:\mathrm{L}\left(\mathrm{I}\right). \\ $$

Question Number 208362    Answers: 4   Comments: 0

P(x) is polynomial P(x) = ((x^4 + 2ax^3 − bx − 5)/((x + 1)^2 )) Find: b = ?

$$\mathrm{P}\left(\mathrm{x}\right)\:\:\mathrm{is}\:\mathrm{polynomial} \\ $$$$\mathrm{P}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{2ax}^{\mathrm{3}} \:−\:\mathrm{bx}\:−\:\mathrm{5}}{\left(\mathrm{x}\:+\:\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{b}}\:=\:? \\ $$

Question Number 208566    Answers: 0   Comments: 0

Question Number 208359    Answers: 1   Comments: 1

Question Number 208354    Answers: 0   Comments: 1

Question Number 208342    Answers: 2   Comments: 0

a,b,c∈N x = 4(2a+5) = 6(b+9) = 9(c−1) find: min(x+a+b+c) = ?

$$\mathrm{a},\mathrm{b},\mathrm{c}\in\mathbb{N} \\ $$$$\mathrm{x}\:=\:\mathrm{4}\left(\mathrm{2a}+\mathrm{5}\right)\:=\:\mathrm{6}\left(\mathrm{b}+\mathrm{9}\right)\:=\:\mathrm{9}\left(\mathrm{c}−\mathrm{1}\right) \\ $$$$\mathrm{find}:\:\:\:\boldsymbol{\mathrm{min}}\left(\mathrm{x}+\mathrm{a}+\mathrm{b}+\mathrm{c}\right)\:=\:? \\ $$

Question Number 208344    Answers: 4   Comments: 4

Question Number 208338    Answers: 0   Comments: 0

Question Number 208335    Answers: 1   Comments: 0

∫_(−1) ^1 (√(1−t^4 ))dt

$$\int_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\mathrm{1}−{t}^{\mathrm{4}} }{dt} \\ $$

Question Number 208334    Answers: 0   Comments: 1

∫_0 ^(4/π) ln(cosx)dx

$$\int_{\mathrm{0}} ^{\frac{\mathrm{4}}{\pi}} {ln}\left({cosx}\right){dx} \\ $$

Question Number 208332    Answers: 1   Comments: 0

Question Number 208328    Answers: 0   Comments: 0

Question Number 208327    Answers: 1   Comments: 0

Question Number 208322    Answers: 1   Comments: 0

Question Number 208318    Answers: 1   Comments: 0

calcul lim n→+∞ ∫_0 ^(+∞) ((cos(nx))/((nx+1)(1+x^2 ) ))dx

$${calcul}\:\:\:{lim}\:{n}\rightarrow+\infty \\ $$$$\int_{\mathrm{0}} ^{+\infty} \:\frac{{cos}\left({nx}\right)}{\left({nx}+\mathrm{1}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\:}{dx} \\ $$

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