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AllQuestion and Answers: Page 204

Question Number 201728    Answers: 1   Comments: 0

cos^2 4x ∙ sin^2 4x = 0,25 for equation [0;90] how many roots are there in the piece?

$$\mathrm{cos}^{\mathrm{2}} \:\mathrm{4x}\:\centerdot\:\mathrm{sin}^{\mathrm{2}} \:\mathrm{4x}\:=\:\mathrm{0},\mathrm{25}\:\mathrm{for}\:\mathrm{equation} \\ $$$$\left[\mathrm{0};\mathrm{90}\right]\:\mathrm{how}\:\mathrm{many}\:\mathrm{roots}\:\mathrm{are}\:\mathrm{there}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{piece}? \\ $$

Question Number 201712    Answers: 0   Comments: 2

Question Number 201707    Answers: 1   Comments: 0

Question Number 201702    Answers: 1   Comments: 1

Find: ∫_1 ^( 3) dx ∫_x ^( x^3 ) (x − y) dy

$$\mathrm{Find}: \\ $$$$\int_{\mathrm{1}} ^{\:\mathrm{3}} \:\mathrm{dx}\:\int_{\boldsymbol{\mathrm{x}}} ^{\:\boldsymbol{\mathrm{x}}^{\mathrm{3}} } \:\left(\mathrm{x}\:−\:\mathrm{y}\right)\:\mathrm{dy} \\ $$

Question Number 201693    Answers: 3   Comments: 0

Question Number 201689    Answers: 1   Comments: 0

Question Number 201683    Answers: 0   Comments: 1

Starting from substituting z=x+iy. Identify the maximal region within which f(z) is analytic f(z)=(1/(z(z+1))). Note. Do not start by just differentiating f(z). Start by doing a substitution of x and iy and then verify Cauchy Rieman theorem.

$${Starting}\:{from}\:{substituting}\:{z}={x}+{iy}.\:{Identify} \\ $$$${the}\:{maximal}\:{region}\:{within}\:{which}\:{f}\left({z}\right)\:{is}\:{analytic} \\ $$$${f}\left({z}\right)=\frac{\mathrm{1}}{{z}\left({z}+\mathrm{1}\right)}.\: \\ $$$$ \\ $$$${Note}.\:{Do}\:{not}\:{start}\:{by}\:{just}\:{differentiating}\:{f}\left({z}\right).\: \\ $$$${Start}\:{by}\:\:{doing}\:{a}\:{substitution}\:{of}\:{x}\:{and}\:{iy}\:{and}\: \\ $$$${then}\:{verify}\:{Cauchy}\:{Rieman}\:{theorem}. \\ $$$$ \\ $$

Question Number 201681    Answers: 1   Comments: 0

f(x+1)−f(x)=3f(x)×f(x+1) D_f =N 2023×f(1402)=1 have equation f(x)=1 solution?

$${f}\left({x}+\mathrm{1}\right)−{f}\left({x}\right)=\mathrm{3}{f}\left({x}\right)×{f}\left({x}+\mathrm{1}\right) \\ $$$${D}_{{f}} ={N} \\ $$$$\mathrm{2023}×{f}\left(\mathrm{1402}\right)=\mathrm{1} \\ $$$${have}\:{equation}\:{f}\left({x}\right)=\mathrm{1}\:{solution}? \\ $$

Question Number 201680    Answers: 1   Comments: 0

lim_(x→0) ((2sin x−sin 2x)/(sin x−x cos x)).

$$\:\: \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:\mathrm{x}−\mathrm{sin}\:\mathrm{2x}}{\mathrm{sin}\:\mathrm{x}−\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}. \\ $$

Question Number 201679    Answers: 5   Comments: 0

Question Number 201660    Answers: 1   Comments: 0

An equilateral triangle inscribed in a parabola y^2 =4x. One of its vertices is at the vertex of the parabola. Find the length of each side of the triangle in units.

$${An}\:{equilateral}\:{triangle}\:{inscribed}\:{in}\:{a}\:{parabola} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{x}.\:{One}\:{of}\:{its}\:{vertices}\:{is}\:{at}\:{the}\:{vertex}\:{of}\:\:{the}\:{parabola}. \\ $$$${Find}\:{the}\:{length}\:{of}\:{each}\:{side}\:{of}\:{the}\:{triangle}\:{in}\:{units}. \\ $$

Question Number 201659    Answers: 2   Comments: 0

Find the shortest distance between point A(3,2) and curve y=(√x) (x>0).

$${Find}\:{the}\:{shortest}\:{distance}\:{between}\: \\ $$$${point}\:{A}\left(\mathrm{3},\mathrm{2}\right)\:{and}\:{curve}\:{y}=\sqrt{{x}}\:\left({x}>\mathrm{0}\right). \\ $$

Question Number 201654    Answers: 6   Comments: 0

Question Number 201653    Answers: 0   Comments: 0

Question Number 201646    Answers: 1   Comments: 0

Question Number 201644    Answers: 0   Comments: 0

Question Number 201631    Answers: 1   Comments: 0

Question Number 201629    Answers: 0   Comments: 3

Question Number 201627    Answers: 1   Comments: 0

Question Number 201657    Answers: 1   Comments: 0

if f(x)= { ((((sin (1+[x]))/([x])) for [x]≠0)),((0 for [x]=0)) :} where [x] represents an integer x greatest ≤ x Find lim_(x→0^− ) f(x).

$${if}\:\:{f}\left({x}\right)=\begin{cases}{\frac{\mathrm{sin}\:\left(\mathrm{1}+\left[{x}\right]\right)}{\left[{x}\right]}\:\:{for}\:\left[{x}\right]\neq\mathrm{0}}\\{\mathrm{0}\:\:{for}\:\left[{x}\right]=\mathrm{0}}\end{cases} \\ $$$${where}\:\left[{x}\right]\:{represents}\:{an}\:{integer}\:\boldsymbol{{x}}\:{greatest}\:\leqslant\:\boldsymbol{{x}} \\ $$$${Find}\:\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}{f}\left({x}\right). \\ $$

Question Number 201615    Answers: 1   Comments: 0

x,y,z ∈ R a,b,c>0 prove that: (x^2 /a) + (y^2 /b) + (z^2 /c) ≥ (((x + y + z)^2 )/(a + b + c))

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\mathrm{a},\mathrm{b},\mathrm{c}>\mathrm{0} \\ $$$$\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}}\:+\:\frac{\mathrm{z}^{\mathrm{2}} }{\mathrm{c}}\:\geqslant\:\frac{\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)^{\mathrm{2}} }{\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}} \\ $$

Question Number 201613    Answers: 2   Comments: 0

x,y,z ∈ R { ((xy + yz + zx = 3)),((x + y + z = 5)) :} → max(z) = ?

$$\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R} \\ $$$$\begin{cases}{\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{zx}\:=\:\mathrm{3}}\\{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:=\:\mathrm{5}}\end{cases}\:\:\:\:\:\rightarrow\:\:\:\:\mathrm{max}\left(\boldsymbol{\mathrm{z}}\right)\:=\:? \\ $$

Question Number 201604    Answers: 2   Comments: 1

Question Number 201599    Answers: 1   Comments: 2

Question Number 201595    Answers: 3   Comments: 0

1) ∣((3+2x)/(3x))∣ ≤1 2) 1≤ ∣ ((x−3)/(1−2x))∣≤ 2 3) ((x^2 +2x−35)/(x+2)) > 0 4) −1 ≤ ((x+1)/(x−2)) ≤2

$$\left.\mathrm{1}\right)\:\:\mid\frac{\mathrm{3}+\mathrm{2}{x}}{\mathrm{3}{x}}\mid\:\leq\mathrm{1} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{1}\leq\:\mid\:\frac{{x}−\mathrm{3}}{\mathrm{1}−\mathrm{2}{x}}\mid\leq\:\mathrm{2} \\ $$$$ \\ $$$$\left.\mathrm{3}\right)\:\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{35}}{{x}+\mathrm{2}}\:>\:\mathrm{0} \\ $$$$ \\ $$$$\left.\mathrm{4}\right)\:−\mathrm{1}\:\leq\:\frac{{x}+\mathrm{1}}{{x}−\mathrm{2}}\:\leq\mathrm{2} \\ $$

Question Number 201582    Answers: 0   Comments: 0

Solve.... y′′(t)−sin(t)y(t)=0 , y^((2)) (0)=0 , y^((1)) (0)=−1 , y(0)=0 L{y′′(t)−sin(t)y(t)}=0 s^2 F(s)−sy(0)−y′(0)−L{sin(t)y(t)}=0 Holy...×uck I already know y′′(t)−ty(t)=0 solution C_1 Ai(t)+C_2 Bi(t) But I Can′t Solve y′′(t)−sin(t)y(t)=0....

$$\mathrm{Solve}.... \\ $$$${y}''\left({t}\right)−\mathrm{sin}\left({t}\right){y}\left({t}\right)=\mathrm{0}\:,\: \\ $$$${y}^{\left(\mathrm{2}\right)} \left(\mathrm{0}\right)=\mathrm{0}\:,\:{y}^{\left(\mathrm{1}\right)} \left(\mathrm{0}\right)=−\mathrm{1}\:,\:{y}\left(\mathrm{0}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\boldsymbol{\mathcal{L}}\left\{{y}''\left({t}\right)−\mathrm{sin}\left({t}\right){y}\left({t}\right)\right\}=\mathrm{0} \\ $$$${s}^{\mathrm{2}} \boldsymbol{\mathrm{F}}\left({s}\right)−{sy}\left(\mathrm{0}\right)−{y}'\left(\mathrm{0}\right)−\boldsymbol{\mathcal{L}}\left\{\mathrm{sin}\left({t}\right){y}\left({t}\right)\right\}=\mathrm{0} \\ $$$$\mathrm{Holy}...×\mathrm{uck} \\ $$$$\mathrm{I}\:\mathrm{already}\:\mathrm{know}\:{y}''\left({t}\right)−{ty}\left({t}\right)=\mathrm{0}\:\:\:\mathrm{solution} \\ $$$$\mathrm{C}_{\mathrm{1}} \mathrm{Ai}\left({t}\right)+{C}_{\mathrm{2}} \mathrm{Bi}\left({t}\right) \\ $$$$\mathrm{But}\:\mathrm{I}\:\mathrm{Can}'\mathrm{t}\:\mathrm{Solve}\:{y}''\left({t}\right)−\mathrm{sin}\left({t}\right){y}\left({t}\right)=\mathrm{0}....\: \\ $$

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