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Question Number 169004 Answers: 1 Comments: 3

lim_(x→(π/2)) (((sinx−1)/(x−(π/2))))=?

$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\left(\frac{{sinx}−\mathrm{1}}{{x}−\frac{\pi}{\mathrm{2}}}\right)=? \\ $$

Question Number 169000 Answers: 1 Comments: 0

Question Number 168982 Answers: 1 Comments: 1

Question Number 168980 Answers: 0 Comments: 7

Question Number 168979 Answers: 0 Comments: 0

f(x,y,z) = (3x^2 y,x^3 +y^3 , 2z) prove that the function has a potential to be determined.

$${f}\left({x},{y},{z}\right)\:=\:\left(\mathrm{3}{x}^{\mathrm{2}} {y},{x}^{\mathrm{3}} +{y}^{\mathrm{3}} ,\:\mathrm{2}{z}\right) \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function}\:\mathrm{has}\:\mathrm{a}\:\mathrm{potential} \\ $$$$\mathrm{to}\:\mathrm{be}\:\mathrm{determined}. \\ $$

Question Number 168978 Answers: 1 Comments: 0

Evaluate (a) ∫((t−2)/(t−3(√(2t−4)) +2)) dt (b) ∫((3z)/((1−4z−2z^2 )^2 )) dz

$$\mathrm{Evaluate}\: \\ $$$$\left(\mathrm{a}\right)\:\int\frac{{t}−\mathrm{2}}{{t}−\mathrm{3}\sqrt{\mathrm{2}{t}−\mathrm{4}}\:+\mathrm{2}}\:{dt}\: \\ $$$$\left(\mathrm{b}\right)\:\int\frac{\mathrm{3}{z}}{\left(\mathrm{1}−\mathrm{4}{z}−\mathrm{2}{z}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dz} \\ $$

Question Number 168977 Answers: 0 Comments: 0

check that the function u(x,t) = exp{−((n^2 α^2 π^2 )/L^2 )t} sin((nπx)/L) n = 1,2,... satisfy the heat equation heat equation α^2 (∂^2 u/∂x^2 ) = (∂u/∂t), 0 < x < L

$$\mathrm{check}\:\mathrm{that}\:\mathrm{the}\:\mathrm{function} \\ $$$${u}\left({x},{t}\right)\:=\:\mathrm{exp}\left\{−\frac{{n}^{\mathrm{2}} \alpha^{\mathrm{2}} \pi^{\mathrm{2}} }{{L}^{\mathrm{2}} }{t}\right\}\:\mathrm{sin}\frac{{n}\pi{x}}{{L}} \\ $$$${n}\:=\:\mathrm{1},\mathrm{2},...\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{heat}\:\mathrm{equation} \\ $$$$\boldsymbol{\mathrm{heat}}\:\boldsymbol{\mathrm{equation}} \\ $$$$\alpha^{\mathrm{2}} \frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }\:=\:\frac{\partial{u}}{\partial{t}},\:\mathrm{0}\:<\:{x}\:<\:{L} \\ $$

Question Number 168952 Answers: 1 Comments: 0

Question Number 168948 Answers: 1 Comments: 0

Question Number 168946 Answers: 2 Comments: 1

Question Number 168942 Answers: 0 Comments: 2

Question Number 168956 Answers: 4 Comments: 0

solve ⌊ x ⌋ + ⌊ (x/6) ⌋ =(2/3) x

$$ \\ $$$$\:\:\:\:{solve} \\ $$$$\:\:\lfloor\:{x}\:\rfloor\:+\:\lfloor\:\frac{{x}}{\mathrm{6}}\:\rfloor\:=\frac{\mathrm{2}}{\mathrm{3}}\:{x} \\ $$$$ \\ $$

Question Number 168937 Answers: 1 Comments: 0

Determine m & n such that: digit-sum(m^2 )=n_(&_(digit-sum(n^2 )=m) ) ^■ digit-sum(abc..^(−) .)=a+b+c+...

$$ \\ $$$$\underline{\mathcal{D}{etermine}\:{m}\:\&\:{n}\:{such}\:{that}:} \\ $$$$\:\:\underset{\underset{{digit}-{sum}\left({n}^{\mathrm{2}} \right)={m}} {\&}} {\:{digit}-{sum}\left({m}^{\mathrm{2}} \right)={n}} \\ $$$$\:^{\blacksquare} {digit}-{sum}\left(\overline {{abc}..}.\right)={a}+{b}+{c}+...\:\:\:\:\:\:\:\: \\ $$

Question Number 168968 Answers: 0 Comments: 0

Question Number 168966 Answers: 2 Comments: 0

lim_(x→0) ((∫_0 ^( x) (∫_0 ^( u^2 ) tan^(−1) (1+t)dt)dt )/(x−x cos x)) =?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\int_{\mathrm{0}} ^{\:{x}} \:\left(\int_{\mathrm{0}} ^{\:{u}^{\mathrm{2}} } \mathrm{tan}^{−\mathrm{1}} \left(\mathrm{1}+{t}\right){dt}\right){dt}\:}{{x}−{x}\:\mathrm{cos}\:{x}}\:=? \\ $$

Question Number 168964 Answers: 2 Comments: 0

lim_(x→0) ∫_0 ^x ((sin 2t)/( (√(4+t^2 )) ∫_0 ^x ((√(1+t))−1)dt)) dt =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{t}}{\:\sqrt{\mathrm{4}+{t}^{\mathrm{2}} }\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\left(\sqrt{\mathrm{1}+{t}}−\mathrm{1}\right){dt}}\:{dt}\:=?\: \\ $$

Question Number 168926 Answers: 2 Comments: 2

Question Number 168921 Answers: 1 Comments: 0

Question Number 168911 Answers: 1 Comments: 1

Question Number 168910 Answers: 0 Comments: 3

Resolve 1) (x−y)ydx−x^2 dy=0 2)(2x−y)dx+(4x−2y+3)dy=0

$${Resolve} \\ $$$$\left.\mathrm{1}\right)\:\left({x}−{y}\right){ydx}−{x}^{\mathrm{2}} {dy}=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\left(\mathrm{2}{x}−{y}\right){dx}+\left(\mathrm{4}{x}−\mathrm{2}{y}+\mathrm{3}\right){dy}=\mathrm{0} \\ $$

Question Number 168909 Answers: 0 Comments: 1

Resolve (1−x^2 y)dx+(x^2 y−x^3 )dy=0 ; μ=μ(x)

$${Resolve}\: \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} {y}\right){dx}+\left({x}^{\mathrm{2}} {y}−{x}^{\mathrm{3}} \right){dy}=\mathrm{0}\:;\:\mu=\mu\left({x}\right) \\ $$

Question Number 168906 Answers: 1 Comments: 0

Question Number 168905 Answers: 1 Comments: 0

Question Number 168898 Answers: 1 Comments: 0

Question Number 168894 Answers: 1 Comments: 1

Question Number 168885 Answers: 0 Comments: 11

Find all the values of n such that: determinant (((digit-sum(n^3 )=n ))) and show that n has at most 2 digits. ^■ digit-sum(abc..^(−) )=a+b+c+...

$$ \\ $$$$\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}: \\ $$$$\:\:\:\:\begin{array}{|c|}{\mathrm{digit}-\mathrm{sum}\left(\mathrm{n}^{\mathrm{3}} \right)=\mathrm{n}\:}\\\hline\end{array} \\ $$$$\mathrm{and} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{n}\:\mathrm{has}\:\mathrm{at}\:\mathrm{most}\:\mathrm{2}\:\mathrm{digits}. \\ $$$$\:\:^{\blacksquare} \mathrm{digit}-\mathrm{sum}\left(\overline {\mathrm{abc}..}\right)=\mathrm{a}+\mathrm{b}+\mathrm{c}+... \\ $$

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