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Question Number 227073    Answers: 1   Comments: 0

Question Number 227054    Answers: 1   Comments: 0

A Segment of a sphere has radius r and maximum height h.Prove that its volume ((𝛑h)/6)(h^2 +3r^2 )

$${A}\:{Segment}\:{of}\:{a}\:{sphere}\:{has}\:{radius}\:{r} \\ $$$${and}\:{maximum}\:{height}\:{h}.{Prove}\:{that} \\ $$$${its}\:{volume}\:\frac{\boldsymbol{\pi{h}}}{\mathrm{6}}\left(\boldsymbol{{h}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{{r}}^{\mathrm{2}} \right) \\ $$

Question Number 226821    Answers: 1   Comments: 0

Differentiate 20sin (x+3)cos (x^2 /2)

$${Differentiate}\:\: \\ $$$$\mathrm{20sin}\:\left({x}+\mathrm{3}\right)\mathrm{cos}\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}} \\ $$

Question Number 226820    Answers: 1   Comments: 0

Differentiate x^x^x

$$ \\ $$$$\:\:\:\:\:{Differentiate}\:\:\:\:{x}^{{x}^{{x}} } \\ $$$$ \\ $$

Question Number 226340    Answers: 0   Comments: 0

Question Number 225651    Answers: 0   Comments: 0

Question Number 225047    Answers: 2   Comments: 0

∫ x^x dx

$$\int\:{x}^{{x}} \:{dx} \\ $$

Question Number 223735    Answers: 0   Comments: 0

for all n ∈ Z , Show that Ο„ ( Ο• ( n )) β‰₯ Ο• (Ο„ (n ))

$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\:{n}\:\in\:\mathbb{Z}\:, \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\tau\:\left(\:\varphi\:\left(\:{n}\:\right)\right)\:\geqslant\:\varphi\:\left(\tau\:\left({n}\:\right)\right) \\ $$$$ \\ $$

Question Number 223006    Answers: 1   Comments: 0

L { tsin((√t) )}=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\mathscr{L}\:\:\left\{\:{tsin}\left(\sqrt{{t}}\:\right)\right\}=? \\ $$

Question Number 222778    Answers: 1   Comments: 0

lim_(xβ†’0) ((2log(1+x)βˆ’((x(3x+2))/((x+1)^2 )))/x^3 )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2log}\left(\mathrm{1}+\mathrm{x}\right)βˆ’\frac{\mathrm{x}\left(\mathrm{3x}+\mathrm{2}\right)}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }}{\mathrm{x}^{\mathrm{3}} } \\ $$

Question Number 222601    Answers: 0   Comments: 1

L 60 1. y=x^2 +5x Find the equation of a line with the slope of 7 that touches y=x^2 +5x. [Sol.] Let f(x)=x^2 +5x Then fβ€²(x)=2x+5 Since 2x+5=7β‡’x=1 then the point is (1, 1^2 +5βˆ™1)=(1, 6) So the equation of a line is yβˆ’6=7(xβˆ’1)β‡’y=7xβˆ’1 2. y=ax^2 +bx (2, 2) a, b Find the values of constants a, b that the slope of the line that touches (2, 2) and y=ax^2 +bx is 5. [Sol.] Let f(x)=ax^2 +bx Then fβ€²(x)=2ax+b and build two equations to solve for a and b { ((f(2)=aβˆ™2^2 +bβˆ™2=4a+2b=2)),((fβ€²(2)=2aβˆ™2+b=4a+b=5)) :} Solving for a, b gives a=2, b=βˆ’3 3. y=x^3 βˆ’3x^2 βˆ’1 Find the equation of a line that is drawn, touches y=x^3 βˆ’3x^2 βˆ’1. [Sol.] The line of the equation is yβˆ’(a^3 βˆ’3a^2 βˆ’1)=(3a^2 βˆ’6a)(xβˆ’a) Calculating gives y=(3a^2 βˆ’6a)xβˆ’(3a^2 βˆ’6a)a+(a^3 βˆ’3a^2 βˆ’1) y=(3a^2 βˆ’6a)x+(βˆ’3a^3 +6a^2 )+(a^3 βˆ’3a^2 βˆ’1) y=(3a^2 βˆ’6a)x+(βˆ’2a^3 +3a^2 βˆ’1) βˆ’2a^3 +3a^2 βˆ’1=0 a=βˆ’(1/2) or a=2 ...a=βˆ’3x, a=((15)/4)x

$$\mathrm{L}\:\mathrm{60} \\ $$$$ \\ $$$$\mathrm{1}.\: {y}={x}^{\mathrm{2}} +\mathrm{5}{x} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{line}\:\mathrm{with}\:\mathrm{the}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{7}\:\mathrm{that}\:\mathrm{touches}\:{y}={x}^{\mathrm{2}} +\mathrm{5}{x}. \\ $$$$\left[\mathrm{Sol}.\right]\:\mathrm{Let}\:{f}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{5}{x}\:\mathrm{Then}\:{f}'\left({x}\right)=\mathrm{2}{x}+\mathrm{5} \\ $$$$\mathrm{Since}\:\mathrm{2}{x}+\mathrm{5}=\mathrm{7}\Rightarrow{x}=\mathrm{1}\:\mathrm{then}\:\mathrm{the}\:\mathrm{point}\:\mathrm{is}\:\left(\mathrm{1},\:\mathrm{1}^{\mathrm{2}} +\mathrm{5}\centerdot\mathrm{1}\right)=\left(\mathrm{1},\:\mathrm{6}\right) \\ $$$$\mathrm{So}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{line}\:\mathrm{is}\:{y}βˆ’\mathrm{6}=\mathrm{7}\left({x}βˆ’\mathrm{1}\right)\Rightarrow{y}=\mathrm{7}{x}βˆ’\mathrm{1} \\ $$$$ \\ $$$$\mathrm{2}.\: \:{y}={ax}^{\mathrm{2}} +{bx}\: \:\left(\mathrm{2},\:\mathrm{2}\right) \:{a},\:{b} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\mathrm{constants}\:{a},\:{b}\:\mathrm{that}\:\mathrm{the}\:\mathrm{slope}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{that}\:\mathrm{touches}\:\left(\mathrm{2},\:\mathrm{2}\right)\:\mathrm{and}\:{y}={ax}^{\mathrm{2}} +{bx}\:\mathrm{is}\:\mathrm{5}. \\ $$$$\left[\mathrm{Sol}.\right]\:\mathrm{Let}\:{f}\left({x}\right)={ax}^{\mathrm{2}} +{bx}\:\mathrm{Then}\:{f}'\left({x}\right)=\mathrm{2}{ax}+{b}\:\mathrm{and}\:\mathrm{build}\:\mathrm{two}\:\mathrm{equations}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{for}\:{a}\:\mathrm{and}\:{b} \\ $$$$\begin{cases}{{f}\left(\mathrm{2}\right)={a}\centerdot\mathrm{2}^{\mathrm{2}} +{b}\centerdot\mathrm{2}=\mathrm{4}{a}+\mathrm{2}{b}=\mathrm{2}}\\{{f}'\left(\mathrm{2}\right)=\mathrm{2}{a}\centerdot\mathrm{2}+{b}=\mathrm{4}{a}+{b}=\mathrm{5}}\end{cases} \\ $$$$\mathrm{Solving}\:\mathrm{for}\:{a},\:{b}\:\mathrm{gives}\:{a}=\mathrm{2},\:{b}=βˆ’\mathrm{3} \\ $$$$ \\ $$$$\mathrm{3}.\: \:{y}={x}^{\mathrm{3}} βˆ’\mathrm{3}{x}^{\mathrm{2}} βˆ’\mathrm{1} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{line}\:\mathrm{that}\:\mathrm{is}\:\mathrm{drawn},\:\mathrm{touches}\:{y}={x}^{\mathrm{3}} βˆ’\mathrm{3}{x}^{\mathrm{2}} βˆ’\mathrm{1}. \\ $$$$\left[\mathrm{Sol}.\right]\:\mathrm{The}\:\mathrm{line}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{is}\:{y}βˆ’\left({a}^{\mathrm{3}} βˆ’\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{1}\right)=\left(\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{6}{a}\right)\left({x}βˆ’{a}\right) \\ $$$$\mathrm{Calculating}\:\mathrm{gives}\:{y}=\left(\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{6}{a}\right){x}βˆ’\left(\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{6}{a}\right){a}+\left({a}^{\mathrm{3}} βˆ’\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{1}\right) \\ $$$${y}=\left(\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{6}{a}\right){x}+\left(βˆ’\mathrm{3}{a}^{\mathrm{3}} +\mathrm{6}{a}^{\mathrm{2}} \right)+\left({a}^{\mathrm{3}} βˆ’\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{1}\right) \\ $$$${y}=\left(\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{6}{a}\right){x}+\left(βˆ’\mathrm{2}{a}^{\mathrm{3}} +\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{1}\right) \\ $$$$βˆ’\mathrm{2}{a}^{\mathrm{3}} +\mathrm{3}{a}^{\mathrm{2}} βˆ’\mathrm{1}=\mathrm{0} \\ $$$${a}=βˆ’\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{or}\:{a}=\mathrm{2} \\ $$$$...{a}=βˆ’\mathrm{3}{x},\:{a}=\frac{\mathrm{15}}{\mathrm{4}}{x} \\ $$

Question Number 222427    Answers: 1   Comments: 0

if lim_(x→0) (((sin2x)/x^3 )+(a/x^2 )+b)=1 find a and b without using LHopial rule

$$\:\:\boldsymbol{{if}}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\boldsymbol{{sin}}\mathrm{2}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{3}} }+\frac{\boldsymbol{{a}}}{\boldsymbol{{x}}^{\mathrm{2}} }+\boldsymbol{{b}}\right)=\mathrm{1}\: \\ $$$$\:\:\:\:\boldsymbol{{find}}\:\boldsymbol{{a}}\:\boldsymbol{{and}}\:\boldsymbol{{b}}\:\:\boldsymbol{{without}} \\ $$$$\:\:\:\:\:\:\boldsymbol{{using}}\:\boldsymbol{{LH}}{opial}\:{rule} \\ $$

Question Number 222389    Answers: 0   Comments: 0

Prove: ∫_0 ^∞ ((cos(nx)cos(p arctan x))/((1+x^2 )^(p/2) ))=(Ο€/2) ((n^(pβˆ’1) e^(βˆ’n) )/(Ξ“(p))) (p>0)

$$\mathrm{Prove}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{cos}\left({nx}\right)\mathrm{cos}\left({p}\:\mathrm{arctan}\:{x}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{{p}}{\mathrm{2}}} }=\frac{\pi}{\mathrm{2}}\:\frac{{n}^{{p}βˆ’\mathrm{1}} {e}^{βˆ’{n}} }{\Gamma\left({p}\right)}\:\left({p}>\mathrm{0}\right) \\ $$

Question Number 222329    Answers: 0   Comments: 1

lim_(xβ†’βˆž) 4x+(√(16x^2 βˆ’3x))

$$\boldsymbol{\mathrm{lim}}_{\boldsymbol{\mathrm{x}}\rightarrow\infty} \:\mathrm{4}\boldsymbol{\mathrm{x}}+\sqrt{\mathrm{16}\boldsymbol{\mathrm{x}}^{\mathrm{2}} βˆ’\mathrm{3}\boldsymbol{\mathrm{x}}} \\ $$$$ \\ $$

Question Number 222284    Answers: 1   Comments: 0

y=(8^x /((in8)^3 )) find (d^6 y/dx^6 )

$$\boldsymbol{\mathrm{y}}=\frac{\mathrm{8}^{\boldsymbol{\mathrm{x}}} }{\left(\boldsymbol{\mathrm{in}}\mathrm{8}\right)^{\mathrm{3}} } \\ $$$$\boldsymbol{\mathrm{find}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{6}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{6}} } \\ $$

Question Number 222113    Answers: 0   Comments: 2

name the following compound

$$\mathrm{name}\:\mathrm{the}\:\mathrm{following}\:\mathrm{compound} \\ $$

Question Number 221044    Answers: 0   Comments: 1

If V be a function of x and y, prove that (βˆ‚^2 V/βˆ‚x^2 )+(βˆ‚^2 V/βˆ‚y^2 )=(βˆ‚^2 V/βˆ‚r^2 )+(1/r) (βˆ‚V/βˆ‚r)+(1/r^2 ) (βˆ‚^2 V/βˆ‚ΞΈ^2 ), where x=r cos ΞΈ , y=rsin ΞΈ

$${If}\:{V}\:{be}\:{a}\:{function}\:{of}\:{x}\:{and}\:{y},\:{prove}\:{that} \\ $$$$\frac{\partial^{\mathrm{2}} {V}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {V}}{\partial{y}^{\mathrm{2}} }=\frac{\partial^{\mathrm{2}} {V}}{\partial{r}^{\mathrm{2}} }+\frac{\mathrm{1}}{{r}}\:\frac{\partial{V}}{\partial{r}}+\frac{\mathrm{1}}{{r}^{\mathrm{2}} }\:\frac{\partial^{\mathrm{2}} {V}}{\partial\theta^{\mathrm{2}} }, \\ $$$${where}\:{x}={r}\:\mathrm{cos}\:\theta\:,\:{y}={r}\mathrm{sin}\:\theta \\ $$

Question Number 220964    Answers: 0   Comments: 0

Find the general solution of the differential equation x^2 (d^3 y/dx^3 ) + x(d^2 y/dx^2 )βˆ’6(dy/dx)+6(y/x)=((x ln x+1)/x^2 ),[x>0]

$${Find}\:{the}\:{general}\:{solution}\:{of}\:{the}\:{differential}\:{equation} \\ $$$${x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{3}} {y}}{{dx}^{\mathrm{3}} }\:+\:{x}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }βˆ’\mathrm{6}\frac{{dy}}{{dx}}+\mathrm{6}\frac{{y}}{{x}}=\frac{{x}\:\mathrm{ln}\:{x}+\mathrm{1}}{{x}^{\mathrm{2}} },\left[{x}>\mathrm{0}\right] \\ $$

Question Number 220863    Answers: 1   Comments: 0

(211) Find the derivative of Ξ”x, where Ξ”x= determinant (((f_1 (x)),(Ο†_1 (x)),(Ξ¨_1 (x))),((f_2 (x)),(Ο†_2 (x)),(Ξ¨_2 (x))),((f_3 (x)),(Ο†_3 (x)),(Ξ¨_3 (x)))) and f_1 (x) ,f_2 (x), f_3 (x),Ο†_1 (x), etc. are different functions of x.

$$\left(\mathrm{211}\right) \\ $$$$\:\: \\ $$$${Find}\:{the}\:{derivative}\:{of}\:\Delta{x},\:{where} \\ $$$$\Delta{x}=\begin{vmatrix}{{f}_{\mathrm{1}} \left({x}\right)}&{\phi_{\mathrm{1}} \left({x}\right)}&{\Psi_{\mathrm{1}} \left({x}\right)}\\{{f}_{\mathrm{2}} \left({x}\right)}&{\phi_{\mathrm{2}} \left({x}\right)}&{\Psi_{\mathrm{2}} \left({x}\right)}\\{{f}_{\mathrm{3}} \left({x}\right)}&{\phi_{\mathrm{3}} \left({x}\right)}&{\Psi_{\mathrm{3}} \left({x}\right)}\end{vmatrix} \\ $$$${and}\:{f}_{\mathrm{1}} \left({x}\right)\:,{f}_{\mathrm{2}} \left({x}\right),\:{f}_{\mathrm{3}} \left({x}\right),\phi_{\mathrm{1}} \left({x}\right),\:{etc}.\:{are}\:{different}\:{functions}\:{of}\:{x}. \\ $$

Question Number 220232    Answers: 1   Comments: 0

prove that (Ο€/(16)) < ∫_0 ^( 1 ) (√((x(1βˆ’x))/(sin(Ο€x)+cos(Ο€x)+2))) dx<(Ο€/8)

$$ \\ $$$$\:\:\:\:{prove}\:{that} \\ $$$$ \\ $$$$\:\:\frac{\pi}{\mathrm{16}}\:<\:\int_{\mathrm{0}} ^{\:\mathrm{1}\:} \sqrt{\frac{{x}\left(\mathrm{1}βˆ’{x}\right)}{{sin}\left(\pi{x}\right)+{cos}\left(\pi{x}\right)+\mathrm{2}}}\:{dx}<\frac{\pi}{\mathrm{8}} \\ $$$$\:\:\:\:\: \\ $$

Question Number 220131    Answers: 2   Comments: 0

If f(x,y)=(((x^2 +y^2 )^n )/(2n(2nβˆ’1)))+xΟ†((y/x))+Ξ¨((y/x)), then using Eulerβ€²s theorem on homogenous functions,show that x^2 ((Ξ΄^2 f)/(Ξ΄x^2 ))+2xy((Ξ΄^2 f)/(Ξ΄xΞ΄y))+y^2 ((Ξ΄^2 f)/(Ξ΄y^2 ))=(x^2 +y^2 )^n

$${If}\:\:\:{f}\left({x},{y}\right)=\frac{\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{{n}} }{\mathrm{2}{n}\left(\mathrm{2}{n}βˆ’\mathrm{1}\right)}+{x}\phi\left(\frac{{y}}{{x}}\right)+\Psi\left(\frac{{y}}{{x}}\right), \\ $$$${then}\:{using}\:{Euler}'{s}\:{theorem}\:{on}\:{homogenous}\:{functions},{show}\:{that} \\ $$$${x}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{x}^{\mathrm{2}} }+\mathrm{2}{xy}\frac{\delta^{\mathrm{2}} {f}}{\delta{x}\delta{y}}+{y}^{\mathrm{2}} \frac{\delta^{\mathrm{2}} {f}}{\delta{y}^{\mathrm{2}} }=\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{{n}} \\ $$

Question Number 219868    Answers: 1   Comments: 0

Prove that; (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = βˆ’cos 2x

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:βˆ’\mathrm{cos}\:\mathrm{2}{x}\:\:\:\: \\ $$$$ \\ $$

Question Number 219451    Answers: 0   Comments: 1

Question Number 219243    Answers: 3   Comments: 2

Question Number 219098    Answers: 2   Comments: 0

΢(α)=Σ_(n=1) ^(+∞) (1/n^α )

$$\zeta\left(\alpha\right)=\underset{{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\alpha} }\:\: \\ $$

Question Number 216800    Answers: 1   Comments: 0

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