Question and Answers Forum

All Questions   Topic List

DifferentiationQuestion and Answers: Page 1

Question Number 206340    Answers: 2   Comments: 0

∫_0 ^( 1) (( ln(1−x )ln(1+x ))/x)dx = Σ_(n=1) ^∞ Ω_n find : Σ_(n=1) ^∞ n Ω_n = ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{ln}\left(\mathrm{1}−{x}\:\right){ln}\left(\mathrm{1}+{x}\:\right)}{{x}}{dx}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\Omega_{{n}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{find}\::\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{n}\:\Omega_{{n}} \:=\:? \\ $$

Question Number 206244    Answers: 1   Comments: 0

ζ

$$\:\:\:\zeta \\ $$

Question Number 205827    Answers: 2   Comments: 0

x^3 +y^3 =1 find the implceat second derivative

$$\mathrm{x}^{\mathrm{3}} +\mathrm{y}^{\mathrm{3}} =\mathrm{1} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{implceat}\:\mathrm{second}\:\mathrm{derivative} \\ $$

Question Number 205393    Answers: 0   Comments: 0

Question Number 205338    Answers: 2   Comments: 0

Question Number 205200    Answers: 1   Comments: 0

∫_1 ^∞ (x/(x^3 +lnx)) dx=?

$$\int_{\mathrm{1}} ^{\infty} \:\frac{{x}}{{x}^{\mathrm{3}} +{lnx}}\:{dx}=? \\ $$

Question Number 205024    Answers: 0   Comments: 0

Question Number 205013    Answers: 2   Comments: 0

if y=(x)^(1/7) prove that y^′ =(1/(7 (x^6 )^(1/7) ))

$${if}\:{y}=\sqrt[{\mathrm{7}}]{{x}}\:{prove}\:{that} \\ $$$${y}^{'} =\frac{\mathrm{1}}{\mathrm{7}\:\sqrt[{\mathrm{7}}]{{x}^{\mathrm{6}} }} \\ $$

Question Number 204909    Answers: 1   Comments: 0

Ω= ∫_(1/e) ^( e) (( arctan(x))/x) dx=?

$$ \\ $$$$\:\:\:\:\:\:\:\Omega=\:\int_{\frac{\mathrm{1}}{{e}}} ^{\:{e}} \frac{\:{arctan}\left({x}\right)}{{x}}\:{dx}=? \\ $$

Question Number 204878    Answers: 2   Comments: 0

prove that: (e)^(1/4) < ∫_0 ^( 1) e^( t^2 ) dt< ((1 + e)/2)

$$ \\ $$$$\:\:{prove}\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\sqrt[{\mathrm{4}}]{{e}}\:<\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {e}^{\:{t}^{\mathrm{2}} } {dt}<\:\frac{\mathrm{1}\:+\:{e}}{\mathrm{2}} \\ $$

Question Number 204826    Answers: 1   Comments: 11

A rectangular enclosure is to be made against a straight wall using three lengths of fencing. The total length of the fencing available is 50m. Show that the area of the enclosure is 50x − 2x^2 , where x is the length of the sides perpendicular to the wall. Hence find the maximum area of the enclosure.

$${A}\:{rectangular}\:{enclosure}\:{is}\:{to}\:{be}\:{made} \\ $$$${against}\:{a}\:{straight}\:{wall}\:{using}\:{three} \\ $$$${lengths}\:{of}\:{fencing}.\:{The}\:{total}\:{length}\:{of} \\ $$$${the}\:{fencing}\:{available}\:{is}\:\mathrm{50}{m}.\:{Show} \\ $$$${that}\:{the}\:{area}\:{of}\:{the}\:{enclosure}\:{is} \\ $$$$\mathrm{50}{x}\:−\:\mathrm{2}{x}^{\mathrm{2}} ,\:{where}\:{x}\:{is}\:{the}\:{length}\:{of}\:{the} \\ $$$${sides}\:{perpendicular}\:{to}\:{the}\:{wall}.\:{Hence} \\ $$$${find}\:{the}\:{maximum}\:{area}\:{of}\:{the} \\ $$$${enclosure}. \\ $$

Question Number 204372    Answers: 2   Comments: 0

If , f : [ 0 , b] →^(continuous) R , g : R →_(b−periodic) ^(continuous) R ⇒ lim_(n→∞) ∫_0 ^( b) f(x)g(nx)dx=^? (1/b) ∫_0 ^( b) f(x)dx .∫_0 ^( b) g(x)dx

$$ \\ $$$$\:\:{If}\:,\:\:\:\:{f}\::\:\left[\:\mathrm{0}\:,\:{b}\right]\:\overset{{continuous}} {\rightarrow}\:\mathbb{R}\: \\ $$$$\:\:\:\:\:\:\:\:,\:\:\:\:{g}\::\:\mathbb{R}\:\underset{{b}−{periodic}} {\overset{{continuous}} {\rightarrow}}\:\mathbb{R} \\ $$$$\:\:\:\:\:\:\Rightarrow\:\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\:{b}} {f}\left({x}\right){g}\left({nx}\right){dx}\overset{?} {=}\frac{\mathrm{1}}{{b}}\:\int_{\mathrm{0}} ^{\:{b}} {f}\left({x}\right){dx}\:.\int_{\mathrm{0}} ^{\:{b}} {g}\left({x}\right){dx} \\ $$$$ \\ $$

Question Number 203835    Answers: 1   Comments: 0

Question Number 203063    Answers: 1   Comments: 0

Question Number 202866    Answers: 1   Comments: 0

calculate ... 𝛗= ∫_0 ^( 1) (( tanh^( −1) (x))/((1 + x )^( 2) )) dx = ?

$$ \\ $$$$\:\:\:\:\:\:\:{calculate}\:... \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \left({x}\right)}{\left(\mathrm{1}\:+\:{x}\:\right)^{\:\mathrm{2}} }\:{dx}\:=\:?\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 201940    Answers: 1   Comments: 0

tan^3 (xy^2 +y)=x find (dy/dx)

$$\boldsymbol{{tan}}^{\mathrm{3}} \left(\boldsymbol{{xy}}^{\mathrm{2}} +\boldsymbol{{y}}\right)=\boldsymbol{{x}}\:\:\boldsymbol{{find}}\:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}} \\ $$

Question Number 201534    Answers: 0   Comments: 1

let f(x)=tanx find f^((n)) (x) with n integr natural

$${let}\:{f}\left({x}\right)={tanx} \\ $$$${find}\:{f}^{\left({n}\right)} \left({x}\right)\:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$

Question Number 201214    Answers: 1   Comments: 0

A ball lies on the function z=xy at the point (1,2,2). Find the point in the xy−plane where the ball will touch it. (an unsolved old question Q200929)

$$\mathrm{A}\:\mathrm{ball}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{function}\:{z}={xy}\:\mathrm{at} \\ $$$$\mathrm{the}\:\mathrm{point}\:\left(\mathrm{1},\mathrm{2},\mathrm{2}\right).\:\mathrm{Find}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in} \\ $$$$\mathrm{the}\:{xy}−\mathrm{plane}\:\mathrm{where}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{will} \\ $$$$\mathrm{touch}\:\mathrm{it}. \\ $$$$ \\ $$$$\left({an}\:{unsolved}\:{old}\:{question}\:{Q}\mathrm{200929}\right) \\ $$

Question Number 201140    Answers: 1   Comments: 0

If R_− =x^2 yi_− −2y^2 zj_− +xy^2 z^2 k_− , find ∣(d^2 R/dx^2 )×(d^2 R/dy^2 )∣ at the point (2,1,−2)

$$\boldsymbol{{If}}\:\underset{−} {\boldsymbol{{R}}}=\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}\underset{−} {\boldsymbol{{i}}}−\mathrm{2}\boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}\underset{−} {\boldsymbol{{j}}}+\boldsymbol{{xy}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} \underset{−} {\boldsymbol{{k}}},\:\boldsymbol{{find}}\:\mid\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{R}}}{\boldsymbol{{dx}}^{\mathrm{2}} }×\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{R}}}{\boldsymbol{{dy}}^{\mathrm{2}} }\mid\:\: \\ $$$$\boldsymbol{{at}}\:\boldsymbol{{the}}\:\boldsymbol{{point}}\:\left(\mathrm{2},\mathrm{1},−\mathrm{2}\right) \\ $$

Question Number 201133    Answers: 0   Comments: 1

Ω= ∫_1 ^( 3) (( 1)/( (√((x−1 )^3 )) + (√((x+1 )^3 )))) dx= ?

$$ \\ $$$$ \\ $$$$\:\:\:\Omega=\:\int_{\mathrm{1}} ^{\:\mathrm{3}} \frac{\:\mathrm{1}}{\:\sqrt{\left({x}−\mathrm{1}\:\right)^{\mathrm{3}} }\:+\:\sqrt{\left({x}+\mathrm{1}\:\right)^{\mathrm{3}} }}\:{dx}=\:?\:\:\: \\ $$$$ \\ $$

Question Number 200738    Answers: 0   Comments: 0

Solve: A smooth sphere A,of mass 2kg and moving with speed 6ms^(−1) collides obliquely with a smooth sphere B of mass 4kg. just before the impact B is stationary and the velocity of A makes an angle of 10° with the lines of centers of the two sphere. The coefficient of restitution between the spheres is (1/2). Find the magnitude and directions of the velovities of A and B immediately after the impact.

$$\boldsymbol{{Solve}}:\:\boldsymbol{{A}}\:\boldsymbol{{smooth}}\:\boldsymbol{{sphere}}\:\boldsymbol{{A}},\boldsymbol{{of}}\:\boldsymbol{{mass}}\:\mathrm{2}\boldsymbol{{kg}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{moving}}\:\boldsymbol{{with}}\:\boldsymbol{{speed}}\:\mathrm{6}\boldsymbol{{ms}}^{−\mathrm{1}} \boldsymbol{{collides}}\:\boldsymbol{{obliquely}}\: \\ $$$$\boldsymbol{{with}}\:\boldsymbol{{a}}\:\boldsymbol{{smooth}}\:\boldsymbol{{sphere}}\:\boldsymbol{{B}}\:\boldsymbol{{of}}\:\boldsymbol{{mass}}\:\mathrm{4}\boldsymbol{{kg}}.\:\boldsymbol{{just}}\:\boldsymbol{{before}}\:\boldsymbol{{the}}\:\boldsymbol{{impact}}\:\boldsymbol{{B}}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{stationary}}\:\boldsymbol{{and}}\:\boldsymbol{{the}}\:\boldsymbol{{velocity}}\:\boldsymbol{{of}}\:\boldsymbol{{A}}\:\boldsymbol{{makes}} \\ $$$$\boldsymbol{{an}}\:\boldsymbol{{angle}}\:\boldsymbol{{of}}\:\mathrm{10}°\:\boldsymbol{{with}}\:\boldsymbol{{the}}\:\boldsymbol{{lines}}\:\boldsymbol{{of}}\:\boldsymbol{{centers}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{two}}\:\boldsymbol{{sphere}}. \\ $$$$\boldsymbol{{The}}\:\boldsymbol{{coefficient}}\:\boldsymbol{{of}}\:\boldsymbol{{restitution}}\:\boldsymbol{{between}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{spheres}}\:\boldsymbol{{is}}\:\frac{\mathrm{1}}{\mathrm{2}}.\:\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{magnitude}}\:\boldsymbol{{and}}\: \\ $$$$\boldsymbol{{directions}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{velovities}}\:\boldsymbol{{of}}\:\boldsymbol{{A}}\:\boldsymbol{{and}}\:\boldsymbol{{B}} \\ $$$$\boldsymbol{{immediately}}\:\boldsymbol{{after}}\:\boldsymbol{{the}}\:\boldsymbol{{impact}}. \\ $$

Question Number 200418    Answers: 1   Comments: 0

calculus ( I ) If , I = ∫_0 ^( π) (( x )/(1 + sin^2 (x))) dx = a ζ ( 2 ) ⇒ a = ? where , ζ (s ) = Σ_(n=1) ^∞ (( 1)/n^( s) )

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{calculus}\:\:\left(\:\:\mathrm{I}\:\:\right)\:\: \\ $$$$\:\:\mathrm{I}{f}\:,\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\pi} \:\frac{\:{x}\:}{\mathrm{1}\:\:+\:\mathrm{sin}^{\mathrm{2}} \left({x}\right)}\:\mathrm{d}{x}\:=\:{a}\:\zeta\:\left(\:\mathrm{2}\:\right)\:\: \\ $$$$\:\:\:\:\:\:\:\Rightarrow\:\:\:\:{a}\:=\:?\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:{where}\:\:,\:\:\:\zeta\:\left({s}\:\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\mathrm{1}}{{n}^{\:{s}} } \\ $$$$ \\ $$

Question Number 200251    Answers: 1   Comments: 0

Question Number 199459    Answers: 2   Comments: 0

What minimum value f(x,y)=x^2 +y^2 −z^2 when x+2y+4z=21

$$\:\:\mathrm{What}\:\mathrm{minimum}\:\mathrm{value}\: \\ $$$$\:\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} −\mathrm{z}^{\mathrm{2}} \:\mathrm{when}\: \\ $$$$\:\:\mathrm{x}+\mathrm{2y}+\mathrm{4z}=\mathrm{21} \\ $$

Question Number 199261    Answers: 0   Comments: 0

Question Number 199176    Answers: 1   Comments: 0

If f(x) =(x^2 −4x) sin 4x find f^((6)) (x).

$$\:\:\:\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)\:=\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4x}\right)\:\mathrm{sin}\:\mathrm{4x}\: \\ $$$$\:\:\:\mathrm{find}\:\mathrm{f}^{\left(\mathrm{6}\right)} \left(\mathrm{x}\right).\: \\ $$

  Pg 1      Pg 2      Pg 3      Pg 4      Pg 5      Pg 6      Pg 7      Pg 8      Pg 9      Pg 10   

Terms of Service

Privacy Policy

Contact: info@tinkutara.com