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Question Number 11608    Answers: 1   Comments: 0

If f(x)=xtan^(−1) ((1/x)) , x≠0 =0 , x=0 show that f is countinous but not differentiable at x=0.

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{xtan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:,\:\:\:\:\:\:\:\mathrm{x}\neq\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{0}\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}=\mathrm{0} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{f}\:\mathrm{is}\:\mathrm{countinous}\:\mathrm{but}\:\mathrm{not}\:\mathrm{differentiable} \\ $$$$\mathrm{at}\:\mathrm{x}=\mathrm{0}. \\ $$

Question Number 11607    Answers: 1   Comments: 0

A parabola with equation y = (x^2 /k) − 5 intersects a circle with equation x^2 + y^2 = 25 at exactly 3 points A, B, C Determine all such positive integers k for which the area of ΔABC is an integer

$$\mathrm{A}\:\mathrm{parabola}\:\mathrm{with}\:\mathrm{equation}\:{y}\:=\:\frac{{x}^{\mathrm{2}} }{{k}}\:−\:\mathrm{5}\:\mathrm{intersects} \\ $$$$\mathrm{a}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{equation}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{25}\:\mathrm{at}\:\mathrm{exactly}\:\mathrm{3}\:\mathrm{points}\:{A},\:{B},\:{C} \\ $$$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{such}\:\mathrm{positive}\:\mathrm{integers}\:{k}\:\mathrm{for}\:\mathrm{which} \\ $$$$\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\Delta{ABC}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer} \\ $$

Question Number 11606    Answers: 1   Comments: 0

What is the smallest positive integer x for which (1/(32)) = (x/(10^y )) for some positive integer y ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\:{x}\:\mathrm{for} \\ $$$$\mathrm{which}\:\frac{\mathrm{1}}{\mathrm{32}}\:=\:\frac{{x}}{\mathrm{10}^{{y}} }\:\:\mathrm{for}\:\mathrm{some}\:\mathrm{positive}\:\mathrm{integer}\:{y}\:? \\ $$

Question Number 11600    Answers: 1   Comments: 0

A number is selected at random from the integers 1 to 100 exclusive, What is the chance of choosing at random a two digit number which is (a) Divisible by 4 (b) The multiple of 5

$$\mathrm{A}\:\mathrm{number}\:\mathrm{is}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{1}\:\mathrm{to}\:\mathrm{100}\:\mathrm{exclusive},\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{chance}\:\mathrm{of}\:\mathrm{choosing}\:\mathrm{at}\:\mathrm{random}\:\mathrm{a}\:\mathrm{two}\:\mathrm{digit}\:\mathrm{number}\:\mathrm{which}\:\mathrm{is} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Divisible}\:\mathrm{by}\:\mathrm{4} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{The}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{5} \\ $$

Question Number 11599    Answers: 1   Comments: 0

A number is selected at random from the integers 10 to 35 exclussive, such that the chances take the subsets of P, Q and R respectively, where P is even , Q is even or prime number, and R is odd prime numbers. What is the probability (a) That exacly one of the events of the subset is choosen ? (b) Of choosen at most two of the subset of the event.

$$\mathrm{A}\:\mathrm{number}\:\mathrm{is}\:\mathrm{selected}\:\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{integers}\:\mathrm{10}\:\mathrm{to}\:\mathrm{35}\:\mathrm{exclussive},\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{chances}\:\mathrm{take}\:\mathrm{the}\:\mathrm{subsets}\:\mathrm{of}\:\mathrm{P},\:\mathrm{Q}\:\mathrm{and}\:\mathrm{R}\:\mathrm{respectively},\:\mathrm{where}\:\mathrm{P}\:\mathrm{is} \\ $$$$\mathrm{even}\:,\:\mathrm{Q}\:\mathrm{is}\:\mathrm{even}\:\mathrm{or}\:\mathrm{prime}\:\mathrm{number},\:\mathrm{and}\:\mathrm{R}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{prime}\:\mathrm{numbers}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{probability}\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{That}\:\mathrm{exacly}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{events}\:\mathrm{of}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{is}\:\mathrm{choosen}\:? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Of}\:\mathrm{choosen}\:\mathrm{at}\:\mathrm{most}\:\mathrm{two}\:\mathrm{of}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{the}\:\mathrm{event}. \\ $$

Question Number 11597    Answers: 1   Comments: 0

y = x^x^(√x) , find (dy/dx)

$$\mathrm{y}\:=\:\mathrm{x}^{\mathrm{x}^{\sqrt{\mathrm{x}}} } ,\:\mathrm{find}\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$

Question Number 11594    Answers: 1   Comments: 0

please how can demonstred sin(2x)_ /cos(2x)=2sin(2x)

$${please}\:{how}\:{can}\:{demonstred}\: \\ $$$${sin}\left(\mathrm{2}{x}\underset{} {\right)}/{cos}\left(\mathrm{2}{x}\right)=\mathrm{2}{sin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 11591    Answers: 1   Comments: 0

what is the intercept of (x^2 /a^2 )−(y^2 /b^2 )=1 on the x-axis ?

$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{intercept}\:\mathrm{of}\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{a}^{\mathrm{2}} }−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{b}^{\mathrm{2}} }=\mathrm{1}\:\:\mathrm{on} \\ $$$$\mathrm{the}\:\mathrm{x}-\mathrm{axis}\:? \\ $$

Question Number 11581    Answers: 2   Comments: 0

cAn you prove the chAin role??

$${cAn}\:{you}\:{prove}\:{the}\:{chAin}\:{role}?? \\ $$

Question Number 11580    Answers: 1   Comments: 0

why (dy/dx)=(dy/du).(du/dx)

$${why}\:\frac{{dy}}{{dx}}=\frac{{dy}}{{du}}.\frac{{du}}{{dx}} \\ $$

Question Number 11571    Answers: 2   Comments: 0

why ((d[f{g(x)}])/dx)=((df[{g(x)}])/(dg(x))).((dg(x))/dx)?

$${why}\:\:\:\frac{{d}\left[{f}\left\{{g}\left({x}\right)\right\}\right]}{{dx}}=\frac{{df}\left[\left\{{g}\left({x}\right)\right\}\right]}{{dg}\left({x}\right)}.\frac{{dg}\left({x}\right)}{{dx}}? \\ $$

Question Number 11567    Answers: 1   Comments: 0

why lim_(x→0) ((f(x))/(g(x)))=lim_(x→0) ((f′(x))/(g′(x)))

$${why}\:\:\:\:\:\:{li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}={li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{f}'\left({x}\right)}{{g}'\left({x}\right)} \\ $$

Question Number 11565    Answers: 1   Comments: 0

if (dy/dx)=p ,then why dy=pdx?

$${if}\:\frac{{dy}}{{dx}}={p}\:\:,{then}\:{why}\:{dy}={pdx}? \\ $$

Question Number 11563    Answers: 1   Comments: 2

if f(x)=g(y) then why (d/dx)(f(x))=(d/dy)(g(y))?

$${if}\:\:{f}\left({x}\right)={g}\left({y}\right) \\ $$$${then}\:{why}\:\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)=\frac{{d}}{{dy}}\left({g}\left({y}\right)\right)? \\ $$

Question Number 11552    Answers: 1   Comments: 0

find (d/dx)(y) where y= ^(√x) (√(√x))

$${find}\:\frac{{d}}{{dx}}\left({y}\right)\:{where}\:\:{y}=\overset{\sqrt{{x}}} {\:}\sqrt{\sqrt{{x}}} \\ $$$$ \\ $$

Question Number 11549    Answers: 1   Comments: 0

proof lim h→0^(((x^h −1)/h)=ln(x))

$$\:\:\:\: \\ $$$${proof} \\ $$$${lim} \\ $$$${h}\rightarrow\mathrm{0}^{\frac{{x}^{{h}} −\mathrm{1}}{{h}}={ln}\left({x}\right)} \\ $$

Question Number 11548    Answers: 1   Comments: 0

find (d/dx)(2^x )

$$ \\ $$$${find}\:\frac{{d}}{{dx}}\left(\mathrm{2}^{{x}} \right) \\ $$

Question Number 11547    Answers: 2   Comments: 0

(d/dx)(x^x )=?[please give the answer with proof]

$$\:\:\:\:\:\:\:\:\:\:\frac{{d}}{{dx}}\left({x}^{{x}} \right)=?\left[{please}\:{give}\:{the}\:{answer}\:{with}\:{proof}\right] \\ $$

Question Number 11546    Answers: 2   Comments: 0

f(x)=^x (√x) find (d/dx)(f(x)) and for what x ,we will get the maximum of the function..?

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{f}\left({x}\right)=^{{x}} \sqrt{{x}}\:{find}\:\frac{{d}}{{dx}}\left({f}\left({x}\right)\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{and}\:{for}\:{what}\:{x}\:,{we}\:{will}\:{get}\:{the}\: \\ $$$$\:{maximum}\:{of}\:{the}\:{function}..? \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 11545    Answers: 2   Comments: 0

find(dy/dx) if x^x y^y =1

$${find}\frac{{dy}}{{dx}}\:{if}\:{x}^{{x}} {y}^{{y}} =\mathrm{1} \\ $$

Question Number 11544    Answers: 1   Comments: 0

Evaluate ∫x^3 e^(2x) dx

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:{Evaluate}\:\int{x}^{\mathrm{3}} {e}^{\mathrm{2}{x}} {dx} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 11543    Answers: 2   Comments: 0

solve for x and y in this equation x^x +y^y =31 x+y=5

$${solve}\:{for}\:{x}\:{and}\:{y}\:{in}\:{this}\:{equation} \\ $$$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$${x}+{y}=\mathrm{5} \\ $$

Question Number 11540    Answers: 1   Comments: 0

Question Number 11536    Answers: 2   Comments: 0

Question Number 11515    Answers: 2   Comments: 0

(((4+x)/x))^(1/5) −(((4−6x)/x))^(1/5) =1. x_1 +x_2 =?

$$\sqrt[{\mathrm{5}}]{\frac{\mathrm{4}+\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}}−\sqrt[{\mathrm{5}}]{\frac{\mathrm{4}−\mathrm{6}\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{x}}}}=\mathrm{1}. \\ $$$$\boldsymbol{\mathrm{x}}_{\mathrm{1}} +\boldsymbol{\mathrm{x}}_{\mathrm{2}} =? \\ $$

Question Number 11517    Answers: 2   Comments: 0

f : R → R If x^2 f(x) + f(1−x) = 2x − x^4 Determine f(x)

$${f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R} \\ $$$$\mathrm{If}\:\:{x}^{\mathrm{2}} {f}\left({x}\right)\:+\:{f}\left(\mathrm{1}−{x}\right)\:=\:\mathrm{2}{x}\:−\:{x}^{\mathrm{4}} \\ $$$$\mathrm{Determine}\:{f}\left({x}\right) \\ $$

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