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Question Number 63372 Answers: 1 Comments: 2

For what values of a and b will the integral ∫_a ^b (√(10−x−x^2 ))dx be at maximum

$${For}\:{what}\:{values}\:{of}\:{a}\:{and}\:{b}\:{will}\:{the} \\ $$$${integral}\:\int_{{a}} ^{{b}} \sqrt{\mathrm{10}−{x}−{x}^{\mathrm{2}} }{dx}\:{be}\:{at} \\ $$$${maximum} \\ $$

Question Number 63373 Answers: 0 Comments: 2

just found this on the web I thought it might help in some cases where quartics appear i.e. Sir Aifour′s geometric questions. sometimes we know the nature of the roots, but how to use this information? ax^4 +bx^3 +cx^2 +dx+e=0 1. divide by a 2. x=z−(b/(4a)) this leads to the reduced z^4 +pz^2 +qz+r=0 now we find the nature of the roots: T_1 =16p^4 r−4p^3 q^2 −128p^2 r^2 +144pq^2 r−27q^4 +256r^3 T_2 =p^2 +12r T_3 =−p^2 +4r T_1 <0 ⇒ 2 distinct real and 2 conjugated complex roots T_1 >0∧(p<0∧T_3 <0) ⇒ 4 distinct real roots T_1 >0∧(p>0∨T_3 >0) ⇒ 2 pairs of conjugated complex roots T_1 =0∧(p<0∧T_3 <0∧T_2 ≠0) ⇒ 1 real double and 2 real simple roots T_1 =0∧(T_3 >0∨(p>0∧(T_3 ≠0∨q≠0))) ⇒ 1 real double and 2 conjugated complex roots T_1 =0∧(T_2 =0∧T_3 ≠0) ⇒ 1 real triple and 1 real simple roots T_1 =0∧(T_3 =0∧p<0) ⇒ 2 real double roots T_1 =0∧(T_3 =0∧p>0∧q=0) ⇒ 2 conjugated complex double roots T_1 =0∧T_2 =0 ⇒ all roots are equal

Question Number 63336 Answers: 2 Comments: 1

Question Number 63324 Answers: 1 Comments: 0

Question Number 63399 Answers: 1 Comments: 1

if α and β are the roots of 4x^(2 ) −6x+1===00====================== =0. find α^3 −β^3 .

$${if}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{roots}\:{of}\:\mathrm{4}{x}^{\mathrm{2}\:} −\mathrm{6}{x}+\mathrm{1}===\mathrm{00}====================== \\ $$$$=\mathrm{0}.\:{find}\:\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} . \\ $$

Question Number 63301 Answers: 1 Comments: 9

find (dy/dx) if x(x +y) = y^2

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

Question Number 63300 Answers: 0 Comments: 2

show that a) 1 + tan ((π/4) + A) = (2/(1−tanA)) b) 2cos2θsinθ + 9sinθ + 3 ≡ 11sinθ − 4sin^3 θ + 3

$${show}\:{that}\:\: \\ $$$$\left.{a}\right)\:\mathrm{1}\:+\:{tan}\:\left(\frac{\pi}{\mathrm{4}}\:+\:{A}\right)\:=\:\frac{\mathrm{2}}{\mathrm{1}−{tanA}} \\ $$$$\left.{b}\right)\:\mathrm{2}{cos}\mathrm{2}\theta{sin}\theta\:+\:\mathrm{9}{sin}\theta\:+\:\mathrm{3}\:\equiv\:\mathrm{11}{sin}\theta\:−\:\mathrm{4}{sin}^{\mathrm{3}} \theta\:+\:\mathrm{3} \\ $$

Question Number 63298 Answers: 0 Comments: 2

A particle P, moves on the curve with polar equation r = e^(kθ) , where (r,θ) are polar coordinates referred to a fixed pole and k is a positive constant. Given that the radial velocity of P is (k/r) show that the transverse acceleration of th particle is zero.

$${A}\:{particle}\:{P},\:{moves}\:{on}\:{the}\:{curve}\:{with}\:{polar}\:{equation}\:\: \\ $$$${r}\:=\:{e}^{{k}\theta} \:,\:{where}\:\left({r},\theta\right)\:{are}\:{polar}\:{coordinates}\:{referred}\:{to}\:{a}\:{fixed} \\ $$$${pole}\:{and}\:{k}\:{is}\:{a}\:{positive}\:{constant}.\:{Given}\:{that}\:{the}\:{radial}\:{velocity} \\ $$$${of}\:{P}\:{is}\:\frac{{k}}{{r}}\:\:{show}\:{that}\:{the}\:{transverse}\:{acceleration}\:{of}\:{th}\:{particle} \\ $$$${is}\:{zero}. \\ $$$$ \\ $$

Question Number 63296 Answers: 1 Comments: 1

A random Variable Y has probability function P, defined by P(y) = { (((y^2 /k) , y= 1,2,3)),(((((y−7)^2 )/k) , y= 4,5,6)),((0 , otherwise.)) :} Find (i) The value of the constant k. (ii) the mean and varriance of Y. (iii) The variance R, where R= 2Y −3.

$${A}\:{random}\:{Variable}\:{Y}\:{has}\:{probability}\:{function}\:{P},\:{defined}\:{by} \\ $$$$\:{P}\left({y}\right)\:=\:\begin{cases}{\frac{{y}^{\mathrm{2}} }{{k}}\:,\:{y}=\:\mathrm{1},\mathrm{2},\mathrm{3}}\\{\frac{\left({y}−\mathrm{7}\right)^{\mathrm{2}} }{{k}}\:,\:{y}=\:\mathrm{4},\mathrm{5},\mathrm{6}}\\{\mathrm{0}\:\:\:\:,\:{otherwise}.}\end{cases} \\ $$$${Find}\: \\ $$$$\left({i}\right)\:{The}\:{value}\:{of}\:{the}\:{constant}\:{k}. \\ $$$$\left({ii}\right)\:{the}\:{mean}\:{and}\:{varriance}\:{of}\:{Y}. \\ $$$$\left({iii}\right)\:{The}\:{variance}\:{R},\:{where}\:{R}=\:\mathrm{2}{Y}\:−\mathrm{3}. \\ $$

Question Number 63273 Answers: 0 Comments: 1

let F(x) =∫_x^2 ^x^3 ((sin(t))/(t+x)) dt 1) calculate lim_(x→0) F(x) and lim_(x→+∞) F(x) 2)calculste lim_(x→0) F^′ (x) and lim_(x→+∞) F^′ (x)

$${let}\:{F}\left({x}\right)\:=\int_{{x}^{\mathrm{2}} } ^{{x}^{\mathrm{3}} } \:\:\:\:\:\frac{{sin}\left({t}\right)}{{t}+{x}}\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {F}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculste}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}^{'} \left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} \:{F}^{'} \left({x}\right) \\ $$

Question Number 63261 Answers: 0 Comments: 6

∫x tan(x) dx

$$\int{x}\:{tan}\left({x}\right)\:{dx} \\ $$

Question Number 63247 Answers: 0 Comments: 1

Prove that (√(abc)) + (√((1−a)(1−b)(1−c))) ≤ 1 for 0 ≤ a,b,c ≤ 1

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\sqrt{{abc}}\:+\:\sqrt{\left(\mathrm{1}−{a}\right)\left(\mathrm{1}−{b}\right)\left(\mathrm{1}−{c}\right)}\:\leqslant\:\mathrm{1} \\ $$$$\mathrm{for}\:\mathrm{0}\:\leqslant\:{a},{b},{c}\:\leqslant\:\mathrm{1} \\ $$

Question Number 63246 Answers: 0 Comments: 2

Arrange these digits: 1 1 2 2 3 3 4 4 So that the 1′s are four digit apart So that the 2′s are three digit apart So that the 3′s are two digit apart So that the 4′s are one digit apart

$$\mathrm{Arrange}\:\mathrm{these}\:\mathrm{digits}:\:\:\:\:\:\:\mathrm{1}\:\:\mathrm{1}\:\:\mathrm{2}\:\:\mathrm{2}\:\:\mathrm{3}\:\:\mathrm{3}\:\:\mathrm{4}\:\:\mathrm{4} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{1}'\mathrm{s}\:\mathrm{are}\:\mathrm{four}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{2}'\mathrm{s}\:\mathrm{are}\:\mathrm{three}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{3}'\mathrm{s}\:\mathrm{are}\:\mathrm{two}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$\:\:\:\:\:\:\mathrm{So}\:\mathrm{that}\:\mathrm{the}\:\mathrm{4}'\mathrm{s}\:\mathrm{are}\:\mathrm{one}\:\mathrm{digit}\:\mathrm{apart} \\ $$$$ \\ $$

Question Number 63256 Answers: 0 Comments: 3

Question Number 63233 Answers: 0 Comments: 4

Question Number 63232 Answers: 0 Comments: 2

let B(x,y) =∫_0 ^1 (1−t)^(x−1) t^(y−1) dt 1) study the convergence of B(x,y) 1) prove that B(x,y)=B(y,x) prove that B(x,y) =∫_0 ^∞ (t^(x−1) /((1+t)^(x+y) )) dt 2) prove that B(x,y) =((Γ(x).Γ(y))/(Γ(x+y))) 3) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) for allx ∈]0,1[

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{x}−\mathrm{1}} {t}^{{y}−\mathrm{1}} \:{dt} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{convergence}\:{of}\:{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$${prove}\:{that}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{{x}+{y}} }\:{dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{B}\left({x},{y}\right)\:=\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:{for}\:{allx}\:\in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$

Question Number 63225 Answers: 0 Comments: 0