Question and Answers Forum

AllQuestion and Answers: Page 507

Question Number 163259 Answers: 1 Comments: 0

Question Number 163257 Answers: 0 Comments: 0

Re ( ∫_0 ^( 1) sin^( −1) ((( 1)/(1− x^( 2) )) )dx )=?

$$ \\ $$$$\:\:\:\:\:\mathscr{R}{e}\:\left(\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {sin}^{\:−\mathrm{1}} \left(\frac{\:\mathrm{1}}{\mathrm{1}−\:{x}^{\:\mathrm{2}} }\:\right){dx}\:\right)=? \\ $$$$ \\ $$

Question Number 163263 Answers: 1 Comments: 0

Question Number 163293 Answers: 1 Comments: 1

Question Number 163226 Answers: 2 Comments: 3

Question Number 163223 Answers: 1 Comments: 0

The arc of parabola y=−x^2 +9 in 0<x<3 is revolved about the line y=c and 0<c<9 to generate a solid. Find the value of c that minimizes the volume of the solid.

$$\mathrm{The}\:\mathrm{arc}\:\mathrm{of}\:\mathrm{parabola}\:{y}=−{x}^{\mathrm{2}} +\mathrm{9}\:\mathrm{in}\:\mathrm{0}<{x}<\mathrm{3} \\ $$$$\mathrm{is}\:\mathrm{revolved}\:\mathrm{about}\:\mathrm{the}\:\mathrm{line}\:{y}={c}\:\mathrm{and}\:\mathrm{0}<{c}<\mathrm{9} \\ $$$$\mathrm{to}\:\mathrm{generate}\:\mathrm{a}\:\mathrm{solid}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{c}\:\mathrm{that}\:\mathrm{minimizes}\:\mathrm{the}\: \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}. \\ $$

Question Number 163222 Answers: 1 Comments: 0

(√(((√(x+4)) +2)/(2−(√(x+4))))) ≤ x−4

$$\:\:\:\sqrt{\frac{\sqrt{{x}+\mathrm{4}}\:+\mathrm{2}}{\mathrm{2}−\sqrt{{x}+\mathrm{4}}}}\:\leqslant\:{x}−\mathrm{4}\: \\ $$

Question Number 163217 Answers: 0 Comments: 0

Question Number 163211 Answers: 0 Comments: 0

Question Number 163210 Answers: 1 Comments: 1

Question Number 163209 Answers: 0 Comments: 0

f : I → (0 ; ∞) ; I ⊂ R f - twice derivable ; f^′ ; f^(′′) - continuous f^(′′) (x) f(x) ≥ (f^′ (x))^2 ; ∀ x ∈ I then prove that: 2f (((x + y)/2)) ≤ f(x) + f(y) ; ∀ x;y ∈ I

$$\mathrm{f}\::\:\mathrm{I}\:\rightarrow\:\left(\mathrm{0}\:;\:\infty\right)\:\:;\:\:\mathrm{I}\:\subset\:\mathbb{R} \\ $$$$\mathrm{f}\:-\:\mathrm{twice}\:\mathrm{derivable}\:\:;\:\:\mathrm{f}\:^{'} \:;\:\mathrm{f}\:^{''} \:-\:\mathrm{continuous} \\ $$$$\mathrm{f}\:^{''} \left(\mathrm{x}\right)\:\mathrm{f}\left(\mathrm{x}\right)\:\geqslant\:\left(\mathrm{f}\:^{'} \left(\mathrm{x}\right)\right)^{\mathrm{2}} \:;\:\:\forall\:\mathrm{x}\:\in\:\mathrm{I} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{2f}\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}}{\mathrm{2}}\right)\:\leqslant\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\mathrm{y}\right)\:\:;\:\:\forall\:\mathrm{x};\mathrm{y}\:\in\:\mathrm{I} \\ $$

Question Number 163205 Answers: 0 Comments: 0

Fourier series expansion for ln(sin(x))

$$\boldsymbol{{F}}{ourier}\:{series}\:{expansion}\:{for}\:{ln}\left({sin}\left({x}\right)\right) \\ $$

Question Number 163197 Answers: 0 Comments: 1