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if y= 3x^4 find the percentage increase in y if x increases at (5/2)% or 2(1/2)%^ use the Binomial Expansion method (that is find new y and new x and simplify)

$${if}\:{y}=\:\mathrm{3}{x}^{\mathrm{4}} \:{find}\:{the}\:{percentage}\:{increase}\:{in}\:{y} \\ $$$${if}\:{x}\:{increases}\:{at}\:\frac{\mathrm{5}}{\mathrm{2}}\%\:{or}\:\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}}\overset{} {\%} \\ $$$${use}\:{the}\:{Binomial}\:{Expansion}\:{method} \\ $$$$\left({that}\:{is}\:{find}\:{new}\:{y}\:{and}\:{new}\:{x}\:{and}\:{simplify}\right) \\ $$

Question Number 33158 Answers: 0 Comments: 2

the matrice which comes from the transformation matrix (((cosθ −sinθ)),((sin θ cosθ)) ) at 90° is?

$${the}\:{matrice}\:{which}\:{comes}\:{from} \\ $$$${the}\:{transformation}\:{matrix}\: \\ $$$$\:\begin{pmatrix}{{cos}\theta\:\:\:\:\:\:\:\:\:−{sin}\theta}\\{{sin}\:\theta\:\:\:\:\:\:\:\:\:\:\:\:\:\:{cos}\theta}\end{pmatrix} \\ $$$${at}\:\mathrm{90}°\:{is}? \\ $$

Question Number 33157 Answers: 0 Comments: 1

find k if kloga= loga + log a^2 + log a^(3 ) + log a^4

$${find}\:{k}\:{if} \\ $$$${kloga}=\:{loga}\:+\:{log}\:{a}^{\mathrm{2}} +\:{log}\:{a}^{\mathrm{3}\:} +\:{log}\:{a}^{\mathrm{4}} \\ $$

Question Number 33155 Answers: 0 Comments: 4

Evaluate ∫_(−∞) ^∞ 3x^2 (x^3 + 1)^2 e^(−x^6 − 2x^3 ) dx

$$\mathrm{Evaluate} \\ $$$$\int_{−\infty} ^{\infty} \:\mathrm{3}{x}^{\mathrm{2}} \left({x}^{\mathrm{3}} \:+\:\mathrm{1}\right)^{\mathrm{2}} \:{e}^{−{x}^{\mathrm{6}} \:−\:\mathrm{2}{x}^{\mathrm{3}} } \:{dx} \\ $$

Question Number 33154 Answers: 0 Comments: 1

it is given that Σ_(r=1) ^n U_n = ((1+3^(2n+2) −2×5^(n+1) )/8) where U_n is the n^(th) term of a sequence find the simplified expression for U_n

$${it}\:{is}\:{given}\:{that}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:{U}_{{n}} =\:\frac{\mathrm{1}+\mathrm{3}^{\mathrm{2}{n}+\mathrm{2}} −\mathrm{2}×\mathrm{5}^{{n}+\mathrm{1}} }{\mathrm{8}} \\ $$$${where}\:{U}_{{n}} \:{is}\:{the}\:{n}^{{th}} \:{term}\:{of}\:{a}\:{sequence} \\ $$$${find}\:{the}\:{simplified}\:{expression}\:{for}\:{U}_{{n}} \\ $$

Question Number 33151 Answers: 0 Comments: 0

Question Number 33147 Answers: 2 Comments: 1

If x^m occurs in the expansion of (x + (1/x^2 ))^(2n) , the coefficient of x^m is

$$\mathrm{If}\:\:\:{x}^{{m}} \:\:\mathrm{occurs}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left({x}\:+\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)^{\mathrm{2}{n}} ,\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{m}} \:\mathrm{is} \\ $$

Question Number 33146 Answers: 0 Comments: 0

If the expression exp{(1+∣cos x∣+cos^2 x+∣cos^3 x∣+cos^4 x +....∞)log_e 4} satisfies the equation y^2 −20y+64=0 for 0< x <π, then the set of values of x is

$$\mathrm{If}\:\mathrm{the}\:\mathrm{expression}\: \\ $$$${exp}\left\{\left(\mathrm{1}+\mid\mathrm{cos}\:{x}\mid+\mathrm{cos}^{\mathrm{2}} {x}+\mid\mathrm{cos}^{\mathrm{3}} {x}\mid+\mathrm{cos}^{\mathrm{4}} {x}\right.\right. \\ $$$$\left.\:\left.\:\:\:\:\:\:\:\:\:\:\:+....\infty\right)\mathrm{log}_{{e}} \mathrm{4}\right\}\:\mathrm{satisfies}\:\mathrm{the}\:\mathrm{equation} \\ $$$${y}^{\mathrm{2}} −\mathrm{20}{y}+\mathrm{64}=\mathrm{0}\:\mathrm{for}\:\mathrm{0}<\:{x}\:<\pi,\:\mathrm{then}\:\mathrm{the}\:\mathrm{set} \\ $$$$\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{x}\:\mathrm{is} \\ $$

Question Number 33145 Answers: 1 Comments: 1

∫_( 0) ^3 (dx/((√(x+1)) + (√(5x+1)))) =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{3}} {\int}}\:\frac{{dx}}{\sqrt{{x}+\mathrm{1}}\:+\:\sqrt{\mathrm{5}{x}+\mathrm{1}}}\:= \\ $$

Question Number 33139 Answers: 1 Comments: 0

α and β are roots of ax^2 +bx +c=0 show that α+β= ((−b)/a) and αβ= (c/a) hence form an equation whose sum of roots and product of roots are respectively −(1/2) and 2.

$$\alpha\:{and}\:\beta\:{are}\:{roots}\:{of}\: \\ $$$$\:{ax}^{\mathrm{2}} +{bx}\:+{c}=\mathrm{0} \\ $$$${show}\:{that}\:\alpha+\beta=\:\frac{−{b}}{{a}}\:{and}\:\alpha\beta=\:\frac{{c}}{{a}} \\ $$$${hence}\:{form}\:{an}\:{equation}\:{whose} \\ $$$${sum}\:{of}\:{roots}\:{and}\:{product}\:{of}\:{roots} \\ $$$${are}\:{respectively}\: \\ $$$$\:\:\:−\frac{\mathrm{1}}{\mathrm{2}}\:{and}\:\mathrm{2}. \\ $$

Question Number 33136 Answers: 1 Comments: 0

given that y= 3x^4 find the percentage increase in y when x increases at (5/2)%.

$${given}\:{that}\:{y}=\:\mathrm{3}{x}^{\mathrm{4}} \:{find}\:{the}\:{percentage} \\ $$$${increase}\:{in}\:{y}\:{when}\:{x}\:{increases}\:{at}\:\frac{\mathrm{5}}{\mathrm{2}}\%. \\ $$$$ \\ $$

Question Number 33131 Answers: 0 Comments: 1

1)find Σ_(n=1) ^∞ (e^(inx) /(n(n+1))) 2) find the value of Σ_(n≥1) ((sin(nx))/(n(n+1))) and Σ_(n≥1) ((cos(nx))/(n(n+1))) .

$$\left.\mathrm{1}\right){find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{e}^{{inx}} }{{n}\left({n}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{sin}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)} \\ $$$${and}\:\sum_{{n}\geqslant\mathrm{1}} \:\:\frac{{cos}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)}\:. \\ $$

Question Number 33130 Answers: 0 Comments: 0

find the value of ∫_0 ^∞ ((1+x cosθ)/(x^2 +2x cosθ +1)) dx .

$${find}\:\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}\:{cos}\theta}{{x}^{\mathrm{2}} \:+\mathrm{2}{x}\:{cos}\theta\:+\mathrm{1}}\:{dx}\:. \\ $$

Question Number 33129 Answers: 0 Comments: 2

1)find the value of u_n =∫_(−∞) ^(+∞) ((cos(nx))/(4 +x^2 )) dx 2) find the nature of Σ u_n .

$$\left.\mathrm{1}\right){find}\:{the}\:{value}\:{of}\:\:\:{u}_{{n}} =\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left({nx}\right)}{\mathrm{4}\:+{x}^{\mathrm{2}} }\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{nature}\:{of}\:\Sigma\:{u}_{{n}} \:. \\ $$

Question Number 33128 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ (dx/((1+x^2 )( 1+x^4 ))) .

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:\mathrm{1}+{x}^{\mathrm{4}} \right)}\:. \\ $$

Question Number 33127 Answers: 0 Comments: 1

find Σ_(n=0) ^∞ ((sin(na))/((sina)^n )) (x^n /(n!)) with 0<a<π .

$$\:{find}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({na}\right)}{\left({sina}\right)^{{n}} }\:\frac{{x}^{{n}} }{{n}!}\:\:{with}\:\mathrm{0}<{a}<\pi\:. \\ $$

Question Number 33126 Answers: 0 Comments: 1

let give f(x)= (1/(2x^2 −3x+1)) 1) find f^((n)) (x) 2) find f^((n)) (0) 3) if f(x)=Σ a_n x^n calculate the sequence a_n

$${let}\:{give}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} \:−\mathrm{3}{x}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{if}\:\:\:\:{f}\left({x}\right)=\Sigma\:{a}_{{n}} \:{x}^{{n}} \:\:{calculate}\:{the}\:{sequence}\:{a}_{{n}} \\ $$

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