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Question Number 66467 Answers: 0 Comments: 1

calculate A_n =∫_0 ^∞ (dx/((n+x^n )^2 )) with n>1

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({n}+{x}^{{n}} \right)^{\mathrm{2}} }\:\:\:{with}\:{n}>\mathrm{1} \\ $$

Question Number 66461 Answers: 0 Comments: 0

x(n)=3n^2 −2n+7 find even and odd component

$${x}\left({n}\right)=\mathrm{3}{n}^{\mathrm{2}} −\mathrm{2}{n}+\mathrm{7} \\ $$$${find}\:{even}\:{and}\:{odd}\:{component} \\ $$

Question Number 66439 Answers: 0 Comments: 4

for a geometric series. can the sun to infinty use the two formulas S_∞ = (a/(1−r)) ∣r∣ <1 and S_∞ = (a/(r−1)) ∣r∣ > 1 ?? please i am getting confused on this.

$${for}\:{a}\:{geometric}\:{series}. \\ $$$${can}\:{the}\:{sun}\:{to}\:{infinty}\:{use}\:{the}\:{two}\:{formulas} \\ $$$${S}_{\infty} =\:\frac{{a}}{\mathrm{1}−{r}}\:\:\mid{r}\mid\:\:<\mathrm{1}\:\:{and}\:{S}_{\infty} \:=\:\frac{{a}}{{r}−\mathrm{1}}\:\mid{r}\mid\:>\:\mathrm{1}\:??\:{please}\:{i}\:{am}\:{getting}\:{confused}\:{on}\:{this}. \\ $$

Question Number 66421 Answers: 1 Comments: 0

show that for a given complex number z z^n = r^n (cosnθ + isinnθ)

$${show}\:{that}\:{for}\:{a}\:{given}\:{complex}\:{number}\:{z} \\ $$$$\:{z}^{{n}} \:=\:{r}^{{n}} \:\left({cosn}\theta\:+\:{isinn}\theta\right)\: \\ $$

Question Number 66420 Answers: 0 Comments: 3

solve the differential equation 2(d^2 y/dx^2 ) + (dy/dx) − e^(−x) = 4

$${solve}\:{the}\:{differential}\:{equation} \\ $$$$\:\mathrm{2}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\frac{{dy}}{{dx}}\:−\:{e}^{−{x}} \:=\:\mathrm{4} \\ $$

Question Number 66399 Answers: 0 Comments: 1

Show that for all real values of x; x^(2/3) + 6x^(1/3) + 10 >0

$${Show}\:{that}\:{for}\:{all}\:{real} \\ $$$${values}\:{of}\:{x};\: \\ $$$$\:\:{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \:+\:\mathrm{6}{x}^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\mathrm{10}\:>\mathrm{0} \\ $$

Question Number 66356 Answers: 1 Comments: 0

Question Number 66293 Answers: 0 Comments: 3

What do we mean by ∫_(−∞) ^(+∞) f(x) dx?

$${What}\:{do}\:{we}\:{mean}\:{by}\:\: \\ $$$$\:\:\int_{−\infty} ^{+\infty} {f}\left({x}\right)\:{dx}? \\ $$

Question Number 66285 Answers: 2 Comments: 0

What is the difference between lim_(x→2^− ) and lim_(x→2^+ )

$${What}\:{is}\:{the}\:{difference}\:{between} \\ $$$$\:\:\underset{{x}\rightarrow\mathrm{2}^{−} } {{lim}}\:\:{and} \\ $$$$\underset{{x}\rightarrow\mathrm{2}^{+} } {{lim}} \\ $$

Question Number 66228 Answers: 0 Comments: 3

prove that ∫_2 ^4 ((6x +1)/((2x−3)(3x−2)))dx = ln 10

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{2}} ^{\mathrm{4}} \frac{\mathrm{6}{x}\:+\mathrm{1}}{\left(\mathrm{2}{x}−\mathrm{3}\right)\left(\mathrm{3}{x}−\mathrm{2}\right)}{dx}\:=\:{ln}\:\mathrm{10} \\ $$

Question Number 66227 Answers: 0 Comments: 0

Using a good counter procedure, prove that (∂y/∂x) = lim_(∂x→0) ((f(∂ + x) −f(x))/∂x) for a given function f(x) in x.

$${Using}\:{a}\:{good}\:{counter}\:{procedure},\:{prove}\:{that}\: \\ $$$$\:\:\:\frac{\partial{y}}{\partial{x}}\:=\:\underset{\partial{x}\rightarrow\mathrm{0}} {{lim}}\frac{{f}\left(\partial\:+\:{x}\right)\:−{f}\left({x}\right)}{\partial{x}} \\ $$$${for}\:{a}\:{given}\:{function}\:\:{f}\left({x}\right)\:{in}\:{x}. \\ $$

Question Number 66226 Answers: 0 Comments: 1

the equation f(x)=0 has real roots in the interval (a, b) if A −f(a)>0 and f(b) >0 B f(a) <0 and f(b) <0 C −f(a) >0 and f(b) =0 D f(a) >0 and f(b) < 0

$${the}\:{equation}\:\:{f}\left({x}\right)=\mathrm{0}\:{has}\:{real}\:{roots}\:{in}\: \\ $$$${the}\:{interval}\:\left({a},\:{b}\right)\:{if} \\ $$$${A}\:\:\:\:−{f}\left({a}\right)>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:>\mathrm{0} \\ $$$${B}\:\:\:{f}\left({a}\right)\:<\mathrm{0}\:{and}\:{f}\left({b}\right)\:<\mathrm{0} \\ $$$${C}\:\:−{f}\left({a}\right)\:>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:=\mathrm{0} \\ $$$${D}\:\:{f}\left({a}\right)\:>\mathrm{0}\:\:{and}\:{f}\left({b}\right)\:<\:\mathrm{0} \\ $$

Question Number 66225 Answers: 1 Comments: 2

Given that f(x)= { ((−x + 1, x≤ 3_ )),((kx −8, x >3)) :} is continuous then f(5) = A 2 B 0 C −2 D −1

$${Given}\:{that}\:\:\:\:\:{f}\left({x}\right)=\begin{cases}{−{x}\:+\:\mathrm{1},\:\:{x}\leqslant\:\mathrm{3}_{} }\\{{kx}\:−\mathrm{8},\:\:\:\:{x}\:>\mathrm{3}}\end{cases} \\ $$$${is}\:{continuous}\:{then}\:\:{f}\left(\mathrm{5}\right)\:=\: \\ $$$${A}\:\:\:\mathrm{2} \\ $$$${B}\:\:\:\mathrm{0} \\ $$$${C}\:\:−\mathrm{2} \\ $$$${D}\:\:−\mathrm{1} \\ $$$$ \\ $$

Question Number 66216 Answers: 1 Comments: 2

∣a ∣ = 3 ,∣b∣= 5 , a.b =−14 ∣a − b∣ = ?

$$\mid{a}\:\mid\:=\:\mathrm{3}\:,\mid{b}\mid=\:\mathrm{5}\:,\:{a}.{b}\:=−\mathrm{14} \\ $$$$\:\:\mid{a}\:−\:{b}\mid\:=\:? \\ $$

Question Number 66815 Answers: 2 Comments: 1

solve the congruence equation 6x ≡ 4 (mod 5) i need help please with some explanations

$${solve}\:{the}\:{congruence}\:{equation}\: \\ $$$$\:\:\mathrm{6}{x}\:\equiv\:\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:\:{i}\:{need}\:{help}\:{please}\:{with}\:{some}\:{explanations} \\ $$

Question Number 66160 Answers: 0 Comments: 0

Question Number 66149 Answers: 2 Comments: 0

f(x) =2x^3 −x−4 show that f(x) =0 has roots between 1 and 2

$${f}\left({x}\right)\:=\mathrm{2}{x}^{\mathrm{3}} −{x}−\mathrm{4}\: \\ $$$${show}\:{that}\:{f}\left({x}\right)\:=\mathrm{0}\:{has}\:{roots}\:{between} \\ $$$$\mathrm{1}\:{and}\:\mathrm{2} \\ $$

Question Number 66140 Answers: 0 Comments: 0

1.Show that: ∫_0 ^(π/2) f(sin 2x)sin x dx=(√2) ∫_0 ^(π/4) f(cos 2x)cos x dx. 2.If f(z)=(d/dz){5^(∣f(z)∣) } then what is the value of f′(e)?

$$\mathrm{1}.\boldsymbol{{Show}}\:\boldsymbol{{that}}:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(\mathrm{sin}\:\mathrm{2}{x}\right)\mathrm{sin}\:{x}\:{dx}=\sqrt{\mathrm{2}}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {f}\left(\mathrm{cos}\:\mathrm{2}{x}\right)\mathrm{cos}\:{x}\:{dx}. \\ $$$$\mathrm{2}.\boldsymbol{{If}}\:\boldsymbol{{f}}\left(\boldsymbol{{z}}\right)=\frac{\boldsymbol{{d}}}{\boldsymbol{{dz}}}\left\{\mathrm{5}^{\mid\boldsymbol{{f}}\left(\boldsymbol{{z}}\right)\mid} \right\}\:\:\boldsymbol{{then}}\:\boldsymbol{{what}}\:\boldsymbol{{is}}\:\boldsymbol{{the}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{f}}'\left(\boldsymbol{{e}}\right)? \\ $$

Question Number 66116 Answers: 0 Comments: 5

Given that f(x) = { ((x, for 0≤x<2)),((0, for 2≤x≤3)) :} is periodic with period 3 units, find the value of f(5) and f(−5) sketch the graph of f(x) for x between −3 and 6 please i really need explanations when solving the first part of the question thanks

$${Given}\:{that}\:\:{f}\left({x}\right)\:=\:\begin{cases}{{x},\:\:{for}\:\mathrm{0}\leqslant{x}<\mathrm{2}}\\{\mathrm{0},\:{for}\:\mathrm{2}\leqslant{x}\leqslant\mathrm{3}}\end{cases} \\ $$$${is}\:{periodic}\:{with}\:{period}\:\mathrm{3}\:{units}, \\ $$$${find}\:{the}\:{value}\:{of}\:\:{f}\left(\mathrm{5}\right)\:{and}\:{f}\left(−\mathrm{5}\right) \\ $$$${sketch}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:{for}\:{x}\:{between}\:−\mathrm{3}\:{and}\:\mathrm{6} \\ $$$$ \\ $$$${please}\:{i}\:{really}\:{need}\:{explanations}\:{when}\:{solving}\:{the}\:{first}\:{part}\:{of}\:{the}\:{question} \\ $$$${thanks} \\ $$

Question Number 66115 Answers: 0 Comments: 4

find ∣z∣ where z = (((1+i(√3) )^3 )/((1−i)^3 )) find the maximum value of 12sinx − 5cosx

$$\:{find}\:\mid{z}\mid\:\:{where}\:{z}\:=\:\frac{\left(\mathrm{1}+{i}\sqrt{\mathrm{3}}\:\right)^{\mathrm{3}} }{\left(\mathrm{1}−{i}\right)^{\mathrm{3}} } \\ $$$${find}\:{the}\:{maximum}\:{value}\:{of}\:\:\:\mathrm{12}{sinx}\:−\:\mathrm{5}{cosx} \\ $$

Question Number 66114 Answers: 1 Comments: 0

∫(((e^(2x) −sin2x)/(e^(2x) +cos2x)))dx = ?

$$\int\left(\frac{{e}^{\mathrm{2}{x}} −{sin}\mathrm{2}{x}}{{e}^{\mathrm{2}{x}} +{cos}\mathrm{2}{x}}\right){dx}\:=\:? \\ $$

Question Number 66108 Answers: 0 Comments: 4

Given that the binomial expansion of ((2 + kx)/((2−5x)^(2 ) )) , ∣x∣ < (2/(5 )) ,in ascending powers of x is (1/2) + (7/4)x + Ax^2 + ..., find the values of A and k

$${Given}\:{that}\:{the}\:{binomial}\:{expansion}\:{of}\:\frac{\mathrm{2}\:+\:{kx}}{\left(\mathrm{2}−\mathrm{5}{x}\right)^{\mathrm{2}\:} }\:,\:\mid{x}\mid\:<\:\frac{\mathrm{2}}{\mathrm{5}\:}\:,{in}\:{ascending} \\ $$$${powers}\:{of}\:{x}\:{is}\:\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{7}}{\mathrm{4}}{x}\:+\:{Ax}^{\mathrm{2}} \:+\:...,\:{find}\:{the}\:{values}\:{of}\:{A}\:{and}\:{k} \\ $$

Question Number 66107 Answers: 0 Comments: 3

Given that S_n = ((a(1 −r^n ))/(1−r)) , r ≠ 1, show that ((S_(3n) −S_(2n) )/(S_n )) = r^(2n) hence given that r =(1/2) find Σ_(n=0) ^∞ (((S_(3n) −S_(2n) )/S_n ))

$${Given}\:{that}\:{S}_{{n}} \:=\:\frac{{a}\left(\mathrm{1}\:−{r}^{{n}} \right)}{\mathrm{1}−{r}}\:,\:{r}\:\neq\:\mathrm{1},\:{show}\:{that}\:\frac{{S}_{\mathrm{3}{n}} \:−{S}_{\mathrm{2}{n}} }{{S}_{{n}} \:}\:=\:{r}^{\mathrm{2}{n}} \\ $$$${hence}\:{given}\:{that}\:{r}\:=\frac{\mathrm{1}}{\mathrm{2}}\:{find}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{{S}_{\mathrm{3}{n}} \:−{S}_{\mathrm{2}{n}} }{{S}_{{n}} }\right) \\ $$

Question Number 66105 Answers: 0 Comments: 3

Find ∫_(−1) ^1 ((9 +4x^2 )/(9−4x^2 )) dx

$${Find}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{\mathrm{9}\:+\mathrm{4}{x}^{\mathrm{2}} }{\mathrm{9}−\mathrm{4}{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 66104 Answers: 0 Comments: 1

f(x)= 2x^3 −x−4 show that the equation f(x) =0 has root between 1 and 2 show that the equation f(x) =0 can be written as x = (√(((2/x) +(1/2)))) use the iteration x_(n+1 ) = (√(((2/x_n ) +(1/2)) ,)) with x_0 = 1.385 to find to 3 decimal places the value of x_1 .

$${f}\left({x}\right)=\:\mathrm{2}{x}^{\mathrm{3}} −{x}−\mathrm{4} \\ $$$${show}\:{that}\:{the}\:{equation}\:{f}\left({x}\right)\:=\mathrm{0}\:{has}\:{root}\:{between}\:\mathrm{1}\:{and}\:\mathrm{2} \\ $$$${show}\:{that}\:{the}\:{equation}\:{f}\left({x}\right)\:=\mathrm{0}\:{can}\:{be}\:{written}\:{as}\: \\ $$$$\:\:{x}\:=\:\sqrt{\left(\frac{\mathrm{2}}{{x}}\:+\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$${use}\:{the}\:{iteration} \\ $$$$\:{x}_{{n}+\mathrm{1}\:} \:=\:\sqrt{\left(\frac{\mathrm{2}}{{x}_{{n}} }\:+\frac{\mathrm{1}}{\mathrm{2}}\right)\:,} \\ $$$${with}\:{x}_{\mathrm{0}} \:=\:\mathrm{1}.\mathrm{385}\:{to}\:{find}\:{to}\:\mathrm{3}\:{decimal}\:{places}\:{the}\:{value}\:{of}\:{x}_{\mathrm{1}} . \\ $$$$ \\ $$

Question Number 66103 Answers: 0 Comments: 2

A binary relation R is defined on N,the set of natural numbers by _x R_y ⇔ ∃ n ∈ Z : x = 2^n y, x,y ∈ N show that R is an equivalence relation

$${A}\:{binary}\:{relation}\:{R}\:{is}\:{defined}\:{on}\:\mathbb{N},{the}\:{set}\:{of}\:{natural}\:{numbers}\:{by}\: \\ $$$$\:_{{x}} {R}_{{y}} \:\Leftrightarrow\:\exists\:{n}\:\in\:\mathbb{Z}\::\:{x}\:=\:\mathrm{2}^{{n}} {y},\:\:{x},{y}\:\in\:\mathbb{N} \\ $$$${show}\:{that}\:{R}\:{is}\:{an}\:{equivalence}\:{relation} \\ $$

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