Question and Answers Forum

AllQuestion and Answers: Page 1386

Question Number 66172 Answers: 0 Comments: 4

let A_n =Π_(k=1) ^n (1+(k^2 /n^2 )) calculate lim_(n→+∞) ((ln(A_n ))/n)

$$ \\ $$$${let}\:{A}_{{n}} =\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{{k}^{\mathrm{2}} }{{n}^{\mathrm{2}} }\right)\:\:\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:\frac{{ln}\left({A}_{{n}} \right)}{{n}} \\ $$

Question Number 66171 Answers: 1 Comments: 1

find lim_(n→+∞) (1/n^2 ){sin((1/n^2 ))+2sin((4/n^2 ))+....(n−1)sin((((n−1)^2 )/n^2 ))}

$${find}\:{lim}_{{n}\rightarrow+\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left\{{sin}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)+\mathrm{2}{sin}\left(\frac{\mathrm{4}}{{n}^{\mathrm{2}} }\right)+....\left({n}−\mathrm{1}\right){sin}\left(\frac{\left({n}−\mathrm{1}\right)^{\mathrm{2}} }{{n}^{\mathrm{2}} }\right)\right\} \\ $$

Question Number 66170 Answers: 0 Comments: 1

calculate ∫_0 ^∞ sin(x^3 )dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:{sin}\left({x}^{\mathrm{3}} \right){dx} \\ $$

Question Number 66169 Answers: 0 Comments: 1

find the values of ∫_0 ^∞ cos(x^2 )dx and ∫_0 ^∞ sin(x^2 )dx(fresnel integrals) by using Γ(z) =∫_0 ^∞ t^(z−1) e^(−t) dt

$${find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}\:{and}\:\int_{\mathrm{0}} ^{\infty} \:{sin}\left({x}^{\mathrm{2}} \right){dx}\left({fresnel}\:{integrals}\right) \\ $$$${by}\:{using}\:\Gamma\left({z}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{z}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:\: \\ $$

Question Number 66168 Answers: 0 Comments: 0

prove without calculus that ∫_0 ^∞ cos(x^2 )dx=∫_0 ^∞ sin(x^2 )dx

$${prove}\:{without}\:{calculus}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:{cos}\left({x}^{\mathrm{2}} \right){dx}=\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 66163 Answers: 1 Comments: 0

Question Number 66162 Answers: 3 Comments: 3

Question Number 66161 Answers: 1 Comments: 0

Question Number 66160 Answers: 0 Comments: 0

Question Number 66150 Answers: 0 Comments: 1

find ∫_0 ^∞ e^(−x^3 ) sin(x^3 )dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{3}} } {sin}\left({x}^{\mathrm{3}} \right){dx}\: \\ $$

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 66126 Answers: 1 Comments: 0

Is there any formula to find sum of 1 + n^2 + n^4 + n^6 + n^8 + ... + n^(2k) + ... where n,k ∈ Z^+

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{formula}\:\mathrm{to}\:\mathrm{find}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{1}\:+\:{n}^{\mathrm{2}} \:+\:{n}^{\mathrm{4}} \:+\:{n}^{\mathrm{6}} \:+\:{n}^{\mathrm{8}} \:+\:...\:+\:{n}^{\mathrm{2}{k}} \:+\:... \\ $$$$\mathrm{where}\:{n},{k}\:\in\:\mathbb{Z}^{+} \: \\ $$

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 66109 Answers: 1 Comments: 1

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 66106 Answers: 1 Comments: 0

In an electrolysis experiment, the ammeter records a steady current of 1A. The mass of copper deposited in 30minutes is 0.66g. Calculate the error in the ammeter reading. the electrical equivalent of cu is 3.3×10^(−4) g/C.

$$\mathrm{In}\:\mathrm{an}\:\mathrm{electrolysis}\:\mathrm{experiment},\:\mathrm{the}\:\mathrm{ammeter} \\ $$$$\mathrm{records}\:\mathrm{a}\:\mathrm{steady}\:\mathrm{current}\:\mathrm{of}\:\mathrm{1A}.\:\mathrm{The}\:\mathrm{mass} \\ $$$$\mathrm{of}\:\mathrm{copper}\:\mathrm{deposited}\:\mathrm{in}\:\mathrm{30minutes}\:\mathrm{is}\:\mathrm{0}.\mathrm{66g}. \\ $$$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{error}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ammeter}\:\mathrm{reading}. \\ $$$$\:\:\:\mathrm{the}\:\mathrm{electrical}\:\mathrm{equivalent}\:\mathrm{of}\:\mathrm{cu}\:\mathrm{is}\:\mathrm{3}.\mathrm{3}×\mathrm{10}^{−\mathrm{4}} \mathrm{g}/\mathrm{C}. \\ $$

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

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

Question Number 66102 Answers: 0 Comments: 3

prove by mathematical induction that 4^n +3^n +2 is a multiple of 3 for all positive integral values of n.

$${prove}\:{by}\:{mathematical}\:{induction}\:{that}\:\:\mathrm{4}^{{n}} +\mathrm{3}^{{n}} +\mathrm{2}\:{is}\:{a}\:{multiple}\:{of}\:\mathrm{3}\:{for}\:{all}\: \\ $$$${positive}\:{integral}\:{values}\:{of}\:{n}. \\ $$

Question Number 66101 Answers: 1 Comments: 1

find (dy/dx) when y = x^2 ln(3x) Given that xsinx − y^2 =0 show that y^2 = 2cosx −2((dy/dx))^2 −2y(d^2 y/dx^2 )

$${find}\:\frac{{dy}}{{dx}}\:\:{when}\:{y}\:=\:{x}^{\mathrm{2}} {ln}\left(\mathrm{3}{x}\right) \\ $$$${Given}\:{that}\:{xsinx}\:−\:{y}^{\mathrm{2}} =\mathrm{0}\:{show}\:{that}\:\:{y}^{\mathrm{2}} \:=\:\mathrm{2}{cosx}\:−\mathrm{2}\left(\frac{{dy}}{{dx}}\right)^{\mathrm{2}} \:−\mathrm{2}{y}\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} } \\ $$

Question Number 66099 Answers: 0 Comments: 1

differentiate y=10^(1−sin^2 3x)

$${differentiate}\:{y}=\mathrm{10}^{\mathrm{1}−{sin}^{\mathrm{2}} \mathrm{3}{x}} \\ $$

Pg 1381 Pg 1382 Pg 1383 Pg 1384 Pg 1385 Pg 1386 Pg 1387 Pg 1388 Pg 1389 Pg 1390

Contact: info@tinkutara.com