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Question Number 185054    Answers: 1   Comments: 0

If , f(x)= (x/(⌊ x ⌋)) ⇒ D_( f) =?(domain) and R_( f ) = ? (range )

$$ \\ $$$$\:\:\mathrm{I}{f}\:,\:\:\:{f}\left({x}\right)=\:\frac{{x}}{\lfloor\:{x}\:\rfloor}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\mathrm{D}_{\:{f}} \:=?\left({domain}\right)\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{and}\:\:\:\:\:\:\:\:\:\:\mathrm{R}_{\:{f}\:} =\:?\:\left({range}\:\right) \\ $$

Question Number 185053    Answers: 1   Comments: 0

Question Number 185052    Answers: 0   Comments: 3

Question Number 185042    Answers: 0   Comments: 0

ABCD is a quadrilateral inscribed in a circle if AB^(⌢) + CD^(⌢) = 307° and AC∙BD = 6(√5) find the area of ABCD.

$${ABCD}\:{is}\:{a}\:{quadrilateral}\:{inscribed}\:{in}\:{a}\:{circle} \\ $$$$\:{if}\:\:\:\:\overset{\frown} {{AB}}\:+\:\overset{\frown} {{CD}}\:=\:\mathrm{307}°\:{and}\:{AC}\centerdot{BD}\:=\:\mathrm{6}\sqrt{\mathrm{5}} \\ $$$$\:{find}\:{the}\:{area}\:{of}\:{ABCD}.\: \\ $$

Question Number 185038    Answers: 1   Comments: 0

Find the range of values of x for which the series (x/(27))+(x^2 /(125))+...+(x^n /((2n+1)^3 ))+... is absolutely convergent. Help!

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{for}\:\mathrm{which} \\ $$$$\mathrm{the}\:\mathrm{series}\:\frac{\mathrm{x}}{\mathrm{27}}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{125}}+...+\frac{\mathrm{x}^{\mathrm{n}} }{\left(\mathrm{2n}+\mathrm{1}\right)^{\mathrm{3}} }+... \\ $$$$\mathrm{is}\:\mathrm{absolutely}\:\mathrm{convergent}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 185036    Answers: 1   Comments: 0

Find the series for cosx. Hence, deduce series sin^2 x and show that, if x is small, ((sin^2 x−x^2 cosx)/x^4 )=(1/6)+(x^2 /(360)) approximately. Help!

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{series}\:\mathrm{for}\:\mathrm{cosx}.\:\mathrm{Hence},\: \\ $$$$\mathrm{deduce}\:\mathrm{series}\:\mathrm{sin}^{\mathrm{2}} \mathrm{x}\:\mathrm{and}\:\mathrm{show}\:\mathrm{that}, \\ $$$$\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{small},\:\frac{\mathrm{sin}^{\mathrm{2}} \mathrm{x}−\mathrm{x}^{\mathrm{2}} \mathrm{cosx}}{\mathrm{x}^{\mathrm{4}} }=\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{360}} \\ $$$$\mathrm{approximately}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 185030    Answers: 1   Comments: 1

Question Number 185024    Answers: 1   Comments: 0

prove that : Σ_(k=1) ^n (1/(n+k))<ln(2)

$$\mathrm{prove}\:\mathrm{that}\::\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}<\mathrm{ln}\left(\mathrm{2}\right) \\ $$

Question Number 185023    Answers: 1   Comments: 0

provet that : Σ_(k=1) ^n (1/(n+k))>ln(((2n+1)/(n+1)))

$$\mathrm{provet}\:\mathrm{that}\::\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{n}+\mathrm{k}}>\mathrm{ln}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right) \\ $$

Question Number 185014    Answers: 1   Comments: 1

The relation y=x^2 +kx+c, where K and C are constant passes through the points (−1, −2) and (1, 8) in the coordinate axes. calculate the value of C and K. M.m

$$\mathrm{The}\:\mathrm{relation}\:\mathrm{y}=\mathrm{x}^{\mathrm{2}} +\mathrm{kx}+\mathrm{c},\:\mathrm{where}\:\mathrm{K} \\ $$$$\mathrm{and}\:\mathrm{C}\:\mathrm{are}\:\mathrm{constant}\:\mathrm{passes}\:\mathrm{through} \\ $$$$\mathrm{the}\:\mathrm{points}\:\left(−\mathrm{1},\:−\mathrm{2}\right)\:\mathrm{and}\:\left(\mathrm{1},\:\mathrm{8}\right)\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{coordinate}\:\mathrm{axes}.\:\mathrm{calculate}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{C}\:\mathrm{and}\:\mathrm{K}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{M}.\mathrm{m} \\ $$

Question Number 185013    Answers: 4   Comments: 3

Question Number 185012    Answers: 0   Comments: 2

Σ_(k=0) ^n sin(kx)=Σ_(k=0) ^n Im(e^(ikx) ) =Im(Σ_(k=0) ^n (e^(ix) )^k ) =Im(((1−e^(i(n+1)x) )/(1−e^(ix) ))) ((1−e^(i(n+1)x) )/(1−e^(ix) ))=((e^(i(n+1)(x/2)) (e^(−i(n+1)(x/2)) −e^(i(n+1)(x/2)) ))/(e^(i(x/2)) (e^(−i(x/2)) −e^(i(x/2)) ))) =e^(in(x/2)) ×((−2sin((n+1)(x/2)))/(−2sin((x/2)))) =e^(in(x/2)) ((sin((n+1)(x/2)))/(sin((x/2)))) Σ_(k=0) ^n sin(kx)=((sin((n+1)(x/2) ))/(sin((x/2))))Im(e^(in(x/2)) ) =((sin((n+1)(x/2)))/(sin((x/2))))sin(((nx)/2))

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\mathrm{kx}\right)=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{Im}\left(\mathrm{e}^{\mathrm{ikx}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\left(\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{k}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\frac{\mathrm{1}−\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} }{\mathrm{1}−\mathrm{e}^{\mathrm{ix}} }\right) \\ $$$$\frac{\mathrm{1}−\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} }{\mathrm{1}−\mathrm{e}^{\mathrm{ix}} }=\frac{\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} \left(\mathrm{e}^{−\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} −\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} \right)}{\mathrm{e}^{\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} \left(\mathrm{e}^{−\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} −\mathrm{e}^{\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} \right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} ×\frac{−\mathrm{2sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{−\mathrm{2sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} \frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} \\ $$$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\mathrm{kx}\right)=\frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\:\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{Im}\left(\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{sin}\left(\frac{\mathrm{nx}}{\mathrm{2}}\right) \\ $$

Question Number 185011    Answers: 0   Comments: 0

Question Number 185010    Answers: 0   Comments: 0

Question Number 185009    Answers: 0   Comments: 0

Question Number 185008    Answers: 1   Comments: 0

Question Number 184994    Answers: 0   Comments: 1

(√(1+(√(2+(√(3+(√(4+....+(√(70))))))))))=?

$$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}+\sqrt{\mathrm{4}+....+\sqrt{\mathrm{70}}}}}}=? \\ $$

Question Number 184992    Answers: 1   Comments: 0

prove that: ∫_o ^1 ((sint)/(e^t −1))=Σ_(n=o) (1/(n^2 +1))

$${prove}\:{that}: \\ $$$$\int_{{o}} ^{\mathrm{1}} \frac{{sint}}{{e}^{{t}} −\mathrm{1}}=\underset{{n}={o}} {\sum}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 185000    Answers: 0   Comments: 0

A man walks along straight path at a speed 4 ft/s. A spotlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is spotlight rotating when the man is 15 ft from the point on the path closest to the light?

$${A}\:{man}\:{walks}\:{along}\:{straight}\:{path}\: \\ $$$$\:{at}\:{a}\:{speed}\:\mathrm{4}\:{ft}/{s}.\:{A}\:{spotlight}\:{is} \\ $$$$\:{located}\:{on}\:{the}\:{ground}\:\mathrm{20}\:{ft}\:{from}\: \\ $$$$\:{the}\:{path}\:{and}\:{is}\:{kept}\:{focused}\:{on}\:{the}\:{man}. \\ $$$$\:{At}\:{what}\:{rate}\:{is}\:{spotlight}\:{rotating} \\ $$$$\:{when}\:{the}\:{man}\:{is}\:\mathrm{15}\:{ft}\:{from}\:{the}\: \\ $$$${point}\:{on}\:{the}\:{path}\:{closest}\:{to}\:{the}\:{light}?\: \\ $$$$\: \\ $$

Question Number 184989    Answers: 1   Comments: 0

2^x =4x x=? solution???

$$\mathrm{2}^{\mathrm{x}} =\mathrm{4x} \\ $$$$\mathrm{x}=? \\ $$$$\mathrm{solution}??? \\ $$

Question Number 184988    Answers: 1   Comments: 0

Question Number 185116    Answers: 1   Comments: 0

Question Number 184980    Answers: 2   Comments: 0

(√7) + (√6) = a (√7) − (√6) = ?

$$\sqrt{\mathrm{7}}\:\:+\:\:\sqrt{\mathrm{6}}\:\:=\:\:\mathrm{a} \\ $$$$\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{6}}\:=\:? \\ $$

Question Number 184969    Answers: 1   Comments: 2

3 + (1/(6+(3^2 /(6+(5^2 /(6+(7^2 /(6+(9^2 /(6+......)) )) )) )) )) = 𝛑 provet that.

$$\:\: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{3}\:+\:\frac{\mathrm{1}}{\mathrm{6}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{7}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{9}^{\mathrm{2}} }{\mathrm{6}+......}\:\:\:}\:\:}\:\:}\:\:\:\:}\:\:=\:\boldsymbol{\pi} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{provet}}\:\:\boldsymbol{\mathrm{that}}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 184960    Answers: 2   Comments: 0

Question Number 184959    Answers: 2   Comments: 0

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