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Write down a series expansion for ln [((1−2x)/((1+2x)^2 ))] in ascending powers of x up to and including the term in x^4 . if x is small that terms in x^2 and higher powers are negleted show that (((1−2x)/(1+2x)))^(1/(2x)) ≅ (1 + x)e^(−3)

$$\mathrm{Write}\:\mathrm{down}\:\mathrm{a}\:\mathrm{series}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\:\mathrm{ln}\:\left[\frac{\mathrm{1}−\mathrm{2}{x}}{\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} }\right]\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:\mathrm{x}\: \\ $$$$\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including}\:\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{4}} .\: \\ $$$$\mathrm{if}\:\mathrm{x}\:\mathrm{is}\:\mathrm{small}\:\mathrm{that}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{and}\:\mathrm{higher}\:\mathrm{powers} \\ $$$$\mathrm{are}\:\mathrm{negleted}\:\mathrm{show}\:\mathrm{that}\:\:\:\left(\frac{\mathrm{1}−\mathrm{2}{x}}{\mathrm{1}+\mathrm{2}{x}}\right)^{\frac{\mathrm{1}}{\mathrm{2}{x}}} \:\cong\:\left(\mathrm{1}\:+\:{x}\right){e}^{−\mathrm{3}} \\ $$$$ \\ $$

Question Number 83296 Answers: 0 Comments: 2

Obtain a maclaurin expansion for a) e^(cos x ) b) e^(cos^2 x)

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{maclaurin}\:\mathrm{expansion}\:\mathrm{for}\: \\ $$$$\left.\mathrm{a}\left.\right)\:\mathrm{e}^{\mathrm{cos}\:{x}\:} \:\:\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{e}^{\mathrm{cos}\:^{\mathrm{2}} {x}} \\ $$

Question Number 83295 Answers: 0 Comments: 0

Expand ln (1 + sinh x) as a series in ascending powers of x up to and including the term in x^3 . Hence , show that (1 + sinh x)^(3/x) ≅ e^2 (1 −x + (x^2 /2))

$$\mathrm{Expand}\:\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)\:\mathrm{as}\:\mathrm{a}\:\mathrm{series}\:\mathrm{in} \\ $$$$\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:{x}\:\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including} \\ $$$$\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{3}} \:.\:\mathrm{Hence}\:,\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)^{\frac{\mathrm{3}}{{x}}} \:\cong\:{e}^{\mathrm{2}} \left(\mathrm{1}\:−{x}\:+\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$

Question Number 83293 Answers: 0 Comments: 0

Let A (((−2)),((−1)) ) ,B ((1),(3) ) , C (((−10)),(( 5)) ) three given points in the brand (O,I,J) such as OI=OJ and (OI)⊥(OJ) D is a point such as AD=AC+2 and CD=2 Prove correctly that BD=13 .Can you find the coordinate of D?

$${Let}\:\:{A}\begin{pmatrix}{−\mathrm{2}}\\{−\mathrm{1}}\end{pmatrix}\:\:,{B}\begin{pmatrix}{\mathrm{1}}\\{\mathrm{3}}\end{pmatrix}\:,\:{C}\begin{pmatrix}{−\mathrm{10}}\\{\:\mathrm{5}}\end{pmatrix}\:{three}\:{given}\:{points}\:{in}\:{the}\:{brand}\:\left({O},{I},{J}\right)\:{such}\:{as}\:{OI}={OJ}\:{and}\:\left({OI}\right)\bot\left({OJ}\right) \\ $$$$\:{D}\:{is}\:{a}\:{point}\:{such}\:{as}\:{AD}={AC}+\mathrm{2}\:\:{and}\:\:{CD}=\mathrm{2}\: \\ $$$${Prove}\:{correctly}\:{that}\:\:{BD}=\mathrm{13}\:.{Can}\:{you}\:{find}\:{the}\:{coordinate}\:{of}\:{D}? \\ $$

Question Number 83289 Answers: 3 Comments: 0

solve log_((24sinx)) (24cosx)=(3/2)

$$\:\:{solve} \\ $$$$\boldsymbol{{log}}_{\left(\mathrm{24}\boldsymbol{{sinx}}\right)} \left(\mathrm{24}\boldsymbol{{cosx}}\right)=\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 83288 Answers: 0 Comments: 0