Question Number 133056 by mr W last updated on 19/Feb/21 Commented by mr W last updated on 18/Feb/21 $${a}\:{ball}\:{is}\:{thrown}\:{from}\:{point}\:{A}\:{with} \\ $$$${speed}\:\boldsymbol{{u}}\:{at}\:{an}\:{angle}\:\boldsymbol{\theta}\:{to}\:{horizontal} \\ $$$${and}\:{strikes}\:{at}\:{a}\:{point}\:{on}\:{the}\: \\ $$$${inclined}\:{plane}\:{and}\:{returns}\:{back}\:{to}…
Question Number 67520 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{f}\left({x},{z}\right)\:=\frac{{z}\:{e}^{{xz}} }{{e}^{{z}} −\mathrm{1}}\:\:\:\:\:\:\left({x}\:{and}\:{z}\:{from}\:{C}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\left({x},{z}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \left({x}\right)\frac{{z}^{{n}} }{{n}!} \\ $$$${with}\:{B}_{{n}} \left({x}\right)\:{is}\:{a}\:{unitaire}\:{polynome}\:{with}\:{degre}\:{n} \\ $$$${determine}\:{B}_{{n}}…
Question Number 1985 by 123456 last updated on 27/Oct/15 $${f}:\left[{a},{b}\right]\rightarrow\mathbb{R} \\ $$$$\underset{{a}} {\overset{{b}} {\int}}{fdx}=\underset{{a}} {\overset{{b}} {\int}}\sqrt{\mathrm{1}+\left(\frac{{df}}{{dx}}\right)^{\mathrm{2}} }{dx} \\ $$$$\mathrm{does} \\ $$$$\pi\underset{{a}} {\overset{{b}} {\int}}{f}^{\mathrm{2}} {dx}=\mathrm{2}\pi\underset{{a}} {\overset{{b}}…
Question Number 67521 by mathmax by abdo last updated on 28/Aug/19 $${calculate}\:\:\:{A}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} \:{cos}\left({nx}\right)}{{n}} \\ $$$${and}\:{B}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} \:{sin}\left({nx}\right)}{{n}} \\ $$ Terms of Service…
Question Number 133053 by pete last updated on 18/Feb/21 $$\mathrm{When}\:\mathrm{the}\:\mathrm{polynomial}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\left(\mathrm{x}−\mathrm{3}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{7}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{f}\left({x}\right) \\ $$$$\mathrm{may}\:\mathrm{be}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{formf}\left({x}\right)=\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)\mathrm{Q}\left(\mathrm{x}\right)+\mathrm{ax}+\mathrm{b}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right).\:\mathrm{If}\:\mathrm{also}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{is}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)=\mathrm{1},\:\mathrm{determine}\:\mathrm{Q}\left(\mathrm{x}\right).…
Question Number 67518 by mathmax by abdo last updated on 28/Aug/19 $${if}\:{z}\:={x}+{iy}\:\:\:{find}\:\:{lnz}\:\:{interms}\:{of}\:{x}\:{and}\:{y} \\ $$$$ \\ $$ Commented by mathmax by abdo last updated on 30/Aug/19…
Question Number 67519 by mathmax by abdo last updated on 28/Aug/19 $${if}\:\frac{{z}}{{e}^{{z}} −\mathrm{1}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \:\frac{{z}^{{n}} }{{n}!} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{B}_{\mathrm{0}} ,{B}_{\mathrm{1}} ,{B}_{\mathrm{2}} ,{B}_{\mathrm{3}} ,{B}_{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{z}\rightarrow\frac{\mathrm{1}}{{e}^{{z}}…
Question Number 67516 by LPM last updated on 28/Aug/19 Commented by Prithwish sen last updated on 28/Aug/19 $$\mathrm{cos}^{\mathrm{2}} \mathrm{x}\:+\left(\mathrm{2cos}^{\mathrm{2}} \mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} =\:\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \mathrm{3x} \\ $$$$\mathrm{Simplyfying}\:\mathrm{we}\:\mathrm{get}, \\…
Question Number 67517 by mathmax by abdo last updated on 28/Aug/19 $${let}\:{z}\:\in{C}\:{and}\:\mid{z}\mid<\mathrm{1}\:\:{prove}\:{that} \\ $$$$\frac{{z}}{\mathrm{1}−{z}^{\mathrm{2}} }\:+\frac{{z}^{\mathrm{2}} }{\mathrm{1}−{z}^{\mathrm{4}} }\:+…..+\frac{{z}^{\mathrm{2}^{{n}} } }{\mathrm{1}−{z}^{\mathrm{2}^{{n}+\mathrm{1}} } }+…=\frac{{z}}{\mathrm{1}−{z}} \\ $$$$\frac{{z}}{\mathrm{1}+{z}}\:+\frac{\mathrm{2}{z}^{\mathrm{2}} }{\mathrm{1}+{z}^{\mathrm{2}} }\:+….+\frac{\mathrm{2}^{{n}}…
Question Number 67514 by mr W last updated on 28/Aug/19 Commented by mr W last updated on 28/Aug/19 $${side}\:{length}\:{of}\:{regular}\:{hexagon}\:{is}\:\mathrm{1}. \\ $$$${find}\:{the}\:{yellow}\:{area}=? \\ $$ Answered by…