Question Number 1369 by 123456 last updated on 25/Jul/15 $$\mathrm{lets}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{differentiable} \\ $$$$\mathrm{compute} \\ $$$${g}\left({x}\right)=\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{f}\left({x}+\Delta{x}\right)} −{e}^{{f}\left({x}\right)} }{{f}\left({x}+\Delta{x}\right)−{f}\left({x}\right)} \\ $$ Answered by prakash jain last updated…
Question Number 132434 by Dwaipayan Shikari last updated on 14/Feb/21 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{cos}\left({n}\right)}{{n}^{\mathrm{2}} } \\ $$ Answered by mnjuly1970 last updated on 14/Feb/21 $$\:\:{solution}: \\…
Question Number 1339 by 123456 last updated on 24/Jul/15 $${f}:\mathbb{C}\rightarrow\mathbb{C},{z}_{\mathrm{0}} \in\mathbb{C}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({z}\right)−{f}\left({z}_{\mathrm{0}} \right)=\left({z}−{z}_{\mathrm{0}} \right){f}\left({z}−{z}_{\mathrm{0}} \right) \\ $$$$\mathrm{does}\:\underset{{z}\rightarrow{z}_{\mathrm{0}} } {\mathrm{lim}}{f}\left({z}\right)={f}\left({z}_{\mathrm{0}} \right)? \\ $$ Commented by…
Question Number 1335 by 123456 last updated on 23/Jul/15 $$\mathrm{if}\:{a}\leqslant{x}\leqslant{b}\:\mathrm{and}\:{c}\leqslant{y}\leqslant{d}\:\mathrm{then}\:\mathrm{did} \\ $$$$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}? \\ $$$${a},{b},{c},{d},{x},{y}\in\mathbb{R} \\ $$$$\mathrm{0}<{c} \\ $$$$\mathrm{0}<{a} \\ $$ Answered by prakash jain last…
Question Number 66868 by John Kaloki Musau last updated on 20/Aug/19 $${A}\:{town}\:{N}\:{is}\:\mathrm{340}{km}\:{due}\:{west}\:{of}\: \\ $$$${town}\:{G}\:{and}\:{town}\:{K}\:{is}\:{due}\:{west}\: \\ $$$${of}\:{town}\:{N}.\:{A}\:{helicopter}\:{Zebra}\: \\ $$$${left}\:{G}\:{for}\:{K}\:{at}\:\mathrm{9}{a}.{m}.\:{Another}\: \\ $$$${helicopter}\:{Buffalo}\:{left}\:{N}\:{for}\:{K} \\ $$$${at}\:\mathrm{11}{a}.{m}.\:{Helicopter}\:{Buffalo} \\ $$$${travelled}\:{at}\:{an}\:{average}\:{speed}\:{of}\: \\…
Question Number 66865 by John Kaloki Musau last updated on 20/Aug/19 $${A}\:{and}\:{B}\:{are}\:{two}\:{towns}\:\mathrm{360}{km}\:{apart}. \\ $$$${An}\:{express}\:{bus}\:{departs}\:{from}\:{A}\:{at} \\ $$$$\mathrm{8}{a}.{m}\:{and}\:{maintains}\:{an}\:{average} \\ $$$${speed}\:{of}\:\mathrm{90}{km}/{h}\:{between}\:{A}\:{and}\:{B}. \\ $$$${Another}\:{bus}\:{starts}\:{from}\:{B}\:{also}\:{at} \\ $$$$\mathrm{8}{a}.{m}\:{and}\:{moves}\:{towards}\:{A}\:{making} \\ $$$${four}\:{stops}\:{at}\:{four}\:{equally}\:{spaced} \\…
Question Number 66856 by John Kaloki Musau last updated on 20/Aug/19 $${Two}\:{towns}\:{T}\:{and}\:{S}\:{are}\:\mathrm{300}\:{km}\:{apart}. \\ $$$${Two}\:{buses}\:{A}\:{and}\:{B}\:{started}\:{from} \\ $$$${T}\:{at}\:{the}\:{same}\:{time}\:{travelling}\:{towards} \\ $$$${S}.\:{Bus}\:{B},\:{travelling}\:{at}\:{an}\:{average} \\ $$$${speed}\:{of}\:\mathrm{10}{km}/{h}\:{greater}\:{than}\:{that} \\ $$$${of}\:{A}\:{reached}\:{S}\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{4}}\:{hours}\:{earlier}. \\ $$$$\left({a}\right)\:{Find}\:{the}\:{average}\:{speed}\:{of}\:{A} \\…
Question Number 1322 by 123456 last updated on 22/Jul/15 $${a}_{\mathrm{0}} =\mathrm{1} \\ $$$${a}_{\mathrm{1}} =\mathrm{2} \\ $$$${a}_{{n}} −{a}_{\mathrm{0}} ={a}_{{n}−\mathrm{1}} −{a}_{{n}−\mathrm{2}} ,{n}\geqslant\mathrm{2} \\ $$$${a}_{\mathrm{10}} =? \\ $$$$\underset{{n}\rightarrow+\infty}…
Question Number 66852 by John Kaloki Musau last updated on 20/Aug/19 Commented by John Kaloki Musau last updated on 20/Aug/19 The cross-section of a head of a bolt is the form of a regular hexagon as shown in the figure below. Determine the area of the cross-section. Commented by John Kaloki…
Question Number 1318 by 123456 last updated on 22/Jul/15 $$\eta:\mathbb{C}^{\mathrm{2}} \rightarrow\mathbb{R}_{+} \\ $$$$\eta\left({x},{y}\right)=\mid\mid{x}\mid−\mid{y}\mid\mid \\ $$$$\eta\left({x},{y}\right)=\mathrm{0}\overset{?} {\Leftrightarrow}{x}={y} \\ $$$$\eta\left({x},{y}\right)\overset{?} {=}\eta\left({y},{x}\right) \\ $$$$\eta\left({x},{z}\right)\overset{?} {\leqslant}\eta\left({x},{y}\right)+\eta\left({y},{z}\right) \\ $$$${x}\in\mathbb{C} \\…