Question Number 69829 by TawaTawa last updated on 28/Sep/19 Commented by TawaTawa last updated on 28/Sep/19 $$\mathrm{The}\:\mathrm{question}\:\mathrm{says}\:\mathrm{do}\:\mathrm{not}\:\mathrm{use}\:\mathrm{Newton}'\mathrm{s}\:\mathrm{law}\:\mathrm{and}\:\mathrm{kinematic} \\ $$ Commented by TawaTawa last updated on…
Question Number 135366 by Dwaipayan Shikari last updated on 12/Mar/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt[{\mathrm{6}}]{\mathrm{6}{x}−\mathrm{15}{x}^{\mathrm{2}} +\mathrm{20}{x}^{\mathrm{3}} −\mathrm{15}{x}^{\mathrm{4}} +\mathrm{6}{x}^{\mathrm{5}} −{x}^{\mathrm{6}} }}{dx}=\frac{\pi}{\mathrm{3}} \\ $$$${Or} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt[{{k}}]{{kx}−\frac{{k}\left({k}−\mathrm{1}\right)}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{{k}\left({k}−\mathrm{1}\right)\left({k}−\mathrm{2}\right)}{\mathrm{6}}{x}^{\mathrm{3}}…
Question Number 4261 by 123456 last updated on 06/Jan/16 $${f}\left({x}\right){f}\left(\mathrm{1}−{x}\right)={f}\left(\mathrm{1}\right) \\ $$$${f}\left({x}\right)=? \\ $$ Commented by Yozzii last updated on 06/Jan/16 $${f}\left({x}\right)=\mathrm{0}\:{for}\:{example}.\: \\ $$$$ \\…
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Question Number 69778 by Rio Michael last updated on 27/Sep/19 $${prove}\:{that}\:{the}\:{equation}\: \\ $$$$\:\:\left({b}^{\mathrm{2}} −\mathrm{4}{ac}\right){x}^{\mathrm{2}} \:+\:\mathrm{4}\left({a}\:+\:{c}\right){x}\:−\mathrm{4}\:=\:\mathrm{0}\:{is}\:{always}\:{real}. \\ $$ Commented by prakash jain last updated on 28/Sep/19…
Question Number 69766 by Rio Michael last updated on 27/Sep/19 $${find}\:\:\frac{{dy}}{{dx}}\:\:{at}\:{the}\:{point}\:\:\left(\mathrm{0},\mathrm{3}\right)\:\:{when}\:\:\mathrm{2}{x}^{\mathrm{2}} {y}\:+\:{y}\:+\:\mathrm{4}{xy}^{\mathrm{2}} \:=\:\mathrm{2}{x}\:+\:\mathrm{3}\: \\ $$ Commented by kaivan.ahmadi last updated on 27/Sep/19 $${f}\left({x},{y}\right)=\mathrm{2}{x}^{\mathrm{2}} {y}+{y}+\mathrm{4}{xy}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}=\mathrm{0}…
Question Number 69765 by Rio Michael last updated on 27/Sep/19 $${Given}\:{that}\:\:{y}\:=\:\sqrt{\mathrm{5}{x}^{\mathrm{2}} \:+\:\mathrm{3}}\:,\:{show}\:{that}\:\:{when}\:{x}^{\mathrm{2}} \:=\:\frac{\mathrm{6}}{\mathrm{5}}\:,\:\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}\:} }\:=\:\frac{\mathrm{125}}{\mathrm{8}} \\ $$ Answered by MJS last updated on 27/Sep/19 $${y}=\sqrt{\mathrm{5}{x}^{\mathrm{2}}…
Question Number 69764 by Rio Michael last updated on 27/Sep/19 $${find}\:\:\:\frac{{dy}}{{dx}}\:\:{if}\:\:{x}\:=\:{sin}^{\mathrm{2}} {t}\:\:{and}\:\:{y}=\:{tan}\:{t}\:{at}\:\:{t}\:=\:\frac{\pi}{\mathrm{4}} \\ $$ Commented by kaivan.ahmadi last updated on 27/Sep/19 $$\frac{{dy}}{{dx}}=\frac{\frac{{dy}}{{dt}}}{\frac{{dx}}{{dt}}}=\frac{\mathrm{1}+{tan}^{\mathrm{2}} {t}}{{sin}\mathrm{2}{t}} \\ $$…
Question Number 69763 by Rio Michael last updated on 27/Sep/19 $${find}\:\frac{{dy}}{{dx}}\:\:{if}\:\:{y}\:=\:\mathrm{3}^{{x}} {e}^{\mathrm{2}{x}\:+\:\mathrm{1}} ,\:{at}\:{x}\:=\mathrm{1} \\ $$ Commented by mathmax by abdo last updated on 16/Oct/19 $${y}\left({x}\right)={e}^{{xln}\mathrm{3}+\mathrm{2}{x}+\mathrm{1}}…
Question Number 69762 by Rio Michael last updated on 27/Sep/19 $${prove}\:{by}\:{mathematical}\:{induction},\:{that}\:{for}\:{all}\:{positive}\:{integers}\:{n}, \\ $$$$\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{r}\left({r}\:+\:\mathrm{1}\right)\:=\:\frac{{n}}{\mathrm{3}}\left({n}\:+\:\mathrm{1}\right)\left(\:{n}\:+\:\mathrm{2}\right) \\ $$ Commented by mathmax by abdo last updated on…