Question Number 1375 by 123456 last updated on 26/Jul/15 $$\mathrm{lets}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{and}\:{g}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{two}\:\mathrm{continuous}\:\mathrm{and}\:\mathrm{differentiable}\:\mathrm{functions} \\ $$$$\mathrm{suppose}\:\mathrm{that}\:{g}\left(\mathrm{0}\right)=\mathrm{0},\:\mathrm{then}\:\mathrm{compute} \\ $$$${h}\left({x}\right)=\underset{\Delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{g}\left({f}\left({x}+\Delta{x}\right)−{f}\left({x}\right)\right)}{\Delta{x}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 132444 by liberty last updated on 14/Feb/21 Answered by bemath last updated on 14/Feb/21 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 66909 by naka3546 last updated on 20/Aug/19 $$\int\:\:\frac{\sqrt{{x}^{\mathrm{2019}} +\mathrm{2019}}\:\:+\:\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2018}} +\mathrm{2018}}}{\:\sqrt[{\mathrm{4}}]{{x}^{\mathrm{2017}} +\mathrm{2017}}\:\:+\:\:\sqrt[{\mathrm{5}}]{{x}^{\mathrm{2016}} +\mathrm{2016}}}\:\:{dx}\:\:=\:\:? \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 66906 by Kunal12588 last updated on 20/Aug/19 $${Intergrate}\:{I}=\int\:\frac{{t}^{\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{4}} }\:{dt} \\ $$ Commented by Prithwish sen last updated on 20/Aug/19 $$\frac{\mathrm{1}}{\mathrm{2i}}\int\frac{\mathrm{1}}{\mathrm{1}−\mathrm{it}^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{it}^{\mathrm{2}} }\:\mathrm{dt}=\frac{\mathrm{1}}{\mathrm{2}}\left[−\mathrm{sin}^{−\mathrm{1}}…
Question Number 132440 by liberty last updated on 14/Feb/21 $$\mathrm{A}\:\mathrm{vessel}\:\mathrm{containing}\:\mathrm{water}\:\mathrm{has}\:\mathrm{the}\: \\ $$$$\mathrm{shape}\:\mathrm{of}\:\mathrm{and}\:\mathrm{inverted}\:\mathrm{right}\:\mathrm{circular} \\ $$$$\mathrm{cone}\:\mathrm{with}\:\mathrm{base}\:\mathrm{radius}\:\mathrm{2m}\:\mathrm{and}\:\mathrm{height}\:\mathrm{5m} \\ $$$$\mathrm{The}\:\mathrm{water}\:\mathrm{flows}\:\mathrm{from}\:\mathrm{the}\:\mathrm{apex} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{at}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{rate}\: \\ $$$$\mathrm{of}\:\mathrm{0}.\mathrm{2}\:\mathrm{m}^{\mathrm{3}} /\mathrm{min}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{at} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{water}\:\mathrm{level}\:\mathrm{is}\:\mathrm{dropping} \\ $$$$\mathrm{when}\:\mathrm{the}\:\mathrm{depth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{water}\:\mathrm{is}…
Question Number 1369 by 123456 last updated on 25/Jul/15 $$\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 1368 by prakash jain last updated on 25/Jul/15 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{ln}\left({ax}+{b}\right)}{\mathrm{ln}\:\left({bx}+{a}\right)}\:=? \\ $$ Commented by 123456 last updated on 25/Jul/15 $${f}\left({x}\right)=\frac{\mathrm{ln}\:\left({ax}+{b}\right)}{\mathrm{ln}\:\left({bx}+{a}\right)} \\ $$$$\frac{\frac{{d}}{{dx}}\left[\mathrm{ln}\:\left({ax}+{b}\right)\right]}{\frac{{d}}{{dx}}\left[\mathrm{ln}\:\left({bx}+{a}\right)\right]}=\frac{\frac{{a}}{{ax}+{b}}}{\frac{{b}}{{bx}+{a}}}=\frac{{a}\left({bx}+{a}\right)}{{b}\left({ax}+{b}\right)}\rightarrow\frac{{ab}}{{ba}}=\mathrm{1} \\…
Question Number 132439 by liberty last updated on 14/Feb/21 $$ \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{known}\:\mathrm{that}\:\mathrm{a}\:\mathrm{particular} \\ $$$$\mathrm{machine}\:\mathrm{will}\:\mathrm{make}\:\mathrm{product}\:\mathrm{with}\: \\ $$$$\mathrm{a}\:\mathrm{qualified}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{90\%}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is} \\ $$$$\mathrm{running}\:\mathrm{well},\:\:\mathrm{but}\:\mathrm{will}\:\mathrm{do}\:\mathrm{so}\:\mathrm{with}\: \\ $$$$\mathrm{a}\:\mathrm{qualified}\:\:\mathrm{rate}\:\mathrm{of}\:\mathrm{only}\:\mathrm{30\%}\:\mathrm{when}\: \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{not}\:\mathrm{running}\:\mathrm{well}.\:\mathrm{The} \\ $$$$\mathrm{probability}\:\mathrm{that}\:\mathrm{machine}\:\mathrm{is}\: \\…
Question Number 132433 by john_santu last updated on 14/Feb/21 Answered by liberty last updated on 14/Feb/21 $$\mathrm{by}\:\mathrm{using}\:\mathrm{Cosine}\:\mathrm{of}\:\mathrm{law}\: \\ $$$$\:\left(\mathrm{r}−\mathrm{0}.\mathrm{5}\right)^{\mathrm{2}} =\mathrm{0}.\mathrm{5}^{\mathrm{2}} +\left(\sqrt{\mathrm{2}}−\mathrm{r}\right)^{\mathrm{2}} −\mathrm{2}×\mathrm{0}.\mathrm{5}×\left(\sqrt{\mathrm{2}}−\mathrm{r}\right)\mathrm{cos}\:\mathrm{45}° \\ $$$$\cancel{\mathrm{r}^{\mathrm{2}} }−\mathrm{r}+\cancel{\frac{\mathrm{1}}{\mathrm{4}}}=\cancel{\frac{\mathrm{1}}{\mathrm{4}}}+\mathrm{2}−\mathrm{2}\sqrt{\mathrm{2}}\mathrm{r}+\cancel{\mathrm{r}^{\mathrm{2}}…
Question Number 132432 by Chhing last updated on 14/Feb/21 $$ \\ $$$$\:\:\mathrm{Solve}\:\:\mathrm{differential}\:\:\mathrm{equations} \\ $$$$\:\:\:\:\mathrm{1}/\:\left(\mathrm{x}^{\mathrm{3}} −\mathrm{1}\right)\mathrm{y}'+\mathrm{xy}=\mathrm{x} \\ $$$$\:\:\:\:\mathrm{2}/\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{y}'−\mathrm{xy}+\frac{\mathrm{2x}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }}=\mathrm{0} \\ $$$$\:\:\:\:\mathrm{3}/\:\mathrm{x}^{\mathrm{2}} \left(\mathrm{lnx}\right)\mathrm{y}''+\mathrm{y}=\mathrm{0}\:,\:\mathrm{know}\:\mathrm{that}\:\mathrm{y}=\mathrm{lnx}\:\mathrm{is}\:\mathrm{the}\:\mathrm{answer} \\ $$$$ \\…