Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 50161 by cesar.marval.larez@gmail.com last updated on 14/Dec/18

Find the function whose first derivative is 8−(5/(x^2 )^(1/3) ) the initial conditions f(8)=−20

$${Find}\:{the}\:{function}\:{whose}\:{first}\: \\ $$$${derivative}\:{is}\:\mathrm{8}−\frac{\mathrm{5}}{\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} }}\:{the}\:{initial}\: \\ $$$${conditions}\:{f}\left(\mathrm{8}\right)=−\mathrm{20} \\ $$

Answered by afachri last updated on 14/Dec/18

F(x) = ∫ f′(x) dx = ∫ 8 − 5x^(−(2/3)) dx = 8x − ((5/((−(2/3)+1))))x^(−(2/3)+1) + C = 8x − 15x^(1/3) + C to find C: F(8) = −20 8(8) − 15(8)^(1/3) + C = −20 64 − 30 + C = −20 C = −54 So : F(x) = 8x − 15x^(1/3) − 54

$${F}\left({x}\right)\:\:=\:\:\int\:{f}'\left({x}\right)\:\:{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\int\:\:\mathrm{8}\:−\:\mathrm{5}{x}^{−\frac{\mathrm{2}}{\mathrm{3}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{8}{x}\:\:−\:\:\left(\frac{\mathrm{5}}{\left(−\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{1}\right)}\right){x}^{−\frac{\mathrm{2}}{\mathrm{3}}+\mathrm{1}} \:+\:\:{C} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{8}{x}\:−\:\:\mathrm{15}{x}\:^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:{C} \\ $$$$\mathrm{to}\:\mathrm{find}\:{C}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{F}\left(\mathrm{8}\right)\:\:=\:\:−\mathrm{20} \\ $$$$\mathrm{8}\left(\mathrm{8}\right)\:\:−\:\:\mathrm{15}\left(\mathrm{8}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \:+\:\:{C}\:\:=\:\:−\mathrm{20} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{64}\:\:−\:\:\mathrm{30}\:\:+\:\:{C}\:\:=\:\:−\mathrm{20} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{C}\:\:=\:\:−\mathrm{54} \\ $$$$ \\ $$$$\mathrm{So}\:\::\:\:{F}\left({x}\right)\:\:=\:\:\mathrm{8}{x}\:\:−\:\:\mathrm{15}{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\:\:\mathrm{54} \\ $$$$ \\ $$$$ \\ $$

Commented by cesar.marval.larez@gmail.com last updated on 14/Dec/18

wow friend my respect i will the others

$${wow}\:{friend}\:{my}\:{respect}\:{i}\:{will}\:{the}\:{others} \\ $$

Commented by afachri last updated on 14/Dec/18

it′s been our pleasure

$$\mathrm{it}'\mathrm{s}\:\mathrm{been}\:\mathrm{our}\:\mathrm{pleasure} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com