Question Number 190249 by alcohol last updated on 30/Mar/23
$${if}\:\mathrm{8\%}\:{error}\:{is}\:{made}\:{on}\:{x},\: \\ $$$${what}\:{is}\:{the}\:{percentage}\:{error} \\ $$$${on}\:\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \:? \\ $$
Answered by mr W last updated on 30/Mar/23
$${y}=\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \\ $$$$\frac{\Delta{y}}{\Delta{x}}\approx\frac{{dy}}{{dx}}=\frac{\lambda}{\mathrm{5}}{x}^{−\frac{\mathrm{4}}{\mathrm{5}}} \\ $$$$\frac{\Delta{y}}{{y}}=\frac{\frac{\lambda}{\mathrm{5}}{x}^{−\frac{\mathrm{4}}{\mathrm{5}}} }{\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} }=\frac{\Delta{x}}{\mathrm{5}{x}}=\frac{\mathrm{8\%}}{\mathrm{5}}=\mathrm{1}.\mathrm{6\%} \\ $$$${i}.{e}.\:{error}\:{on}\:\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \:{is}\:\mathrm{1}.\mathrm{6\%}. \\ $$
Commented by mr W last updated on 30/Mar/23
$${or} \\ $$$${y}=\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \\ $$$${y}+\Delta{y}=\lambda\left({x}+\Delta{x}\right)^{\frac{\mathrm{1}}{\mathrm{5}}} \approx\lambda{x}^{\frac{\mathrm{1}}{\mathrm{5}}} \left(\mathrm{1}+\frac{\Delta{x}}{\mathrm{5}{x}}\right)={y}\left(\mathrm{1}+\frac{\Delta{x}}{\mathrm{5}{x}}\right) \\ $$$$\Delta{y}\approx{y}\frac{\Delta{x}}{\mathrm{5}{x}} \\ $$$$\frac{\Delta{y}}{{y}}\approx\frac{\Delta{x}}{\mathrm{5}{x}}=\frac{\mathrm{8\%}}{\mathrm{5}}=\mathrm{1}.\mathrm{6\%} \\ $$