Question

If 𝑅 𝑢 = 𝑢 − 𝑢 2 𝑖 + 2𝑢 3 𝑗 − 3𝑘, find �𝑑� �� �� ׬ .�� 𝑏. න 1 2 𝑅(𝑢)𝑑�

Answer 1

Ans:-

$𝑅 (𝑢 )= 𝑢 − 𝑢 2 𝑖 + 2𝑢 3 𝑗 − 3𝑘\\$

$\dfrac{dR}{du}=\lim_{u\to0}\dfrac{R(u+\Delta u-R(u))}{\Delta u}\\$

$=\lim_{u\to0}\dfrac{[x(u+\Delta u)2i+y(u+\Delta u)6j]+z(u+\Delta u)3k]-[x(u)i+y(u)j+z(u)k]}{\Delta u}$

$=\lim_{u\to0}\dfrac{ x(u+\Delta u)-x(u)}{\Delta u}2i+\dfrac{ y(u+\Delta u)-y(u)}{\Delta u}6j+\dfrac{ z(u+\Delta u)-z(u)}{\Delta u}3k$

$=\dfrac{dx}{du}2i+\dfrac{dy}{du}6j+\dfrac{dz}{du}3k$