Find the principal unit normal vector to the curve defined by r(t)=<t, t^2>.
T(t)=1/sqrt(1+4t^2)i+2t/sqrt(1+4t^2)j
I need to find T'(t) using quotient rule but don't know how. Show your steps.
Find the principal unit normal vector to the curve defined by r(t)=<t, t^2>.
T(t)=1/sqrt(1+4t^2)i+2t/sqrt(1+4t^2)j
I need to find T'(t) using quotient rule but don't know how. Show your steps.
T = T(t)=1/sqrt(1+4t^2)i+2t/sqrt(1+4t^2)j
T' = i (-1/2)(8t)(1 + 4t^2)^(-1/2)/(1 + 4t^2) + J (2(1 + 4t^2)^(1/2) - (2t)(1/2)(8t)(1 + 4t^2)^(-1/2))/(1 + 4t^2)
= (i (-4t)/(1 + 4t^2)^(3/2) + j 2(1 + 4t^2)^(1/2) - (8t^2)(1 + 4t^2)^(-1/2))/(1 + 4t^2)
Multiply the second term (j) by (1 + 4t^2)^(1/2)/(1 + 4t^2)^(1/2), This makes both denominators, (i) and (j) equal and one term becomes simpler (-8t^2).
= ( -4t i + j ( 2(1 + 4t^2) - 8t^2))/(1 + 4t^2)^(3/2)
= ( -4t i + 2 j )/(1 + 4t^2)^(3/2)
N(t) = (-4t i + 2 j)/(1 + 4t^2)^(1/2)