Let vector r(t) = t i + t^2 j represent a plane curve.
Find T(t), T(1) and N(1). Sketch the plane curve and graph the vectors T(1) and N(1) at the point t = 1.
Please see the attached file.
To determine unit normal vector for the curve r(t) = t i + t2 j ....(1)
Step 1 : Determine the tangent vector by differentiating (1)
dr/dt = T(t) = i + 2tj ...........(2)
Step 2 : Determine unit tangent vector by dividing right hand side of (2) by the magnitude of the vector
Ť(t)= (i + 2tj)/√1 + 4t2 ..........(3)
Step 3 : Determine normal vector by differentiating (3)
N(t) = dŤ(t)/dt
To differentiate (3) we use the following rule :
d/dx[f(x)/g(x)] = [g(x)df(x)/dx - f(x)dg(x)/dx]}/g(x)2
= (Denominator x derivative of numerator - Numerator x derivative of denominator)/Denominator2
Derivative of numerator = d/dt (i + 2tj) = 2j ......(4)
Derivative of denominator = d/dt √1 + ...
The vector functions and plane curve representations are determined. The expert sketches the plane curve and graphs the vectors.