Given rt sketch the plane curve represented by r indicate

Given: r(t) = , sketch the plane curve represented by r (indicate the orientation), find the velocity, acceleration and speed functions, find and sketch the vectors: r(0), v(0), and a(0) (on your graph), find T(t) and N(t), find the curvature function K(t) and lim_t rightarrow infinity K(t), and find the equation of the circle of curvature at t = 0.

Solution

a) it is the function y = e^2t

as t increases y increases,

Its an increasing function

b) velocity

y\' = dy/dt = 2e^2t = 2y

as y increases , velocity increases

accleration = d^2y/dt^2 = 4e^2t = 4y

c) at t =0 , y = 1

at position 1 at the start

v(0) = 2y(0) = 2

a(0) = 4y(0) = 4