Solve the following:
Q1) A particle is moving in R^3 so that at time t its position is r(t) = (6t, t^2,t^3).
a. Find the equation of the tangent line to the particle's trajectory at the point r(1).
b. The particle flies off on tangent at t0 = 2 and moves along the tangent line to its trajectory with the same velocity that it had at time 2. (Note: the tangent line maybe different than a.) Where will the particle be at time t = 5?
Q2) The temperature of the air in some region is given by the function f(x, y, z) = x^2 - y^2 - z.
a. A bird is at the point (2, 1, 4) and its velocity is currently (1, 5, 1). At what rate is the temperature of the air around the bird changing at this time?
b. What velocity should the bird have (speed is the same) so that the temperature around the bird increase as fast as possible? What is the rate of change of the temperature?
c. If the bird wants that the temperature around it decreases as fast as possible, what velocity (same speed) should it have?