Consider the following position functions r and R for two objects. a. Find the interval [c, d] over which the R trajectory is the same as the r trajectory over [a,b]. b. Find the velocity of both objects. c. Graph the speed of the two objects over the intervals [a,b] and [c,d], respectively. r(t) = (cost,3 sin t), [a,b] = [0,2π] R(t) = (cos 5t,3 sin 5t) on [c, d] a. The interval is

The position of a honeybee flying up and down a straight flowerbed is given by (the euqation ) where t is measured in seconds and s is measured in meters. Positive values of s indicate that the bee is to the right of the center of the flowerbed, while negative values of s indicate that it is to the left of the center. (a) Use the graph to guess the time(s) at which the velocity of the bee is 0, and then use calculus to confirm the answer (b) How fast and in what direction is the bee flying at t = 3.5 seconds?  (c) Find the velocity of the bee at all times when its acceleration is zero.

Find the distance in between the point P(0, 2, – 1) and the point Q(0, 1, 2).

Suppose that a bee is approaching a flower following the trajectory x = t° – 3t, y = t². Please note: Parts (a) and (b) should be written on one page, parts (c) and (d) - on the next page. (a) Find equation(s) of the tangent line(s) at (0, 3); (b) Find the coordinates of the points where the bee is flying horizontally and vertically; (c) Where is the trajectoty concave up and down? (d) Draw a rough sketch of the trajectory, indicating your findings about tangent lines. Hint: Use values of t = +2, ±V3, ±1,0.