(1) Solve the following differential equation. (8 pts.)
dy = (8r++3)e-34, =
(2) Find the area of the surface obtained by rotating the curve given by
x (t) =t T3, y(t) = t2. , 0 < t < 2 about the x-axis. (8 pts.)
(3) A bacteria culture doubles in size every 5 hours, how long will the culture
triple in size? (8 pts.)
(4) Find the length of the curve given by r = e20, , O(5) Find the centroid of the region bounded by the curves y = 3 - x2 and y = - -1. (8
(6) Consider the parametric equations x = -2 COS 0, y = sin² 0, , 0 < 0 < TT.
(a) Sketch the curve represented by the parametric equations. (2 pts.)
(b) Describe the motion of a particle with position (x, y) as 0 varies in the interval
0, \T]. (2 pts.)
(c) Find the equation of the tangent line to the curve at 0 = 37/4. (2 pts.)
(d) Find the at 0 = 37/4. (2 pts.)
(7) (a) Graph r = 2 + COS 0 and r = 2 on the same polar axis. (3 pts.)
(b) Find the area inside the curve r = 2 and outside the curve r = 2 + cos 0. (5 pts.)
Note cos2 0 = 1/1 (1 + COS 20).
(8) Find polar or Cartesian coordinates for the given points. (2 pts. each)
(a) (x, y) = (-2,2) (r,0) =
(b) (x,y) = (1, - V 3)
(c) (r,0) = (2,7x/6) (x,y) =
(d) (r,0) = = (212,57/4)
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