Reference: given by school Course: AP Calculus BC

3. The graphs of the polar curves r = 5 and r = 4 + 3cos0 are shown to the right. Let S be the region that inside the graph of r = 5 and outside the graph of r = 4 + 3cos0, and let R be the region that is inside both graphs. 3 a) Set up and evaluate an integral to find the area of S. (3 pts.) 3 b) Set up and evaluate an integral to find the area of R. (3 pts.)c) An ant named Scott walks along a path traced by the polar curver = 4 + 3cos0. Find the first angle 0 for which Scott's x-coordinate is equal to 1. (2 pts.) d) Scott walks in such a way that, at time / seconds, his angle with the polar axis is given by 0 = t . Find the first time in the interval 1 at time t 2 0 such that its velocity is given by at dx = -212 1+3: and = cos (1 - 4e ). At time t = 1, the object is at the point ( - 2, 3). a) Find the acceleration vector and the speed of the object at t = 1. (3 pts.) b) Find the position (x(t), y(t)) of the curve when t = 3. (4 pts.) c) Find the values of t where the curve has a horizontal tangent. If it does not have one, explain why. (2 pts.) d) Let m(t) denote the slope of the tangent line to the curve at the point (x(t), y(t)). Write an expression in terms of t and use it to evaluate lim,_m(t). Interpret what this tells us about the plane curve. (3 pts.) e) Write an integral expression for lim _,y(t). Use it to find lim, , y(t) and interpret what this tells us about the plane curve. (3 pts.)