Let m = 2k Z be an even number.

Consider the collection of closed curves in R 2 given in polar coordinates by: C2k : r = 1 + 2 sin 2k, 0 2.

(a) Draw the curves C2k for k = 1, 2, 3 on separate graphs. [Use of software is acceptable, but graphs should be drawn by hand and the right features should be present.] [3 points]

(b) How many small petals and how many big petals does C2k have? [2 points]

(c) Find a formula for the area enclosed by C2k. [3 points]

(d) Compute its limit as k . [1 point]

(e) Let I2k denote the arc-length of C2k. Write down a definite integral representing I2k. [2 points]

(f) The integral you wrote down in part (d) does not have a closed form. Nevertheless, find a function f(k) with the property that: lim k I2k f(k) = 1. Make sure to explain in full sentences why your function f(k) satisfies the limit above. [3 points]

(g) What is interesting about your answers to parts (c) and (f)? [1 point]