II: Calculus Final Exam Practice 1. Given the polar equation r=32+cos2. (a) (4 points) Graph the equation in a rectangular plane.
(b) (8 points) Find the area bounded by the polar curve.
(c) (8 points) Find its arc length.
2. Given the parametric curve of the cycloid x=at-sint and y=a1-cost. (a) (4 points) Find dydx. When do vertical tangent lines happen on 0t2?
(b) (8 points) Find d2ydx2. What is the concavity of the curve on 0t2?
(c) (8 points) Find the area the of region bounded by the cycloid on 0t2.
3. Determine whether each infinite series converges or diverges. If possible, find its sum. (a) (5 points) n=11+3n+5n4n
(b) (5 points) n=12nn+12
(c) (5 points) n=1-1nn2n+1
(d) (5 points) n=1n2-15n2+2
4. Determine whether each infinite series converges absolutely, converges conditionally, or diverges. (a) (5 points) n=0-8nn(n+1)!
(b) (5 points) n=1ln1nn
(c) (5 points) n=1-1n+2ln(n)n2
(d) (5 points) n=0cos3(n)n2+3