Final Exam Calculus II Name:
Time: 90 min
For maximum credit, show all your work cleanly and legibly in the space provided.
For problems 1 and 3, evaluate the integrals:
1. (10 points) xcos5x dx
1]______
2. (10 points) x-1x2+6x+5dx
2]______
3. (10 points) cosx 3 dx
3]______
4. (10 points) Use the method of cylindrical shells or the disk or washer method to find the volume of the solid generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell or a typical disk or washer.
y=14x2 , y=x about the x-axis
4]______
5. (5 points) Determine whether the following series converge or diverge. Use our known tests for convergence & show all work in how you arrive at your conclusion, including correct mathematical notation & naming the test/tests used.
n=1∞n-1.1+n-1.5
5]______
6. (5 points) Determine whether the following series is convergent or divergent. Use our known tests for convergence & show all work in how you arrive at your conclusion, including correct mathematical notation & naming the test/tests used.
If it is convergent, find its sum.
n=0∞1 3 n
6]______
7. (5 points) Determine if n=1∞(-1)n nnn! converges absolutely.
7]______
8. Given f(x)=cosx ;
a) (5 points) Write the Maclaurin Series for f(x)=cosx
8a]______
b) (5 points) Use Taylor/Maclaurin Series to evaluate the limit:
lim x→0 cosx - x + 16 x3x5
.
8b]______
9. (15 pts) A curve C is defined by the parametric equations x=t2 and y=t3-3t.
a) Show that C has two tangents at the point (3, 0) and find their equations.
9a]______
b) Find the points on C where the tangent is horizontal or vertical.
9b]______
c) Determine where the curve is concave upward or downward.
9c]______
10. (10 points) Find a Cartesian equation for the polar equation r=3sinθ .
10]______
11. (10 points) For the cardioid r=1+sinθ ;
a) Find the slope of the tangent line when θ= π3 .
11a]______
b) Find the points on the cardioid where the tangent line is horizontal or vertical.
11b]______