.
Each problem is worth 10 points, show all work and give adequate
explanations for full credit. Please keep your work as legible as possible.
1. Find the Taylor polynomial of degree 5 for the function f(x)=sin
x centered at a=.
2. Find the vertex, focus, and directrix of the parabola 8x2
= -y and sketch its graph, indicating the positions of the focus and directrix
on the graph.
3. Find the interval of convergence of the power series 
.
4. A moose walks along a path given by the parametric equations x=t2+t, y=t2-t (where t=0 of course represents the moment when the moose sees Officer Rebel). Set up an integral for the distance traveled by the moose (i.e., the arc length) between t=-1 and t=1.
| Moose on campus scares Dartmouth students
Hanover, N.H. -- Amid the normal flurry of activity on the Dartmouth College Green Thursday morning, students were surprised to find a special visitor running through campus -- a moose. The moose ran through the West side of campus Thursday morning. Although no one was hurt in the incident, there are some safety concerns when moose are running through populated areas, according to Safety and Security Officer Rebel Roberts, who responded to the call. (taken from the Oklahoma Daily) |
5. Write an integral for the area under one arch of the
cycloid with parametric equations x = t - sin t, y = 1 - cos t. The plot
below shows the curve for 0 
t
6.
6. Set up an integral for the area inside one loop of the curve
r = 3cos 5.
7. Eric is a calculus student at the University of Nebraska, and he's having some trouble with MacLaurin polynomials. Eric says, "Uh, wow. Those MacLaurin things are tough. My teacher keeps talkin' about how they tell us lots of things, but they don't really tell me much. On the last test we were supposed to use 'em to show that if you do sine of minus x it's the same as if you do the sine of x, but I don't have a clue how you'd do that."
Explain clearly to Eric how the MacLaurin series for sin (x) and sin
(-x) relate to each other.
8. Find the slope of the line tangent to r = cos when = /3.
9. Use the sixth degree MacLaurin polynomial for f(x) =
to approximate the value of 
.
10. Find the coordinates of the lowest point on the curve x=t3-3t,
y=t2+t+1 [Hint: What should the slope of the tangent line be
at the lowest point?].
Extra Credit [5 points possible]:
Find the area bounded by the parametric curve with equations x = cos t, y = sin 2t [Hint: Depending on how you proceed, you might find the trig identity sin 2x = 2sin x cos x useful].