Math 153
Practice Final
You have 3 hours to complete this practice final. Dont freak out, and approach each
problem carefully. If you get stuck, make sure to move on to the next problem. You MUST
show all of your work and state your reasoning, unless instructed otherwise. You do not need
to simplify any numerical answer (such as reducing fractions) Good luck! =)
1. Mark each statement as True or False (remember true means always true). No need
to show work.
X
(a) If an 0, then
an converges.
(b) If an 0 for all n and
an converges, then
a2n converges.
(c) Let f (x) be represented by a power series with interval of convergence I. Then
the power series for f 0 (x) also has interval of convergence I.
(d) The polar curves r1 = sin (4) and r2 = cos (4( 8 )) have the same shape.
(e) The parametric equations x(t) = sin t and y(t) = cos t trace out a circle counterclockwise.
(f) In R2 , let ~a = ha1 , a2 i and ~b = hb1 , b2 i. Then ~a ~b = a1 b2~i a2 b1~j.
(g) In R2 , ~a ~b 6= ~a.
(h) There is exactly one plane containing two non-intersecting lines.
(i) If ~r(t) is a differentiable vector function, then
d
|~r(t)| = |~r0 (t)|.
dt
(j) The curve given by the vector function ~r(t) = h2t3 , t3 , 5t3 i is a line.
Math 153
Practice Final
2. Determine the sum of the following series:
(a)
X
2n + 2
n=2
(b)
3n
X
n=1
n2
3
+ 2n
3. Determine whether the series converges absolutely, converges conditionally, or diverges.
X
4n (n!)2
(a)
(n + 2)!
n=1
X
2 2n + 7
(b)
n+5
n=1
Math 153
(c)
Practice Final
X
(1)n ln n
n=2
5n
X
2n
(d)
n+1
n=3
(e)
nen
n=1
X
n1
(f)
2n + 1
n=1
Math 153
Practice Final
4. Expres the following as a Maclaurin series. Determine the radius and interval of convergence.
(a) x sin x2
(b)
x2
(3 x)2
5. Sketch the curve given by the parametric equations below, indicating the direction that
the curve is traced out. Then find the equations of the two tangent lines at (0, 0).
x(t) = sin t cos t
y(t) = cos t
Math 153
Practice Final
6. Give the polar or Cartesian (whichever is not given) equation for the curve:
(a) x = y 2
(b) r = 2 sin + 2 cos
7. Sketch the polar curves
(a) r = 1 + 2 sin
(b) r = 1 + cos 2
Math 153
Practice Final
8. Find the vector projection of ~b = h1, 1, 2i onto ~a = h2, 3, 1i.
9. A bolt is tightened by applying a 40-N force to a 0.25-m wrench at an angle of 75 .
Find the magnitude of the torque about the center of the bolt.
10. Find a vector equation for the line connecting the two points P (5, 1, 3) and Q(6, 5, 1)
11. Find the angle between the planes given by the following equations:
2x y + z = 1
x+z+3=0
Math 153
Practice Final
12. Find the unit tangent and unit normal of the vector function
~r(t) = ht, t2 i
Math 153
Practice Final
13. Reparametrize the curve below with respect to arc length from the point where t = 0.
~r(t) = het cos (2t), 2, et sin (2t)
Math 153
Practice Final
14. Determine the curvature of a helix:
~r(t) = ha cos t, a sin t, cti
