APPM 1360
Final, Part 2
Summer 2013
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name, (2) 1360/FINAL, part 2, (3) instructor’s name/class time and (4) Summer 2013 on the front of your bluebook. Also
make a grading table with room for 5 problems and a total score. Work all problems. Start each problem on
a new page. Box your answers. A correct answer with incorrect or no supporting work may receive no credit,
while an incorrect answer with relevant work may receive partial credit. SHOW ALL WORK
sin(x2 )
. What is the radius of convergence?
x
(b) (4 pts) What is the 5th -order Taylor polynomial for f (x) centered at x = 0?
1. (a) (6 pts) Find the Maclaurin series for f (x) =
(c) (6 pts) Use the 3rd -order Taylor polynomial for ex centered at x = 0 to find an approximation to e.
(d) (8 pts) Find upper bounds on the errors in the approximations in (b) and (c), if 0 ≤ x ≤ 2. (You do not
need to justify using these theorems.)
2. Consider the integral
Z
4
2
e−x dx.
0
(a) (4 pts) Find an approximation to this integral, M4 , using the Midpoint Rule and n = 4 subintervals.
Do not simplify your answer.
(b) (4 pts) Find an approximation to this integral, T4 , using the Trapezoid Rule and n = 4 subintervals Do
not simplify your answer.
(c) (12 pts) Determine reasonable upper bounds on the error in the above approximations. Don’t forget to
find bounds for both of the approximations! Hint: |x + y| ≤ |x| + |y|
3. Consider the parametric curve defined by x = t2 − 1, y = sin t, 0 ≤ t ≤ 2π.
(a) (5 pts) Find an expression for the slope of the tangent line to this curve as a function of t.
(b) (5 pts) What is the equation of the tangent line to the curve at x = 0?
(c) (6 pts) At what t values is the tangent horizontal? Vertical? Remember: 0 ≤ t ≤ 2π
4. Consider the polar curve defined by r =
π
, π ≤ θ ≤ 2π.
θ
(a) (4 pts) Provide a rough sketch of the curve. Indicate direction with an arrow.
(b) (8 pts) Set up but do not evaluate an integral for the length of this curve. Simplify as much as
possible.
(c) (8 pts) What is the area between this curve and the circle r = 1? Simplify as much as possible.
5. Always true or False? (20 pts)
♥
(a) All intersections between polar curves r1 = f (θ) and r2 = g(θ) may be found by setting f (θ) = g(θ).
(b) The parametric curves x = t, y = t2 and x = t2 , y = t4 have the same graph.
2 2 d2 y
d x
(c) Given the parametric curve x = f (t), y = g(t), dx2 = ddt2y
.
dt2
P
P
(d) Given the sequence of partial sums, {Sn }, of a series
an , if lim Sn 6= 0, then
an is divergent.
n→∞
FORMULAS ON THE NEXT PAGE.
FORMULA SHEET/Final (1360Sum13)
Z
Some identities
cos(2x) = cos2 (x) − sin2 (x)
sin(2x) = 2 sin(x) cos(x)
2 cos2 (x) = 1 + cos(2x)
2 sin2 (x) = 1 − cos(2x)
Z
a
Center of Mass: x = My /M, y = Mx /M
Volume of a Solid of Revolution
Disk Method: ∆V = πR2 ∆h
Washer Method: ∆V = π[R2 − r2 ]∆h
Shell Method: ∆V = 2πrh∆r
Some useful limits
ln n
=0
n
lim x1/n
n→∞ lim
n→∞
= 1, x > 0
x n
1+
= ex , any x
n
lim
n→∞
√
n
X
1
=
xn , |x| < 1
1−x
n=0
∞
X
xn
ex =
, |x| < ∞
n!
n=0
∞
X
(−1)n x2n+1
, |x| < ∞
sin(x) =
(2n + 1)!
n=0
∞
X
(−1)n x2n
cos(x) =
, |x| < ∞
(2n)!
n=0
∞
X
(−1)n x2n+1
arctan(x) =
, |x| ≤ 1
2n + 1
n=0
∞
X
k n
(1 + x)k =
x , |x| < 1
n
n=0
n=1
lim xn = 0, |x| < 1
xn
lim
= 0, any x
n→∞ n!
n→∞
Frequently used Maclaurin series
∞
f (x)dx ≈ ∆x[f (x̄1 ) + f (x̄2 ) + · · · + f (x̄n )]
Trapezoidal Rule
Moments, Mass and Center of Mass of a thin
metal plate with density Zρ
Z
Mass: M = density · area = ρ · dA = dm
Z
Z
Moments: Mx = ỹ dm, My = x̃ dm
lim
b
a
cosh2 (x) − sinh2 (x) = 1
2 cosh2 (x) = cosh(2x) + 1
2 sinh2 (x) = cosh(2x) − 1
n→∞
Midpoint Rule
b
f (x)dx ≈
∆x
[f (x0 ) + 2f (x1 ) + · · · + 2f (xn−1 ) + f (xn )]
2