18.01 Section, December 2, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
.
Midterm 4 review, part 2: partial fractions, integral test, work problems
Review
• Physics problems
Rb
◦ Mass: a ρ(x)A(x) dx (where ρ is density – which is assumed to be constant on a slice –
and A(x) is cross-sectional area)
Z b
1
◦ Centroid (center of mass) has x-coordinate =
xρA(x) dx.
mass a
◦ Work = Force × distance
• Fourier series: a + a1 cos(x) + b1 sin(x) + a2 cos(2x) + b2 sin(2x) + . . . where
Z 2π
Z
Z
1 2π
1 2π
1
f (x) dx
ak =
cos(kx)f (x) dx
bk =
sin(kx)f (x) dx
a=
2π 0
π 0
π 0
• Integral comparisons
R∞
◦ Dominated convergence
for
integrals:
if
a g(x) dx converges and |f (x)| ≤ g(x) in the
R∞
interval [a, ∞), then a f (x) dx converges.
R∞
R∞
◦ And the reverse: if a |g(x)| dx diverges and f (x) ≥ |g(x)| for x in [a, ∞), then a f (x) dx
diverges.
◦ Integral convergence criterion
function with f (x) ≥ 0, then:
R ∞ for series: if f is a decreasing
P∞
k=a f (k) converges.
a f (x) dx converges ⇐⇒
R∞
◦ More precisely,
if
convergence
holds
then
the
value
of
the
series
is
between
a f (x) dx and
R∞
f (a) + a f (x) dx.
• Probability
R∞
◦ Probability density function: p(x) ≥ 0 such that −∞ p(x) dx = 1.
Rb
◦ Probability of being between a and b = a p(x) dx
R∞
◦ Expectation: E = −∞ x p(x) dx
R∞
◦ Variance: −∞ (x − E)2 p(x) dx
√
◦ Standard deviation: Variance
R x+h
• Moving average of f with window size h: (T f )(x) = h1 x f (u) du
• Partial fractions
1
◦ To expand (x+2)(x+3)(x+4)
, write
1
A
B
C
??? · x2 + ?? · x + ?
=
+
+
=
(x + 2)(x + 3)(x + 4)
x+2 x+3 x+4
(x + 2)(x + 3)(x + 4)
write the ?’s in terms of A, B, C, and use the fact that ??? = 0, etc. to solve for A, B, C.
◦ (Repeated factors in denominator) To expand
1
,
(x+2)2 (x+3)
write
1
A
B
C
=
+
+
2
2
(x + 2) (x + 4)
x + 2 (x + 2)
x+3
◦ (Quadratic factors in denominator) To expand
(x2
1
,
(x2 +2x+3)(x+4)
write
1
Ax + B
C
= 2
+
+ 2x + 3)(x + 4)
x + 2x + 3 (x + 4)
1
Problems
Z
1. Calculate
1
dx.
(x − 1)(x + 2)
2. A tank full of water is shaped like a cone (point down) with maximum radius 12 m and
height 1 m. The density of water is 1000 kg/m3 . A spigot on the cone point is opened, and
all the water drains out. How much work does gravity do to accomplish this?
3. Does
P∞
n=1
e−n
n
4. (a) Calculate
converge?
P∞
1
n=1 n3
to an error of no more than 0.0001.
(b) If I just add up 10 terms of the above sum, do I land within 0.01 of the answer?
2