18.01 Review, November 11, 2015
Section leader: Eva Belmont (ebelmont@mit.edu, E18-401B)
Midterm 3 review
Taylor approximation
• Taylor series you need to know:
x2 x3 x4
+
+
+ ...
2!
3!
4!
x5 x7
+
−
+ ...
5!
7!
x4 x6
+
−
+ ...
4!
6!
ex ≈ 1 + x +
at a = 0
x3
3!
x2
cos x ≈ 1 −
2!
1
≈ 1 + x + x2 + x3 + x4 + . . .
1−x
1
≈ 1 − x + x2 − x3 + x4 − . . .
1+x
same as saying Taylor series for y1 at a = 1 is 1 − (y − 1) + (y − 1)2 − (y − 1)3 + . . . .
sin x ≈ x −
1
1
1
ln x ≈ (x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + . . .
2
3
4
• Mean Value Theorem:
f (b)−f (a)
b−a
at a = 0
at a = 0
at a = 0
at a = 0
at a = 1
= f 0 (x) for some x in the interval [a, b].
• Taylor’s Theorem (with approximation):
f (x) = f (a) + f 0 (a)(x − a) +
1
1
1 00
f (a)(x − a)2 + · · · + f (n) (a)(x − a)n +
f (n+1) (ξ)(x − a)n+1
2!
n!
(n + 1)!
for some ξ between x and a. (The boxed term is the error term.)
◦ Common problem: given f , a, and x, come up with error bounds for the linear (or quadratic, or nth order,
. . . ) approximation of f (x) centered at a.
Series convergence
•
•
•
•
P∞
i
1
1−r
for |r| < 1 (diverges otherwise)
P1
P (−1)n
Harmonic series:
converges (alternating series test)
n diverges;
n
P
P
ai converges absolutely if
|ai | converges. Absolute convergence is stronger than convergence.
P
P
Dominated
absolute convergence (series
comparison): “if
bi looks worse than
ai in terms of convergence,
P
P
and
bi converges, then so does
ai .” More precisely, if:
◦ |ai | ≤ bi for all i (or, for large enough i);
P
◦
bi converges;
P
P
then
|ai | (and hence
ai ) converges.
P
Alternating series test: i ai converges if:
◦ the ai alternate in sign;
• Geometric series:
i=0 r
=
◦ |ai | ≥ |ai+1 | for all i (good enough to have this just for large i);
◦ limi→∞ ai = 0.
• Ratio test (don’t
this with the ratio test for power series!)
confuse
P
an+1 an converges.
◦ If limn→∞ an < 1 then
P
◦ If limn→∞ an+1
an diverges.
an > 1 then
1
• A test for divergence:
P
ai diverges if limi→∞ ai 6= 0 (or if this limit doesn’t exist).
• Power series:
P
◦ Every power series
ai (x − a)i has a radius of convergence R:
I the series converges for |x − a| < R,
I
diverges for |x − a| > R,
I
and if |x − a| = R, it could go either way.
◦ Ratio test for power
series (shows absolute convergence):
ai exists, then it is the radius of convergence.
I If lim i→∞ ai+1 ◦ It’s worth being familiar with the fact that sin x, cos x, ex centered around zero have infinite radii of
convergence (they converge for all x, no matter how far from zero).
Integration: techniques and applications
•
Rb
f (x) dx is the (signed) area under the graph y = f (x).
Rb 0
• (Fundamental theorem
of
calculus)
a f (x) dx = f (b) − f (a)
R
◦ Upshot: to find f (x) dx, you need to find what function, when differentiated, gives f .
R
• Indefinite integrals f (x) dx
◦ Don’t forget the +C!
R
R
• Integration by parts: f g dx = F g − F g 0 dx
a
• Integration by u-substitution
• How to solve differential equations of the form
dy
dx
= f (x)g(y):
dy
dy
= f (x)g(y) =⇒
= f (x) dx =⇒
dx
g(y)
Z
dy
=
g(y)
Z
f (x) dx
Don’t forget the +C!
R
• Volume of a solid: A(x) dx where A(x) is the area of the (2D) cross-section at x.
Rb
• Volume of f (x) rotated around the x axis: a πf (x)2 dx
Rbp
• Arc length formula: L = a 1 + f 0 (x)2 dx
2
Problems
1. Evaluate the following integrals:
Z 1
e2x
(a)
dx
4x
0 1+e
Z
(b)
x4 ln x dx
2. Find the volume of the solid formed by rotating y = x2 for 0 ≤ x ≤ 2 around (a) the x-axis; (b) the y-axis.
3. Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 1.
4. Find the general solution for
dy
dt
= y2.
5. Do the following series converge?
Xn−1
n3
(b)
X 3n n2
n!
3
(c)
X
(−1)n ln(n)
6. Estimate sin(1) using Taylor series, to within an error of 0.01 (and state an interval within which you are
sure sin(1) lies).
Bonus problems
Z
1. Evaluate
ex sin x dx.
2. This circle of radius 1 in the xz plane, centered at (3, 0), is rotated around the z-axis, forming a (solid)
donut. Write (but don’t evaluate) an integral expressing its volume.
3. (Don’t work on this problem if you have a slightly shaky grasp of series; do book problems instead.) Use
dominated convergence
to come up with a version of the power series ratio test that works for power series
P
of the form
a2n x2n .
4