*References are to Stewart, Calculus - Concepts and Contexts, 2nd Ed.*

6.3 Arclength. Be able to set up
the integral giving the length of the
curve traced out by the point with coordinates `(x(t), y(t))`
between `t = a` and `t = b` [Formula 1, Example 1 p.468,
Exercises 3, 5, 6].
Special case: be able to set up the integral
giving the length of the graph of `y = f(x)` between `x = a`
and `x = b` [Formula 2, Example 2 p.469, Exercises 4,8]. Since
these integrals are *in general* difficult or impossible to
calculate analytically (using anti-differentiation) be able to apply
approximation methods to estimate them [Exercise 13 - use `n`=2
if doing by hand].

6.4 Average value. Be able to calculate the average value of
the function `f(x)` on the interval `[a,b]`:
divide the integral by `(b-a)`. [Box p.473, Example 1, Exercises 1, 2].

Complex Numbers - Appendix I. Understand the identification
of the complex number `x + iy` with the point `(x,y)`
in the plane. Be able to add and multiply complex numbers.
[Exercises 1, 2, 3, 4].
Be able to write the complex number `x + iy` in "polar"
form `r e ^{i theta}`. [Box, p.A72, Example 4 p.A73,
Exercises 25, 26]. Understand the geometric explanation of
multiplication (multiply the

Second Order Differential Equations (we only study an important
special case: linear homogeneous equations with constant coefficients).
Review the Notes. Be able to
substitute `y = e ^{lambda x}` for the unknown
function

8.1 Know what a sequence is, and be able to check convergence/divergence
in simple cases: for rational functions of `n` as in Example 3
[Exercises 10, 11]; using Theorem 2 p.565 and l'Hôpital's rule
[Example 4, Exercises 16, 19, 20].

8.2 Understand that the sum of an infinite series only makes sense as
the limit of the sequence of partial sums (Definition 2 p.574).
Fundamental example: the *geometric series* `a + ar +
ar ^{2} + ar^{3} + ...
` (Example 1) converges to

8.3 Be able to apply the integral test, and determine convergence or
divergence of a series from covergence or divergence of the corresponding
improper integral (discussion on pp. 583, 584) [Exercises 6-8].

Be able to
apply the comparison test: suppose 2 series, the sum of
`{a _{n}}` and and the sum of

Be able to apply the integral test to the "tail" (the sum from the

8.4 Understand that an *alternating* series (terms are alternately
positive and negative) will converge if the terms are decreasing in
absolute value and if their limit is 0 (Box, page 593) [Example 1,
Exercises 3, 4, 5] and the "Alternating Series Estimation Theorem"
(p. 594) [Example 4, Exercises 12, 13, 17, 18].

Understand what *absolute convergence* means, and that it implies
convergence [Example 7, Exercises 19, 21].

VERY IMPORTANT FOR POWER SERIES: Be able to apply
the *ratio test* to a series (Box, p. 597) [Examples 8,9, Exercises
31, 33].

8.5 Power series. Understand what a power series is ("an infinite
polynomial") and that in general the convergence of
`c _{0} + c_{1}x + c_{2}x^{2} + ...`

8.6 A power series in `x` defines a function `f(x)`.
[Example 1: `1/(1+x ^{2}) = 1 - x^{2} + x^{4} - ...
`; note that the domain of definition of the function (here it is the
whole line) may be different from the interval of convergence of the series
(here it is

Be able to calculate new power series by term-by-term differentiation or integration of old ones (Theorem 2 p. 607) [Examples 5, 6, 7, Exercises 11, 12, 13].

8.7 Taylor series. If a function `f(x)` has derivatives of every
order at `0`, then the power series

`f(0) + f'(0)x + f''(0)x ^{2}/2 + ... +
f^{(n)}(0)x^{n}/n! + ...`

is called the Taylor series for `f(x)` about `0`; also
called the Maclaurin series for `f(x)`. More generally if
`f(x)` has derivatives of every
order at `x=a`, then the power series

`f(a) + f'(a)(x-a) + f''(a)(x-a) ^{2}/2 + ... +
f^{(n)}(a)(x-a)^{n}/n! + ...`

is called the Taylor series for `f(x)` about `a`
[Examples 1, 3, Exercises 3, 4, 6]. Be able to apply Taylor's
Inequality (p. 614) -you do not need to memorize it- to show
that `e ^{x}, sin x, cos x` are equal to the sum of
their Maclaurin series [Example 2, Exercise 15, 16]. Be able to
use Taylor series to integrate functions that can't be handled
by ordinary methods, like

8.10 Be able to apply the power series method to solve a
differential equation of order 1 or 2. Be sure you know how this
works for `y'- y = 0` and `y''+ y = 0`.
[Exercises 1, 2, 7, 8]. For an initial value problem, using
the initial values can simplify the calculation.