*References are to Stewart, Calculus - Concepts and Contexts, 3nd Ed.*

7.4 Exponential growth and decay. `y' = ky` is an important
special case of a separable equation. The general solution
is `y=Ce ^{kt}`, called "exponential growth" if

Compound interest. Understand that $1 invested for one year at annual interest t (in dollars) gives $1+t if compounded annually, $(1 + t/2)

7.5 The Logistic Equation. Understand the concept of
*carrying capacity* `K` and why `dP/dt = kP(1-P/K)`
is a natural way of encoding `K` into the growth
equation. Recognize the shape of the slope fields that come
from this kind of equation (`k` always positive in this
context!). [Example 1]. Understand the concept of
*equilibrium solution* [Problem 1d].
Be able to write down the equation from
data giving `k` and `K`. Be able to use Euler's
method (small number of steps) to estimate `P(t)` given
`k, K, P(0)` and `t`. [Problem 3a].

7.6 Predator-Prey Systems. Understand the meaning of the
constants `k, a, r, b`
in Equation 1 p. 545 [Example 1, Exercise 1]. Understand why, when the
constants are positive, the Phase portrait of the system
has the shape given in Figures 1, 2 p. 546; explanation p. 547
[Problem 3]. Be able to locate the equilibrium solutions [Problem 9].

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.559 and l'Hôpital's rule
[Example 4, Exercises 15, 19, 21].

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.568).
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. 577, 578) [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 587) [Example 1,
Exercises 3, 5, 7] and the "Alternating Series Estimation Theorem"
(p. 588) [Example 4, Exercises 12, 13, 17].

Understand what *absolute convergence* means, and that it implies
convergence [Theorem 1 p.590, Example 7, Exercises 19, 21, 23].

VERY IMPORTANT FOR POWER SERIES: Be able to apply
the *ratio test* to a series (Box, p. 591) [Examples 8,9, Exercises
31, 33, 35].

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. 601) [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. 608) -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 11, 13, 15]. Be able to
use Taylor series to integrate functions that can't be handled
by ordinary methods, like

8.8 The Binomial Series. Understand that the Binomial Theorem

`(a+b) ^{k} = a^{k} + C(k,1)a^{k-1}b
+ C(k,2)a^{k-2}b^{2} + ... +
C(k,k-2)a^{2}b^{k-2} + C(k,k-1)ab^{k-1}
+ b^{k}`

where

8.9 Applications of Taylor Polynomials. Understand how to estimate
how good an approximation you get using the `n`-th
Taylor polynomial `T _{n}(x)` for a function