Finding absolute extrema for nice functions Math 8 2009, Chris Lyons Lets suppose youre given a function f : [a, b] R and you know that its continuous. Even this little bit of information already tells you something: Theorem 1 (Thm 3.12 in Apostol). If g
Some Useful Logic: Part 1 Math 8 2009, Chris Lyons Logical statements and negation Lets denote a true-or-false statement using letters P , Q, R, etc. For example, we could have the statement P be P : Im eating an apple. We say that a statement P is equiva
The Limit Game Math 8 2009, Chris Lyons Let cfw_an be a sequence of real numbers, and let L R. This is a two-player game, and its played like this: 1. Player A gives Player B a (small) number > 0. 2. Player B tries come up with some N 1 so that the follo
A small warning about taking limits Math 8 2009, Chris Lyons Consider the following evaluation of a limit: 11 1 = lim = 2 k k k k k lim 1 k k lim 1 k k lim = 0 0 = 0. (1)
Heres the basic principle that we used in this calculation: lim an bn = lim an lim b
CALTECH Control & Dynamical Systems
Key Points, Vector Calculus
Math1C - Spring 2010 Jerry Marsden and Eric Rains
Control and Dynamical Systems and Mathematics, Caltech
For computing resources: www.cds.caltech.edu/~marsden
The Geometry of Eucli
A mathematical mystery. Math 8 2009, Chris Lyons Using the exponential function ex , we can create two simple functions: cosh x = 1x (e + ex ), 2 sinh x = 1x (e ex ). 2
Here cosh x is called the hyperbolic cosine and sinh x is called the hyperbolic sine.
Intuition for the convergence of certain types of series Math 8 2009, Chris Lyons Lets start with a few examples: Example 1. Look at the series
1 . k 3 + 4k 1
Does this converge or diverge? Discussion. Obviously this series (1) is not the same thi
Passing the limit through a continuous function Math 8 2009, Chris Lyons Heres a helpful trick that allows you to pass a limit from outside a continuous function to the inside. Proposition 1. Suppose that (i) g (y ) is continuous at y = L, (ii) limxa f (x
Algebraic topology can be roughly dened as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechani
A possible algebraic solution to one of our in-class examples Math 8 2009, Chris Lyons On October 16, we gave a proof of the following, as one of our in-class examples: Theorem 1. Suppose we have a polynomial f (x) = a0 + a1 x + . . . + ad xd with d > 0.
A reminder about absolute values Math 8 2009, Chris Lyons For x R, we dene the absolute value of x to be |x| = x if x 0 .
x if x < 0
From this denition, one can show that if C 0 then |x| C C x C. How do you show this? Heres one direction, namely |x| C =
Some Useful Logic: Part 2 Math 8 2009, Chris Lyons Very often in Math 1a, well consider true-or-false statements about the elements in some set. We may want to consider whether all elements in the set have some property, or whether at least one element ha