MATH 535: Introduction to Partial Dierential Equations Spring11 (F.J. Sayas) Some dierential equations (Refresher for Section 1.2)
Linear dierential equations of order one
The general form of a linear
Section 6.1: Antidifferentiation
Basic Antiderivatives Constants
Example: 5
The antiderivative of a constant is just the constant multiplied by .
5 =
*Notice you can take the derivative of your answ
Section 2.3: The First- and Second-Derivative Tests and Curve Sketching
1. First Derivative
The first derivative tells us a lot of things:
a. When the first derivative is positive (f>0), the graph is
Jordan Inver
Chapter 9: Congress
Lecture Notes: 11/4, 11/7
Basic Facts:
The United States Congress:
o Composed of the House of Representatives and the Senate
o House of Representatives: 435 members e
Antitrust Laws
The Sherman Antitrust Act (1890): Every person who shall monopolize, or attempt to monopolize, or
combine with any other persons to monopolize any part of the trade or commerce shall be
WRITING A THESIS
STATEMENT
(courtesy Jennifer Hollstein, CSW)
What is a thesis statement?
- A summary of the argument you'll make in
the rest of the paper
- The center and most important sentence of
y
Figure 1: Profit Maximization
Step 1. Locale
the paint whcn: I Profit=area DCPl-X
MR = MC . AIE
Au: C - ATC
Price = AR
Step 3. Dctcrminc
P and (7 along the
dashed Kine.
Step 2. ldmlil'y cfw_he
pmt num
Differentiable Functions
Let S Rn be open and let f : Rn R. We recall that, for xo = (xo , xo , , xo ) S 1 2 n the partial derivative of f at the point xo with respect to the component xj is defined a
Convex Sets
In this section, we introduce one of the most important ideas in the theory of optimization, that of a convex set. We discuss other ideas which stem from the basic definition, and in parti
Convex Functions
Our final topic in this first part of the course is that of convex functions. Again, we will concentrate on the situation of a map f : Rn R although the situation can be generalized i
Section 3.2: The Chain Rule
CHAIN RULE
If = , then = 1
1. What is when:
(a) = ( 5 2 3 4 )4
In this case, = 5 2 3 4 and = 5 4 6 2 4.
Remember you have to take the derivative of the outside then the in