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 dierential equation of order one is y + a(x)y = f (x).
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 answer to check if it is correct.
1
Example:
4
Basic Antid
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 increasing.
b.
c.
When the first derivative is negative
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 elected in Congressional
districts allotted to each stat
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 deemed
guilty of a felony.
Microsoft Case:
1. In May o
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
your whole paper
Where does it go?
- Usually, your thesi
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 numimiaing
output. q.
-u.-_.-.-._
10 step pres-es:
L 51
U
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 as f (xo , xo , , , xo , xo + h, xo , , xo ) f (xo ) 1 2
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 particular, the notion of a convex function which will be im
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 immediately by replacing Rn with any real vector space V
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 inside.
= 4( 5 2 3 4 )41 (5 4 6 2 4)
=4( 5 2 3 4 )3 (5 4