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
The antiderivative of a constant is just the constant multiplied by .
*Notice you can take the derivative of your answer to check if it is correct.
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.
When the first derivative is negative
Chapter 9: Congress
Lecture Notes: 11/4, 11/7
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
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.
1. In May o
WRITING A THESIS
(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
Step 2. ldmlil'y cfw_he
10 step pres-es:
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
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
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
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