The Shape of a Graph, Part I
In the previous section we saw how to use the derivative to
determine the absolute minimum and maximum values of a
function. However, there is a lot more information about
Dr. Z.s Calc5 Lecture 11 Handout: Sturm-Liouville Problem
By Doron Zeilberger
Calc4 Reminders
The general solution of
y + y = 0 is
y + 2 y = 0 is
y = c1 ex
.
y = c1 cos x + c2 sin x
.
Important Deniti
Dr. Z.s Calc5 Lecture 6 Handout:
Using the Laplace Transform to solve Systems of Linear Dierential Equations
By Doron Zeilberger
Systems of ODEs
When we have one dierential equation, with one unknown
Dr. Z.s Calc5 Lecture 17 Handout: Laplaces Equation
By Doron Zeilberger
Important Problem (Laplaces Equation in a Rectangle)
Solve
uxx + uyy = 0 ,
0<x<a
,
0<y<b
,
subject to various types of boundary
Dr. Z.s Calc5 Lecture 13 Handout
By Doron Zeilberger
Important Denition
A linear second-order partial dierential equation in two variables, for the dependent
variable (alias function) u(x, y) ,dependi
Dr. Z.s Calc5 Lecture 12 Handout: Fourier-Legendre Series and Linear Recurrences
By Doron Zeilberger
Important Denition
The Legendre polynomials cfw_Pn (x) are dened by the generating function
n=0
Pn
Dr. Z.s Calc5 Lecture 5 Handout: The Dirac Delta Function
By Doron Zeilberger
Important Denition: The Dirac Delta Function, (t) is a function that is zero everywhere
but at t = 0. Its shift, (t t0 ) i
Dr. Z.s Calc5 Lecture 16 Handout: The Wave Equation
By Doron Zeilberger
Version of Nov. 13, 2014 (thanks to Genci Pustina) .
Important Problem
The Wave equation for the displacement, u(x, t), of a sti
Dr. Z.s Calc5 Lecture 0 Handout: Numerical Solutions of Partial Dierential Equations
By Doron Zeilberger
Important Denitions: Discretization
The discrete approximations of the second derivatives with
NAME: (print!)
E-Mail address:
MATH 421 (1), Dr. Z. , FINAL EXAM, Mon., Dec. 22, 2014, 8:00-11:00am,
SEC 212
No Calculators!, You can only use the ocial cheatsheet downloaded from
http:/www.math.rutge
Dr. Z.s Calc5 Lecture 8 Handout: Fourier Series
By Doron Zeilberger
Important Dention: If a function f (x) is dened over the interval (, ), then its Fourier
series is
a0
+
an cos nx +
bn sin nx ,
2
n=
Dr. Z.s Calc5 Lecture 10 Handout: Complex Fourier Series
By Doron Zeilberger
Important Denition (Fundamental Interval): The complex Fourier series of a function
f dened on an interval (, ) is given by
Proof of Trig Limits Study Guide for Midterm
In this section were going to provide the proof of the two limits
that are used in the derivation of the derivative of sine and
cosine in the Derivatives o
FALL 2016
CHAPTER 2
NOMENCLATURE- PRACTICE SHEET-2
1. Write the chemical formula for the following acids:
a) nitric acid
b) phosphoric acid
c) sulfurous acid
d) perchloric acid
e)hypobromous acid
2. W
The Nature of
Politics
Fall 2016
Professor Andrew Murphy
[email protected]
138 Hickman Hall, TTh 5:35-6:55 pm
Organization of the course
Lecture (Hickman 138)
Meets
Some
on days specified
Upload to Assignments on Sakai by 11 am
and due in class on Monday, 11/3/16
Name:_
NetID:_
Nonverbal Observations
Exercise 2, Introduction to Communication, Fall 2016
This exercise is designed to help
Name: _
NetID:_
Third Short Assignment
Introduction to Communication, Fall 2016
How have the new communication technologies changed COMMUNICATION
IN FAMILIES?
Choose an individual who lived part of th
#16. Find the rectangle of largest area that can be inscribed in a semicircle of
radius R, assuming that one side of the rectangle lies on the diameter of the
semicircle.
1 We introduce a Cartesian co
Solutions to Dr. Z.s Math 421(1), Exam #1
1. (15 points) Using the denition nd the Laplace transform Lcfw_f (t) (alias F (s) of
1,
if 0 t 2;
3, if t 2.
f (t) =
Sol.:
=
est 2 est
+3
s 0
s
Ans. to 1: 1
Dr. Z.s Calc5 Lecture 9 Handout: Fourier Cosine and Sine Series
By Doron Zeilberger
Important Denitions
A function f (x) is even if
f (x) = f (x) .
2
(Examples: 1, x2 , x4 , x6 , . . ., cos x, ex )
A
Dr. Z.s Shortcut Methods for Solving Boundary Value Problems for PDEs
By Doron Zeilberger
Fourier Series (over (, )
Every function dened on the interval (, ) can be written as a nite or (more often in
Dr. Z.s Calc5 Lecture 7 Handout: Orthogonal Functions
By Doron Zeilberger
Important Denition 1
Two functions f (x) and g(x) dened on an interval [a, b] are orthogonal if
b
f (x)g(x) dx = 0 .
a
Importa
Dr. Z.s Calc5 Lecture 2 Handout: The Inverse Laplace Transform and Derivatives
By Doron Zeilberger
Theory: The Laplace Transform is a dictionary that goes from functions of t (usually time) to
functio
Dr. Z.s Calc5 Lecture 1 Handout: Denition of the Laplace Transform
By Doron Zeilberger
Theory:
The Denition of the Laplace Transform.
Input: A function f (t) dened on the non-negative real axis [0, ).
Dr. Z.s Calc5 Lecture 4 Handout: Operational Properties for the Laplace Transform
By Doron Zeilberger
Important Formula
Once we know the Laplace transform F (s) of some function f (t) we can immediate
Dr. Z.s Calc5 Lecture 15 Handout: The Heat Equation
By Doron Zeilberger
The Heat Equation is the following pde
k
2u
u
=
x2
t
,
0<x<L
,
t>0
for some constant k, that must be positive. In subscript nota
Dr. Z.s Calc5 Lecture 21 Handout: Fourier Transform
By Doron Zeilberger
Important Denition: The Fourier Transform and The Inverse Fourier Transform
Fourier Transform:
f (x)eix dx = F () .
Fcfw_f (x) =
Dr. Z.s Calc5 Lecture 19 Handout:
Applications of the Laplace Transfrom for solving Partial Dierential Equations
By Doron Zeilberger
Important Formula
Recall that the Laplace Transform of a function f
Dr. Z.s Calc5 Lecture 18 Handout: Laplaces Equation in Polar Coordinates
By Doron Zeilberger
Version of Nov. 23, 2014 (two minor typos corrected, thanks to Dr. Z. (Nov. 17, 2014), thanks
to Alex Sandb