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 a
graph that can be determined from the first derivati
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 Denition
A Regular Sturm-Liouville Problem on an interval [a,
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 function y(t) and we use the Laplace
Transform method,
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 conditions, involving the function itself or its deriva
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) ,depending on the independent variables, x, y, is something
of
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 (x)tn = (1 2xt + t2 )1/2
.
n=0
Another way to dene them
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 ) is zero everywhere and innity at t = t0 .
Important Form
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 sting of length L is, held tight at both ends
on the inter
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 mesh-size h are:
uxx
1
[u(x + h, y) 2u(x, y) + u(x h,
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.rutgers.edu/zeilberg/calc5/cheatsheet.pdf
Write the nal answ
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=1
n=1
where the number a0 is given
a0 :=
1
f (x) dx
,
a
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
cn einx
,
n=
where
cn =
1
2
f (x)einx dx
n = 0, 1, 2,
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 of Trig Functions section of the
Derivatives chapter.
Pr
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. Write the chemical names for the following compounds con
The Nature of
Politics
Fall 2016
Professor Andrew Murphy
armurphy@polisci.rutgers.edu
138 Hickman Hall, TTh 5:35-6:55 pm
Organization of the course
Lecture (Hickman 138)
Meets
Some
on days specified on the syllabus
weeks, lecture cancelled when
discussi
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 you develop your observational skills, and to increase
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 their adult life BEFORE
the widespread use of the cell ph
#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 coordinate system and draw the semicircle so that its
dia
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
s
=
2
est (3)
est (1) +
2
0
0
2
est f (t) dt =
Lcfw_f
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 function f (x) is odd if
f (x) = f (x) .
2
(Examples: x
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 innite)
linear combination of pure sine-waves and pure co
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
Important Denition 1
Two functions f (x) and g(x) dened on an
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
functions of s. It is often necessary to be able to translate
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, ).
Output: Another function, of s, given by :
f (t)ets dt
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 immediately gure out the
Laplace transform of tn f (t) for any p
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 notation it is:
kuxx = ut
.
There are lots of boundary cond
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) =
Inverse Fourier Transform:
F 1 cfw_F () =
1
2
F ()eix
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 (t) of a single variable t (t > 0, t is usually
time),
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 Sandberg (correcting a typo in p. 4)
Important Formula
The L