16. Determine which relation is a function.
[A]
[C]
@ or the following graph, give the domain and range.
0963:. Dowigwarw Domain: All real numbers - [B] Domain : x 2 0
' 5 RangeAleeahim 5;le Range :y 2 0
' : [C] Domain :All real numbers . [D] Doma
30. Solve 3-/=_ 31. Solve ggii=2i
3 3 y K
7?le jay kg;
66%;: X" 1.931395% (If:
32. Given the pomts (25, 136) and (32, 264), write an egponential function to model this
situation. I5 "032" ,3; s) 3? shx x.
32.-15: 2(91 (if L44:
\Bblo 04] x45
305132! b "
Hon Geom notes Finding Equations to A Medians Nw me Aqswu kg,
\
Given points R(-6, 8), 5(6, 4), and T(-6, 10),
1] Plot points R, S, and T and draw ARST.
2] To find the equation ofthe median from R: = - I
3] Find the midpoint of ST, label it M.
=(9'l'6 ;
Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
85
Lesson 4: More Examples of Functions
Student Outcomes
Students classify functions as either discrete or not discrete.
Classwork
Discussion (5 minutes)
Consider two functions we discussed in previous lesso
THE LAPLACE EQUATION
The Laplace (or potential ) equation is the equation
u = 0.
where is the Laplace operator
2
in R
x22
2
=
+ 2
in R2
x2
y
2
2
2
+
+
in R3
=
x2
y 2
z 2
The solutions u of the Laplace equation are called harmonic functions and play
an imp
THE METHOD OF SEPARATION OF VARIABLES
To solve the BVPs that we have encountered so far, we will use separation of
variables on the homogeneous part of the BVP. This separation of variables leads
to problems for ordinary dierential equations (some with en
FOURIER SERIES PART I:
DEFINITIONS AND EXAMPLES
To a 2-periodic function f (x) we will associate a trigonometric series
a0
+
an cos(nx) + bn sin(nx) ,
2
n=1
or in terms of the exponential eix , a series of the form
cn einx .
nZ
For most of the functions
BESSEL EQUATIONS AND BESSEL FUNCTIONS
Bessel functions form a class of the so called special functions. They are important in math as well as in physical sciences (physics and engineering). They
are especially important in solving boundary values problems
FOURIER SERIES PART II:
CONVERGENCE
We have seen in the previous note how to associate to a 2-periodic function f
a Fourier series
a0
+
an cos(nx) + bn sin(nx) .
2
n=1
Now we are going to investigate how the Fourier series represents f . Let us rst
intro
FOURIER SERIES PART III:
APPLICATIONS
We extend the construction of Fourier series to functions with arbitrary periods,
then we associate to functions dened on an interval [0, L] Fourier sine and Fourier
cosine series and then apply these results to solve
THE WAVE EQUATION
The aim is to derive a mathematical model that describes small vibrations of a
tightly stretched exible string for the one-dimensional case, or of a tightly stretched
membrane for the dimensional case. The derivation of these models is m
LEGENDRE POLYNOMIALS AND APPLICATIONS
We construct Legendre polynomials and apply them to solve Dirichlet problems
in spherical coordinates.
1. Legendre equation: series solutions
The Legendre equation is the second order dierential equation
(1 x2 )y 2xy
THE HEAT EQUATION
The main equations that we will be dealing with are the heat equation, the wave
equation, and the potential equation. We use simple physical principles to show how
these equations are derived. We start the discussion with the heat equati
STURM-LIOUVILLE PROBLEMS:
GENERALIZED FOURIER SERIES
1. Regular Sturm-Liouville Problem
The method of separation of variables to solve boundary value problems leads
to ordinary dierential equations on intervals with conditions at the endpoints of
the inte
FOURIER-BESSEL SERIES AND
BOUNDARY VALUE PROBLEMS IN
CYLINDRICAL COORDINATES
The parametric Bessels equation appears in connection with the Laplace operator in polar coordinates. The method of separation of variables for problem with
cylindrical geometry
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR
SECOND ORDER LINEAR PDE
1. Linear Partial Differential Equations
A partial dierential equation (PDE) is an equation, for an unknown function
u, that involves independent variables, x, y, , the function u,
NONHOMOGENEOUS BOUNDARY VALUE PROBLEMS
AND PROBLEMS IN HIGHER DIMENSIONS
We illustrate how eigenfunctions expansions can be used to solve more general boundary value problems. These include some nonhomogeneous problems and
problems in higher dimensions.
1
Aday 1
Erick Aday
Professor Uszerowicz
ENC 1102
4/10/2017
My First Regret: My Worst Accident
Maybe all one can do is hope to end up with the right regrets (Arthur Miller). I have
had many scarring accidents from when I was young. I have burn scars through
6
Paper: Geall et al. (2012) Nonviral delivery of self-amplifying RNA
vaccines. PNAS 109 (36 ):14604-09. doi/10.1073/pnas.1209367109.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general terms, what is the plan Geall et al. came up with to test
7
Paper: Chen et al. (2016) DNA methylation-based measures of biological
age: meta-analysis predicting time to death. Aging 8:1844-1859 +
supplementary data.
.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general terms, what is the plan Chen et
8
Paper: Zhu et al, (2017) Immunosuppression via Loss of IL2r Enhances LongTerm Functional Integration of hESC-Derived Photoreceptors in the Mouse
Retina. Cell Stem Cell 20:374-384 plus supplemental information.
Q1:
What is the overall hypothesis of the p
1
Paper: Bridgham et al. (2013) Evolution of Hormone-Receptor Complexity
by Molecular Exploitation. Science 312:97-101 + supplementary data.
Accompanied by a commentary/Perspectives piece.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general te
3
Paper: Hoshino, A., et al. (2015). "Tumour exosome integrins
determine organotropic metastasis." Nature 527(7578): 329-335.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general terms, what is the plan Hoshino et al. came up with to test
their
5
Paper: Ridauraetal. (2013) Gut Microbiota from Twins Discordant for Obesity
Modulate Metabolism in Mice. Science 341:1079- + supplemental info and
an associated commentary.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general terms, what is t
2
Paper: Hammond et al. (2016) A CRISPR-Cas9 gene drive system
targeting female reproduction in the malaria mosquito vector Anopheles
gambiae. Nature Biotechnology 34:78-83 + supplementary data.
Accompanied by a commentary/Perspectives piece.
Q1:
What is
4
Paper: Ren, et al. (2017). " Ependymal cell contribution to scar
formation after spinal cord injury is minimal, local and dependent
on direct ependymal injury." Scientific Reports 7: 41122.
Q1:
What is the overall hypothesis of the paper?
Q2:
In general
Chapter 4 Study Guide
3.7
Find antiderivatives
Find position and/or velocity function when given acceleration
function
Section 3.7, problems 1 33 odd, 39 42
4.1 4.2
You will not have to draw and approximate area using rectangles
You WILL have to inte
Test Guide for Calculus Chapter 3
(with suggested book problems)
3.1 Max & Min
Find critical numbers (23-34)
Find absolute max/min using closed interval method (35-43)
3.2 MVT
Use the Mean Value Theorem (9-14, do not graph)
3.3 Derivatives and Shapes o
Review list of topics for CHM1045 Exam 2
Please note You are responsible to review ALL the material covered/ mentioned in your textbook!
The below list does NOT mean ONLY questions with this information will be on the test! Chemistry is
a cumulative subje
Math 1920, Prelim 2
November 8, 2016, 7:30 PM to 9:00 PM
Problem 1. (20 points) Let f (x, y) = x2 + 2xy
D = cfw_(x, y) | x2 + y 2 1.
y 2 be defined in the domain
(a) Find all the critical points f in the interior of D and determine whether they are relati