Chapter 13
Fibonacci Numbers and the Golden Ratio
13.1 Fibonacci Numbers
The sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
The Fibonacci Numbers are the numbers in this sequence
Let Fn denote the nth Fibonacci number
Example
F1=1
F2 =1
F3 =2
Fact
Chapter One
1-1
Mathematics of Elections
o Preference ballots and preference schedules consider an election with
multiple candidates
Alice, Bob, Charly, etc
o A single choice ballot is a ballot in which voters select their favorite
candidate
o A preferen
Partners: Cayla Lamkin, Melitza Argote, Stacy Castro, Greg Inegbe
Keplers Law of Equal Area
Introduction:
Kepler explains that the planets move faster when closer to the sun using a combination of three
laws of planetary motion, which he discovered using
Newtons Universal Law of Gravitation
Names: Cayla Lamkin and Greg Inegbe
Remember to explain all written answers in full sentences and complete the math problems by
identifying the data, stating the equation, showing your work, and stating the correct ans
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University of Regina
Department of Mathematics and Statistics
MATH 401/890 Matrix Lie Groups Winter 2015
Homework Assignment no. 3 - Solutions
0. Read the proof of Theorem 2.7 in the textbook. Exercises 2 and 5, page 59 in the textbook are
used in that pr
CCSS Math Standards
Common Core State Standards (CCSS) for Mathematics
Go to http:/www.corestandards.org/Math (Links to an external site.) and read the home page then the
Introduction (How to read the grade level standards).
Next review the domains for ea
Standards for Math Practice
Grade: Kindergarten
Standard: CCSS.MATH.CONTENT.K.CC.A.2
Count forward beginning from a given number within the known sequence (instead of having to
begin at 1).
Activity: For a warm up activity, the students will watch a brief
Lesson Plan 1
Name:
Lesson Title: Identifying Shapes
Grade/Group Age :Kindergarten
Objectives
The students will be able to identify many different shapes regardless of their orientation.
The students will be able to sort the shapes given, correctly and in
Echelon form
any rows of 0s at the bottom of the matrix
leading entry of any row is left of the leading entry of the next row
all the entries below a leading entry are zero
Reduced echelon form
additionally, all leading entries are 1
all entries abov
Echelon form
any rows of 0s at the bottom of the matrix
leading entry of any row is left of the leading entry of the next row
all the entries below a leading entry are zero
Reduced echelon form
additionally, all leading entries are 1
all entries abov
A payoff table is given as; what choice should be made by an optimistic decision maker?
s1
s2
s3
d1
250
750
500
d2
300
-250
1200
d3
500
500
600
d3
d2
d1
d2 or d3
The following is a regret table.
What decision should be made based on the minimax regret cri
THE TAYLOR POLYNOMIAL ERROR FORMULA
Let f (x) be a given function, and assume it has derivatives around some point x = a (with as many derivatives as we nd necessary). For the error in the Taylor
polynomial pn(x), we have the formulas
1
f (x) pn(x) =
(x a
PROPAGATION OF ERROR
Suppose we are evaluating a function f (x) in the machine. Then the result is generally not f (x), but rather
e
an approximate of it which we denote by f (x). Now
suppose that we have a number xA xT . We want
e
to calculate f (xT ), b
SOME DEFINITIONS
Let xT denote the true value of some number, usually
unknown in practice; and let xA denote an approximation of xT .
The error in xA is
error(xA) = xT xA
The relative error in xA is
error(xA)
xT xA
=
rel(xA) =
xT
xT
Example: xT = e, xA =
COMPUTING ANOMALIES
These examples are meant to help motivate the study
of machine arithmetic.
1. Calculator example : Use an HP-15C calculator,
which contains 10 digits in its display. Let
x1 = x2 = x3 = 98765
There are keys on the calculator for the mea
EVALUATING A POLYNOMIAL
Consider having a polynomial
p(x) = a0 + a1x + a2x2 + + anxn
which you need to evaluate for many values of x. How
do you evaluate it? This may seem a strange question,
but the answer is not as obvious as you might think.
The standa
SUMMATION
How should we compute a sum
S = a1 + a2 + + an
with a sequence of machine numbers cfw_a1, ., an. Should
we add from largest to small, should we add from
smallest to largest, or should we just add the numbers
based on their original given order?
ROOTFINDING
We want to nd the numbers x for which
f (x) = 0, with f a given function. Here, we denote
such roots or zeroes by the Greek letter . Rootnding problems occur in many contexts. Sometimes they
are a direct formulation of some physical situtation
INTERPOLATION
Interpolation is a process of nding a formula (often
a polynomial) whose graph will pass through a given
set of points (x, y ).
As an example, consider dening
x1 = ,
x0 = 0,
4
and
yi = cos xi,
x2 =
2
i = 0, 1, 2
This gives us the three point
MULTIPLE ROOTS
We study two classes of functions for which there is
additional diculty in calculating their roots. The rst
of these are functions in which the desired root has a
multiplicity greater than 1. What does this mean?
Let be a root of the functi
FIXED POINT ITERATION
We begin with a computational example. Consider
solving the two equations
E1: x = 1 + .5 sin x
E2: x = 3 + 2 sin x
Graphs of these two equations are shown on accompanying graphs, with the solutions being
E1: = 1.49870113351785
E2: =
ROOTFINDING : A PRINCIPLE
We want to nd the root of a given function f (x).
Thus we want to nd the point x at which the graph of
y = f (x) intersects the x-axis. One of the principles
of numerical analysis is the following.
If you cannot solve the given p
THE SECANT METHOD
Newtons method was based on using the line tangent
to the curve of y = f (x), with the point of tangency
(x0, f (x0). When x0 , the graph of the tangent
line is approximately the same as the graph of y =
f (x) around x = . We then used t
Chapter 1 What are Partial Differential Equations?
Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u du + u2 = c
Chapter 4 Separation of Variables
There are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes v
Chapter 12 Partial Differential Equations in Space
At last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and