MATH 411
Chapter 5
0.1
Definition 1: Lipschitz Condition
A function f (t, y) is said to satisfy a Lipschitz condition in the variable y on a set D R2 if a constant L > 0 exists with |f (t, y1 ) - f (t, y2 )| L|y1 - y2 | whenever (t, y1 ), (t, y2 ) D. The
1.
2.
3.
4.
Y = x3
Y = -x3
Y = x3+4x2
Cubic functions always have a vertex where the graph changes directions. This occurs in
the middle.
5. A cubic function is never not one-to-one. It never ends, and it never grows in a linear,
non-exponential way.
6. T
Experiencing Exponents
In 1974 Richard M. Nixon resigned as president of the United States, Hank Aaron hit 715
home runs to break Babe Ruths record, and one gallon of milk cost $1.57. Since 1974
the average annual rate of inflation has been about 5%. This
Properties of Exponents
Part 1: Simplify each exponent expression below:
1. 5 4 53
57
2. 5 4
3
5. 50 1
6. 5 3 1/53
4
2
24/54
5
3.
(5 x ) 4 54x4
7.
4.
57
54
53
8. 5 3
2
Part 2: Complete each exponent property listed below:
1. a n a m
5. a 0
2.
(a n
1.
2.
3.
4.
5.
Y = 2x
Y = -2x
It never has a vertical asymptote; this is shown by the limitless domain.
It always has a horizontal asymptote. This is shown by the limited range.
Exponential functions are never linear, they never have an unlimited range an
Day 8: Unit 1 Vocabulary Quiz
Instructions:
Please answer the following questions. On your first attempt, try to complete this quiz
without the use of notes, or reference materials. This will let you see how well you
currently know the vocabulary. After m
Applying Exponents
1. I have $5000 dollars to invest at 3% compounded annually (The bank pays me interest at
the end of each year. That interest becomes part of my principle for the next year.)
a. How much money will I have after 10 years (principle and i
The Composition of Functions
Youve seen in previous activities that you can combine functions using arithmetic
operations. You can also create new functions from old ones by first applying one
function and then applying another.
Given two functions f and
Polynomial Characteristics
Use the Polynomial Factors app to investigate the characteristics of polynomials and
to answer the following questions (be sure to look at the various sheets or tabs at
the bottom):
1. Are the equations in the Polynomial Factors
My two equations were the standard equation of y=0.25x2-3x+10.5 and the transformation
equation of y=0.25(x-6)2+1.5. To convert from the standard, you would do this:
y=0.25x2-3x+10.5
y=.25(x2-12x)+10.5
y=.25(x2-12x+36-36)+10.5
y=.25(x-6)2-36)+10.5
y=.25(x
Function Fact Sheet
Name of Function Family:_Linear_
Formula
The general formula for this type
of function (if there is one):
If there is not a general formula for this type of function, how do you tell if a given formula is
this type of function?
y = mx
1. y = x
2. This would look like this:
3. y =
3 x
4. This would look like an S like this:
5. Radical functions never have an end, a maximum, and are never not one-to-one,
6. When you swing back and forth on a swing at the park, how long it takes you to co
Solving the Simplest
1. Solve each of these matrix equations by first turning them into a system of linear
equations by using the multiplication algorithm that we developed in class (please
read the reference: Matrices and Linear Systems) and then using o
Transformations
Practice Identifying
This PowerPoint allows you to practice what you know about
transformations. Get a piece of paper (and maybe even some graph
paper). Write down your answers to each question. Only then
should you go on to the next slide
At the bottom of this file you will find the tabs for each of the nine function families that we will learn about in Unit 1.
To explore, choose the family that you want to learn about by clicking on the appropriate tab.
As you change the numbers in the gr
Day 9: Review for Unit 1 Test
Instructions:
Add instructions.
Answer the following six questions. For each statement below identify
whether it is true or false.
1. True or false? A relation is also a function.
2. True or false? The domain for (x)=2x2-3x+5
Quadratic
Cubic
Example equation:
y
Rational
Example equation:
y
Example Data:
Example Data:
Example Application:
Example equation:
p ( x)
y
q( x)
Example Data:
Example Application:
Independent variable is
squared.
Graph is a parabola
To graph: find the
v
Function Compare/Contrast Assignment
I. Complete the following chart by filling in each cell with an example and a description of how to
recognize each function when given in that representation.
Linear
Symbolic
y = mx + b
The highest power
on the variabl
1. y = log2(x+1)
2. A logarithmic function always has a vertical asymptote. The domain tells us this because
it never reaches one point, but always gets closer
3. They never have horizontal asymptotes. The range confirms this because it is evergrowing.
4.
1.
List the Rule of Four for representing functions and give examples of each.
A formula, a graph, a real world example, and a table of numbers
Y=x,
6
4
2
0
, the amount you are paid compared to the hours
worked,
1
1
2
2
5
5
2.
Apply the vocabulary of fun