2. In Cognitive Tutor, you will learn composition of linear
functions. However, it is possible to do composition
of functions with any type of functions. Here is an example
between linear and quadratic functions:
Example
Let and – .
So,
( )
Careful: is not the same as . Order matters! See the
example below using the same
functions above.
( )
One-to-One Functions
Previously, you learned that the basic idea of a function is that
“every input has exactly one output,” or
“every x value has exactly one y value.”
4. Test).
Understanding one-to-one functions is essential to understand
inverse functions, which you will learn in
Cognitive Tutor. A real-life example of one-to-one functions is
the following: Every person has a unique
social security number and each social security number
corresponds to a person.
Exponential Functions
When a constant (a fixed number) is raised to a variable (say,
x), we are working with exponential
functions. Algebraically, exponential functions have the form
, where a > 0 but cannot be 1.
For example, . When the value of is greater than 1, the
graph of these functions have this
form:
From the graph above, as the x increases, y increases by a factor
of a. For such reason, we call the
constant a the growth factor. When the value of a is between 0
and 1 (in other words, a rational
number), the graph of these functions have the form
6. The most often used logarithm is where , so means
. This is referred to as
the common logarithm. The base 10 is implied if no subscript
number is given in the logarithmic
expression.
So, is typically expressed as .
The follow charts offers some common logarithms for numbers
that are powers of 10:
10
-3
(0.001)
10
-2
(0.01)
10
-1
(0.1)
10
0
9. This sequence has infinite number of terms where = 3, = 5,
= 7, = 9, and so on.
If you look closely, the next term of this sequence is 2 more
than the previous term. When a sequence is
created by adding a constant amount, the sequence is considered
an arithmetic sequence. Any term of
an arithmetic sequence can be found using the formula:
where is the first term of the sequence, is the position of the
term in the sequence, and is the
common difference between any two terms. For the sequence
above, = 3 and = 2; so,
Do the following to find the 10th term in the sequence infinite
3, 5, 7, 9 …
When a sequence is created by multiplying a constant amount,
the sequence is considered a geometric
sequence. For example,
2, 4, 8, 16, …
If you look closely, the next term of this sequence is 2 times the
11. A series is the sum of the numbers in a sequence. The partial
sum of a sequence is denoted with the
notation . For example, to find the sum of the first 4 terms of
the infinite sequence 3, 5, 7, 9, …, we
write it as
or we can use summation notation and the formula to find each
term of the sequence:
∑
Fortunately, there are formulas to quickly find the partial sum
of arithmetic and geometric sequences:
Partial Sum for an Arithmetic Sequence:
12. is the first term and is the last term of the sequence. For
example, the partial sum of the arithmetic
sequence 3, 5, 7, 9 is
Partial Sum for a Geometric Sequence:
is the first term, is the number of terms in the sequence, and
is the common ratio. For example,
the partial sum of the geometric sequence 2, 4, 8, 16 is
WEEK TEN
Grammar Exercise: Sentence Revisions (Due Date: March 26)
Reading: Chapter 17 (Definition) plus the sample essay listed as
Extended Definition Reading in the Assignment Guidelines
(“What is Poverty?”)
13. Writing Assignment: Create your own “extended definition” of
at least 5 paragraphs. Use something that is an abstract term or
one that is not the same definition to everyone. Consider
personal connotation and interpretation of the term. Suggested
topics: Expert, Rain check, Morality, Militant, Liberal, Fitness,
Innovation, Fulfillment, Middle Age, Affirmative Action, etc
Like the author of “What is Poverty?”, make your own
definition very specific and detailed. (Due Date: March 30)