Infinite Geometric Series
I.
Definition of an Infinite Geometric Series
We learned that a geometric series has the form
Definition
The series
is called the infinite geometric series.
II.
Calculating t
Sequences and Series
I.
Sequences
Example: Find the next term and describe the pattern:
A.
2, 4, 6, 8, 10, .
B.
1, 4, 9, 16, 25, .
C.
3, 7, 15, 31, 63, .
D.
1, -1/2, 1/6, -1/24, 1/120, -1/720, .
Solut
The Circle
I.
Conic Sections
A conic section is formed by intersecting a plane with a cone. The different possible
conic sections are the circle, parabola, ellipse, and the hyperbola.
II.
Circles
A ci
The Ellipse
I.
Definition of the Ellipse (Geometric)
Let P and Q be two points (the foci) in the plane. The ellipse is the set of all points R in
the plane such that PR + QR is a fixed constant. An el
Logarithms
The Definition of the Logarithm
I.
Definition
The function logbx is defined as the inverse function of y = bx
Recall that by definition, if f and g are inverse functions then
II.
f(g(x) = g
Properties of Logarithms
I.
Properties of Logarithms and their proofs
Property 1:
logbxy = ylogbx
II.
Proof:
We have
III.
logbxy = logb(blogb(x)y
= logb(bylogb(x) = ylogbx
Property 2:
logb(xy) = logbx
Linear Programming
I.
Linear Programming (An Example)
Example
Maximize
P = 2x + 5
subject to the constraints
x + 3y < 15
4x + y < 16
x>0
y>0
First we graph the system of inequalities.
For
x + 3y = 15
Linear Systems
Geometry of Systems of Equations
I.
We know that for two by two linear systems of equation, the geometry is that of two lines
that either intersect, are parallel, or are the same line.
Matrices
I.
Definition of a Matrix
An m by n matrix is an array of numbers with m rows and n columns.
Example 1:
4 5
0 15
-9 3
is a 3 by 2 matrix.
Example 2:
Consider the system of equations
2x - y +
The Hyperbola
I.
Definition of the Hyperbola (Geometric)
The final conic that we will study is called the hyperbola.
Definition
Let P and Q be two fixed points, and c be a constant. Then the
set of po
The Parabola
Algebraic Definition of The Parabola
Recall that the standard equation of the parabola is given by
y = a(x - h)2 + k
If we are given the equation of a parabola
y = ax2 +bx + c
we can comp
Factorials and Their Applications
I.
Definition of the Factorial
We define n! recursively by
0! = 0,
1! = 1,
n! = n(n - 1)!
Example:
5! = 5(4)(3)(2) = 120
Example:
Suppose that we are interested in ho
Geometric Sequences And Series
I.
Geometric Sequences
Example: Find the General Element
i) Recursively
ii) Explicitly
A) 3, 6, 12, 24, 48, .
B) 5, 15, 45, 135, .
C) -3, 30, -300, 3000, .
D) 2, 2/3, 2/
Induction
I.
Evaluating a Series
If we have a finite series there are two ways of evaluating it. The first way is
computation by hand.
Example:
n = 15 (2n-1)
is
1 + 3 + 5 + 7 + 9 = 25.
Notice that we
Exponentials
I.
Example of an Exponential Function
A biologist grows bacteria in a culture. If initially there were three grams of bacteria,
after one day there are six grams of bacteria, and after tw
Exponential and Log Equations
I.
Equations that Involve Logs
Step by Step Method
Step 1: Contract to a single log.
Step 2: Get the log by itself.
Step 3: Exponentiate both sides with the appropriate b
Arithmetic Sequences and Series
I.
Arithmetic Sequence
Examples
Find the general term for the following sequences both recursively and explicitly:
A.
2,6,10,14,18,22, .
B.
-5,-3,-1,1,3,.
C.
1,4,7,10,1
Determinants and Inverses
I.
Determinants:
Consider row reducing the standard 2x2 matrix. Suppose that a is nonzero.
a b
c d
1/a R1 -> R1
R2 - cR1 -> R2
1 b/a
c d
1
b/a
0 d - cb/a
Now notice that we c