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 the Infinite Geometric Series
Example
Suppose that a run
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, .
Solution:
E.
We see that the next term is 12. We can get to
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 circle is the set of points in a plane a fixed distance f
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 ellipse can be constructed using a
piece of string. Fix t
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(f(x) = x
Hence we have the following two properties:
I
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 + logby
IV.
Property 3:
logb(x/y) = logbx - logby
V.
E
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
we use (0,5) and (15,0) and note that the arrows point
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. If they intersect then there is
exactly one solution, i
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 + 3z = 5
x +
4z = 3
5x - 7y + 3z = 7
Then the matrix
2 -1
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 points in the plane such that
|QR - PR| = c
is a hyperbol
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 complete the square to get the parabola in standard form.
G
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 how many ways there are in scrambling the letters of
the
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, 2/9, 2/27, .
Solution
A.
We see that to get to the next t
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 just plugged in the values 1,2,3,4, and 5 for n and add
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 two days, there are twelve grams,
how many grams will the
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 base.
Step 4: Solve.
Step 5: Check your solution for dom
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.
1,4,7,10,13,16,.
D.
-1,10,21,32,43,54,.
E.
3,0,-3,-6,-9,-12,.
Sol
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 cannot make the lower right corner a 1 if
d - cb/a = 0
o