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
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
Circles and Distance
The Distance Formula
Recall that the Pythagorean Theorem states that if a, b, c are sides of a right triangle with c the
hypotenuse, then
a2 + b2 = c2
Let (x,y) be a point in the plane. Then if we draw the triangle with one vertex at
Arithmetic Sequences and Series
Arithmetic Sequences
Exercise: Find the next term and the general formula for the following:
A. cfw_2, 5, 8, 11, 14, .
B. cfw_0, 4, 8, 12, 16, .
C. cfw_2, -1, -4, -7, -10, .
For each of these three sequences there is a comm
Composition and Inverses
I. Composition of Functions
Example:
Sociologists in Holland determine that the number of people y waiting in a water ride at an
amusement park is given by
y = 1/50C2 + C + 2
where C is the temperature in degrees C. The formula to
Ellipses and Hyperbolae
Draw an Ellipse With a String and Two Fixed Points
Geometrically an ellipse is defined as follows: Let P and Q be fixed points in the plane and let k be a
positive real number. Then a point R is in the ellipse if the sum of the dis
Geometric Sequences and Series
Geometric Sequences
Find the general term of the following:
A. 1, 2, 4, 8, 16, .
B. 27, 9, 3, 1, 1/3, .
C. 3, 6, 12, 24, 48, .
D. 1/2, -1, 2, -4, 8, .
Definition of a Geometric Sequence
A Geometric Sequence is a sequence whi
LOGARITHMS
The inverse of the exponential function- Logarithms
Below is the graph of
y = 2x
and its inverse which we defined as
y = log2 x
We see that the logarithm function y = logbx has the following properties:
1. The x - intercept is (1,0).
2. There i
More Parabolas
Horizontal Shifting
Consider the graphs
y=
1. (x + 0)2
2. (x + 1)2
3. (x + 2)2
4. (x + 3)3
We see that adding a number inside the
parenthesis shifts the graph left or right. The
rules below state this precisely.
Rule1: f(x - a) = f(x) shift
Relations and Functions
Relations
A relation is a rule that takes an input from a set (called the domain) and gives one or more
outputs of another set (called the range).
Examples
1.
(0,0), (4,4), (0,3), (2,1)
2.
y = 2x
3.
4. The circle is a relation. We
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