SEQUENCE a list of numbers in a particular order.
A recursive formula gives an+1 in terms of an:
cfw_an = cfw_a1, a2, a3, an,
an+1 = an + C
An explicit formula gives an as a function of n:
terms of the sequence
an = 2n
Sequences
Find the (a) next three t
Fundamental Counting Principle.
1. How many odd 3-digit positive integers can be written using the digits 2, 3, 4, 5 and 6?
2. A student council has 5 seniors, 4 juniors, 3 sophomores and 2 freshmen as members. In how many ways can a 4-member council
comm
1. Summation Notation
Evaluate the following summations.
PARTIAL SUMS and SUMMATION (SIGMA) NOTATION
upper limit
4
(a)
(2i 1)
i
1
5
(b)
(1 r
2
lower limit
)
index of summation
r
3
2.
Little Gauss Problem
When little Carl Gauss (1777-1855) was in the thi
1.
An Epidemic Model
Rudy is one of 1,200 employees at the Saline Visteon Plant, which operates seven days a week. He arrives at the plant on Day 1 (a
Monday) with a slight fever. By noon, Rudy became so ill that he had to go home from work. It turns out
GEOMETRIC SEQUENCE a sequence characterized by a common ratio (r).
Explicit formula : an a1 r n 1
common ratio: r = an+1 an
Recursive formula : S n 1 ran
st
1 term: a = a1
NOTE: a geometric sequence is also known as a geometric progression.
Geometric Sequ
Define: Random Variable
Binomial Experiment
Example 1. Probability Distribution
Probability
Probability Distribution
3
/8
1
/4
1
Let X be a random variable that represents the number of heads
when 4 coins are tossed.
/8
0
1
2
3
4
Number of heads
(a) Make
Probability the likelihood that an event will occur (0 < p < 1)
Theoretical Probability
What should happen.
P ( A)
number of favorable outcomes
total number outcomes
Odds: The odds in favor of event A =
Experimental Probability
What did happen after man
Example 1. Counting Principle.
(a) A customer in a computer store can choose 1 of 3 monitors, 1 of 2 keyboards and
1 of 4 printers. Assuming all components are compatible, how many different
systems can be chosen?
Fundamental Counting Principle.
If one ev
Example 1
1. If you draw two cards from a standard deck of 52 cards find each
probability:
Probability of A and B
If A and B are any two independent events,
then the probability of A and B is:
P(A & B) = P(A) P(B)
(a) You draw a heart and a club if you re
Example 1. Mutually Exclusive Events.
(a) A six-sided die is rolled; what is the probability that the number rolled is
less than 3 or greater than 5?
Probability of A or B
If A and B are any two events, then the
probability of A or B is:
P(A or B) = P(A)
Example 1. A Subset of elements.
How many three letter subsets can be chosen from the set cfw_a, b, c, d, e?
Combination
An r element subset of a set with n
elements. (Order does not matter)
Example 2. Order or Not?
For each of the following examples, dec
The Hyperbola- The locus of points in a plane whose difference in distances from two distinct points (the foci) is constant (2a).
1. The Hyperbola.
Draw and label a hyperbola with a center
2. Graphing Hyperbolas.
Sketch and label each hyperbola.
(a)
x2 y2
1. Distance and Midpoint Formulas.
The Distance Formula
Given two points in a plane (x1, y1) and (x2, y2)
the distance between the two points is:
Consider the points (2, -3) and (-4, -1)
d ( x1 x 2 ) 2 ( y1 y 2 ) 2
(a) Find the distance between the points
1. Solve a Non Linear Inequality.
Solve.
(a) x 2 4 x 12 0
(b) x 3 6 x 2 9 x
2. Solve a Rational Inequality by Graphing.
Consider the function f ( x)
x2 4
x 3
(a) Sketch the graph of y = f(x)
(b) For what values of x is the function greater than zero?
(c)
The Ellipse-The locus of points in a plane whose sum of the distances from two distinct points (the foci) is constant (2a).
1. The Ellipse.
Draw and label an ellipse with a center at
the origin.
2. Graphing Ellipses.
Sketch and label each ellipse.
(a)
x2
1. Circle.
Standard Form of a Circle
Derive the standard form of the circle with a center at the origin and a radius r.
2. Graphing Circles.
Graph each circle. Identify the radius.
(a) x2 + y2 = 16
(b) x2 4 = -y2
(c) -6 + 1/2x2 = -1/2y2
(d) x2 + y2 < 10
3
1. y varies inversely as x. y = 6 when x = 8, find x when y = 20.
2. y varies directly as x and inversely as z. y = 10 when x = 8 and z = 4, find x when y = 6 and z = 12.
3. The amount of time it takes run a marathon is inversely proportional to the rate
1. Solve a Rational Equation.
(a)
z2 z
1
3 6
(b)
3
1
u 2 u 2
(c)
5
2 1
2
x 2
2x
(d)
1
1
4
y 2 y 2 y 2 4
(e) 2
5
x 3
x 2 x 6 x 2
2. Application. Distance Problem.
Tim paddled his kayak 12 km upstream against a 3 km/hr current and back again in 5 h and
1.
2.
Simplifying Rational Expressions
Simplify each of the following. Give any restrictions on your simplification.
x 2 2x
x 2 4 x 12
2 x 2 18
(a)
(b)
(c)
x 2
x2 4
3 2 x x 2
Multiplying and Dividing Rational Expressions
Simplify each of the following.
x
Inverse Variation
Two variables xand y show an inverse variation if y
decreases as x increases.
1. Inverse Variation.
(a) If y is inversely proportional to x, and y = 8 when x = 6, find x when
y = 15.
Variation Equation: y
k
x
where k is the constant of
Rational Function = The ratio of two polynomials.
R( x)
P ( x)
, where P and Q are polynomials.
Q( x)
The domain of R is cfw_x: Q(x) 0.
Vertical Asymptotes and Holes
1. If Q(c) = 0 and P(c) 0, then R(x) has a vertical asymptote at x = c.
2. If Q(c) = 0 a
1. Horizontal Asymptotes of a Rational Function.
2
Consider the function f ( x)
x1
Horizontal Asymptotes for Rational Functions
R( x)
(a) State the domain of f.
a m x m . a 2 x 2 a1 x a o
bn x n . b2 x 2 b1 x bo
if m n
0
H . A lim R( x). a m / bn if m n
1. Adding and Subtracting Rational Expressions.
Simplify
(a)
7 3 1
3 4 6
(d)
2
3
x 2 x
(b)
7
5
12 x 12 x
(e)
5
6
t 2 t 2
(c)
2 x 1 3 x 1
2x
3x 2
(f)
3
5
2
x x 2 x x 6
2
2. Complex Fractions.
Simplify.
2 4
3 9
(a)
11 1
6 2
6
1
x 5 x
(b)
3
2
x x 5
(c)
u
Reference: McDougal Littell: 7.3 Use Functions Involving e. Pages 492-495 See Examples 1-5
1. The Euler Number.
1
n
The Natural Base e
The Natural Base e is an irrational number. It is
defined as:
n
Given f (n) 1 , evaluate the following:
(a) f(10)
(b) f(
Reference: McDougal Littell: 7.1 & 7.2 Graph Exponential Growth and Decay Functions. Pages 480-481 See Examples 4,5 Page 488 Example 4
1. A new ice cream company began in 2010 and was able to produce 1000 gallons of ice cream. In order to grow the
company
1. Once upon a time a brave knight returned from battle a hero. The king, to show his appreciation, offered him his gold
plated chess set. The brave knight, knowing that his fellow countrymen are starving, had a better plan. Instead of the
chess board, he
Reference: McDougal Littell: Graph and Solve Quadratic Inequalities. Pages 300-303. Examples 1-7
1. Quadratic Inequality in Two Variables.
Graph the solution set to each inequality or system of inequalities.
(a) y > x2 + 2x 8
(b) y < 2(x 1)2 3
(c) y > x2
Reference: McDougal Littell: Use the Quadratic Formula and the Discriminant. Pages 292-295. Examples 1-5
1. Using the Quadratic Formula.
Solve using the quadratic formula.
(a) y2 4y 13 = 0
The Quadratic Formula
For a quadratic equation of the form ax2 + b