Summation Notation (Sigma Notation)

Has a base of the capital version of the greek letter "sigma"

When used in mathmatics, it looks like this

What does all of the stuff around it mean?

n=upper limit
i=1 = lower limit
= explicit formula =

EXAMPLE PROBLEM:

=5 + 7 + 9 + 11 = 32
^^^^^^
To do this on a calculator

• Sum(Seq(explicit formula, variable, lower limit, upper limit))
  

To get to Sum push List then go over to math

To get to seq push List then go over to OPS

When Dealing with an infinite upper limit:

it looks like this

FOR EXAMPLE

3 PROPERTIES OF SUMS

1.

2. & 3. (the same for adding and subtracting)

This is a good explanation of summation notation
http://www.columbia.edu/itc/sipa/math/summation.html

9.1 Sequences And Series

Sequence- an ordered list of numbers

ex. 8, 13, 18, 23, 28, 33, . . .

(each number here is increased by 5)

Recursive Formula (for above example)

(you must labelwereevery sequence begins)

(above ex. labeled recursively)

next term=known term +5

Explicit Formula (for above example)

*only for above sequence*

Important Notes

• each sequence has its own recursive and explicitformulas
• explicit formulas can look like anything whereas recursive formulas follow the above format
• use theexplicit formula of a sequence to find the nth term of the sequence

EX. a1=1

a2=2

a3=4

a4=8

EX. -1/2, 4/4, -9/8, 16/16, -25/32, 36/64

Factorials

A factorial is a positive integer n, denoted by n!, that is the product of all positive integers less than or equal to n

ex. 5!= (1)(2)(3)(4)(5)

Solving Factorials

ex.

(you can reduce this by canceling out like terms and reducing others)

ex.

(cancel all like terms out now)

Solving Systems of Equations

Substitution Method:

A solution of a system of equations is an ordered pair that satisfies each equation in the system. To check if an ordered pair is a solution to a system, substitute the number for x and y and see if they equal eachother.

example-

{2x+y=5

{3x-2y=4

(2,1)

2(2)+1=5 3(2)-2(1)=4

5=5 4=4

The ordered pair is a solution

Solving a System of Equations:

example-

{x+y=4 equation 1

{x-y=2 equation 2

Solve equation 1 for y.

y= 4-x

Substitute (4-x) into equation 2 for y.

x-y=2

x-(4-x)=2

x-4+x=2

2x=6

x=3

Solve for y by substituting x=3 into the equation y=4-x

y=4-x

y=4-3

y=1

The solution is the ordered pair of (3,1).

Check in equation 1.

x+y=4

3+1=4

4=4

Check in equation 2.

x-y=2

3-1=2

2=2

Solving a System of Equations by Graphing

example-

{y=lnx

{x+y=1

(1,0) is the only point of intersection on the graph, therefore it is the only solution.

Check.

0=ln1

1+0=1

Review of Exponential and Logrithmic Functions

Exponential Funcions

Using the One-to-one property:

When the bases on both sides of an equation are equal, their powers can be set equal to each other.

Ex 1:

Using the Inverse Property ("The Key to Everything"):

This property allows you to get rid of exponents by taking the log or natural log of both sides of an equation. By doing so, the exponent can be moved to the ouside of the log or natural log, and the log will be able to be changed to 1 (because ln(e)=1 and you can any kind of log so that it changes to 1)

Ex 2:

Logarithmic Functions

Using the One-to-One property:

Ex 3:

Using the Inverse Property ("The Key To Everything"):

Ex 4:

Solving Exponential and Logarithmic Equations- Sec. 3.4

        Friday's lesson consisted of learning the procedures used in solving exponential and logarithmic equations. Before we delved into that, though, we reviewed what it meant for a function to be "one-to-one."

One-to-one:

Where    implies  

for example    3^x=3^2x-5 therefore x=2x-5

and when it comes to logs with the same base implies x=7

Solving Exponential Equations

Example one: Solve for x when

The first thing to recognize is that you can take either an algebraic or logarithmic approach when solving this equation.

If solving this equation algebraically:first take the natural log of both sides
                                                             natural log of e is simply e, so your next step becomes
                                                               x=ln17         which you plug into your calculator to find that
                                                               x=2.83        (remember to round to the nearest hundredth..or else)

If solving this equation logarithmically: first rewrite as the equivalent logarithmic, then get
                                                               x=ln17          (since log base e is representative of ln)
                                                               x=2.83         "  "

Example two: Using log algebraically:  2 + 5^(x-3) = 12  
                                                                 5^x-3=10           subtract 2
                                                           log[5^(x-3)]=log10   take log of both sides
                                                                 (x-3)log5=1        move x-3 out front and log base10 of 10=1
                                                                 x-3=1/log5         divide by log 5
                                                                x=3+(1/log5)       add 3 to each side
                                                                    x=4.43            round to nearest hundredth

                                                                                         (remember that you can easily 
                                                                                               your answer by plugging it
                                                                                                into the original equation)    

The final example Mr.Wilhelm showed us was one of an exponential equation in need of solving that looked eerily similar to a quadratic equation.

Example three:            e^(2x) - 3e^x + 2 = 0         first let u=e^x
                                           u^2 - 3u +2 = 0
                                            (u-2) (u-1) = 0    
                                         e^x = 2    e^x = 1           substitute e^x back in
                                 ln e^x = ln 2     ln e^x = ln 1    take the natural log of all four sides
                                   x = ln 2                 x=0          ln 1 implies e^x must equal 1 so therefore x is 0 and
                                   x= .69                                    

Homework: Section 3.4 problems 3,5,12,14,16-40 multiples of 4, 46, 47, 50

                                                                    
                             

                                                           

                                                               
                                                         

              

Properties of Logrithms

Properties of Logarithms

Assuming and , the Law of everything, which says, , we can conclude that , and Furthermore, we know that and sooo

This is the logic by which the 3, and only 3, properties of logrithms are derived. They are as follows: