limits

Rules to the Limits

 fraction with two polynomials divided, use the rules:
1. If the highest exponent of the top is equal to the highest exponent of the bottom, then you put the leading coeff over the leading coeff.

2. If the highest exponent of the top is greater than the highest exponent of the bottom, then you use plus or minus infinity (plug in to see if you get positive or negative)

3. If the highest exponent of the top is less than the highest exponent of the bottom, then it is equal to zero.

**If it doesn't follow the rules,
1. Plug into y=

2. Use the 2nd funtion, then table on your calculator.
3. Plug in 10, 100, 1000, 10000, 100000....

(In the table, if e is negative, then it is equal to zero. If e is positve, then it is equal to infinity.)

Until you see a pattern.

**If it is a geometric(a number raised to n) and |r| <1, then limit=0
If it is greater than 1, then it is infinity.

Example 1:

lim (n^2 -n)/n^3
= 0

lim [3n^2/n]
= infinity

lim (7n^2+5n)/(10n^2)
= 7/10

Limits

This past week of school we were introduced to Calculus. Our first lesson was in Chapter 19, on Limits. The lesson was pretty simple because we did lessons on limits in advanced math.

Notes to take:

A limit is a singular y-value.

-To find a limit, plug into the equation. If you have 0 on the bottom and 0/0, you have to try doing something else.

Other options:

1-Graph (infinite/-infinite is still considered a (y) value.

2-Use the table:

x-.1, x-.01, x-.001, x, x + .001, x + .01, x + .1

à              ß                              

   

3-Trying to factor and cancel something (algebra).

Examples:

11)   Lim x->4

x-4 / x^2-3x-4

x         3.9     3.99    3.999  4.001   4.01    4.1

f(x)   .209   .2009  .2009   1.9996 1.996  1.96

From doing the table, you can see that as the table values approach x, it gets closer to .2. Therefore, .2 is the answer.

Analytic geometry

Definition of Circle

Definition: A circle is the set of all points that are the same distance, r, from a fixed point.

General Formula: X ² + Y ²=r² where r is the radius

• Unlike parabalas, circles ALWAYS have X ² and Y ² terms.
• X² + Y²=4 is a circle with a radius of 2 ( since 4 =2²)
• Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.

Find the equation of a circle having endpoints (3,7) and (5,-2).

center:  (3-5/2, 7-2/2)=(-1,5/2)

D:  square root of (-5-3)^2+(-2-7)^2

=square root of 64+81

=square root of 145

=12.04

=12.04/2=6.02

(x-h)^2+(y-k)^2=r^2

(x+1)^2+(y-5/2)^2=36.24

Review! :)

What is the formula for an arithmetic sequence?
****tn=t1+(n-1)d

Find the formula for the sequence 10,15,20,25,30,.....
You use the tn=t1+(n-1)d
Tn=10+(n-1)5
Tn=10+ 5n-5
****Tn=5n+5


how many positive two digit numbers are there that are multiples of three?
12,15,.....99
Plug it into your tn=t1+(n-1)d
99=12+(n-1)3
99=12+3n-3
99=9+3n
90=3n
****N=30

what is the geometric sum of the series formula?
***sn=t1(1-r^n)/1-r


Find the slope of the parallel line to: 4x-y=6.
Plug it into -a/b.
-4/-1
****4

Find the slope of the perpendicular line to:3x+2y=4.
-a/b
-3/2
***2/3

Hyperbolas & Identifying graphs

*Standard form
x^2/a^2-y^2/b^2=1

1. the major axis is the variable with the largest denominator (positive)
2. the minor is the variable with the smallest denominator (negative)
3. The vertex is the square root of the largest denominator (put in point form)
4. The other value is the square root of the smallest denominator (put in point form)
5. Skip
6. Skip
7. Focus - F^2=largest denominator + smallest denominator
8. Horizontal Asymptotes - x= + or - square root of y denominator / square root of x denominator 
9. Graph

Ax^2+Bxy+Cy^2+Dx+Ey+F=o *standard form*

-To find the shape of the graph without using the standard form, plug into B^2-4AC

• If it is a circle, you get a negative number,  A=C & B=o
• If it is an ellipse, you get a negative number, A doesn't = C, *& B doesn't = 0
• If it is a parabola you get 0
• If it is a hyperbola, you got a positive number

Example 1: Identify the graph of the equation : X^2-2xy+3y^2-1=0

A=1   B=-2   C=0

B^2-4AC

(-2)^2-4(1)(3)

4-12=-8

Ellipse

1) Take the coefficients of your A, B, & C term

2) Plug them into your formula

3) Solve, then look back at your notes when you find your answer to identify what the graph is

Review of sums of infinite series

This week we haven’t been doing much but reviewing lessons from the beginning of school since finals are approaching. So Im going to do this blog on reviewing lesson 13-5, Sums of infinite series since this lesson is in our review packet!

Notes to know:

-Only geometrics where lrl < 1 have an infinite sum.

The sum formula for geometric is S= t1/1-r

-To find where an infinite geometric converges, set lrl < 1 and solve for x.

-To write a repeating decimal as a fraction, do this:

Whats repeating/place – 1

Examples:

1 1)      Find the sum of the infinite geometric series.

9-6+4-…

To find your R: -6/9= -2/3 , 4/-6= -2/3 (r=-2/3)

S=9/1 – (-2/3)= 9/1+2/3= 27/5

2 2)      For what values does the series converge?

1+ (x-2) + (x-2)^2 + (x-2)^3

l x-2 l < 1

-1 < x-2 < 1

Add 2 to each side and your answer is 1<x<3

3 3)      Repeating decimals: Write .454545.. as a rational number

45/100-1 = 45/99= 5/11

Arithmetic & Geometric Sequences!

*Sequence-a list of numbers. Two types of sequences: --Arithmetic sequence-made by adding the same positive or negative interger to generate the sequence -Fromula tn=t1=(n-1) d -t1=first term -n=term # -d=difference (what is being added or subtracted) -tn=last term --Geometric sequence-made by mulitiplying by the same whole number or fraction to generate a sequence. -Fromula tn=t1 * r ^(n-1) -r=ratio the number that is mulitiplied to generate the sequence -t1=first term -n=term # (tn= ___ term) There are many ways to generate a sequence using these formulas or to find the formula for a sequence of numbers. _____________________________________________________ Ex.1: Find a formula for tn. 1, 4, 7, 10, ... d=3 t1=1 n=n tn=t1+(n-1)d 1+(n-1)3 =1+3n-3 tn=3n-4 Ex.2: Is the following sequence arithmitic or geometric? Find the formula for the nth term. 3,5,7,9,11... This would be arithmitic because we are adding 2. Therefore d = 2. To find the formula for the nth term we need to plug numbers into the formula: tn = t1 + (n-1)d tn = 3 + (n-1)2 tn = 3 + 2n - 2 tn=2n+1

Combination and Permutation

Permutation means when the order is importance
The formula used is : nPr=n!/(n-r)!
When order isnt important you use combination  
The formula used is:Cr=n!/(n-r)!r!


Example:what order could 16 pool balls be in?After choosing, number "14" you can't choose it again.

Ex 2:Permutation: 16 × 15 × 14 × 13 × ... = 20,922,789,888,000

Combination:Picking a team of 3 people from a group of 10. 


C(10,3) = 10!/(7! * 3!) = 10 * 9 * 8 / (3 * 2 * 1) = 120.


Ex 3:Permutation: Picking a President, VP and Waterboy from a group of 10. 
P(10,3) = 10!/7! = 10 * 9 * 8 = 720.


Combination: Choosing 3 desserts from a menu of 10. 
C(10,3) = 120.


Permutation: Listing your 3 favorite desserts, in order, from a menu of 10.


 P(10,3) = 720.


Ex 4:Given 4 people, Bob, Mike, Sue and Alice, how many different ways can these three people be arranged where order matters?

Finding all possible arrangements of Bob, Mike, Sue and Alice where order matters would be:
BMSA,    BMAS,    BSMA,    BSAM,    BAMS,    BASM
MBSA,    MBAS,    MABS,    MASB,    MSBA,    MSAB
SBMA,    SBAM,    SMBA,    SMAB,    SABM,    SAMB
ABMS,    ABSM,    AMBS,    AMSB,    ASBM,    ASMB

There are 24 ways to arrange the four people four at a time,   or   4!

A series is a list of numbers being added. A sequence is a list of numbers. An arithmetic sequence is a sequence that is generated by adding the same number each time. Arithmetic Sequence formula tn=t1 + (n-1)d t1= first term n= term number d= difference  tn= ___ term A geometric sequence is a sequence that is generated by multiplying the same number each time. (to divide, we use fractions) formula: tn=t1(r)^(n-1) t1= first term r= what is being multiplied n= term number tn= ___ term Example: 1.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn= 4n+12 n=1 4(1)+12=16 n=2 4(2)+12=20 n=3 4(3)+12=24 n=4 4(4)+12=28 It is arithmetic because you are adding 4 each time! Example: 2.) Find the first four terms of the sequence and state whether the sequence is arithmetic, geometric, or neither. tn=-2n+8 n=1 -2(1)+8=6 n=2 -2(2)+8=4 n=3 -2(3)+8=2 n=4 -2(4)+8=0 It is arithmetic because you are adding -2 each time

Vectors in Three Dimensions!

Helloo,
I hope everyone had a wonderful weekend!

distance: square root of (x2-x1)^2  +(y2-y1)^2 + (z2-z1)^2
midpoint: ( x1+x2/2 , y1+y2/2 , z1+z2/2 )
sphere: (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2
vector equation: ( x1 y1 z1 ) = ( x0 , y0 , z0 ) + t( a , b , c )
parametric: x=x0+at  y=y0+bt  z=z0+ct
lml (magnitude): square root of x^2 + y^2 + z^2
u * v: x1 x2 + y1 y2 + z1 z2

Example 1:
Find the length and midpoint of AB.
A = ( 2 , 5 , -3 ) and B = ( 0 , 3 , 1 )
Midpoint: ( 2 + 0 / 2 , 5 + 3 / 2 , -3 + 1 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( 0 - 2 ) ^ 2 + ( 3 - 5 ) ^ 2 + ( 1 + 3 ) ^ 2 = square root of 4 + 4 + 16 = square root of 24 = 2 square root of 6

Example 2:
Find the length and midpoint of AB.
A = ( 3 , -5 , 0 ) and B = ( -1 , 1 , 2 )
Midpoint: ( 3 + 1 / 2 , -5 + 1 / 2 , 0 + 2 / 2 ) = ( 1 , -2 , 1 )
Distance: square root of ( -1 - 3 ) ^ 2 + ( 1 + 5 ) ^ 2 + ( 2 + 0 ) ^ 2 = square root of 16 + 36 + 4 = square root of 56 = 2 square root of 14

Example 3:
Simplify the expression.
a. ( 3 , 8 , -2 ) + 2 ( 4 , -1 , 2 )
( 3 , 8 , -2 ) + ( 8 , -2 , 4 ) = ( 11 , 6 , 2 )
b. ( 1 , -8 , 6 ) * ( 5 , 2 , 1 )
5 + -16 + 6 = -5
c. magnitude of ( 3 , 5 , 1 )
square root of 3^2 + 5^2 + 1^2
= square root of 9 + 25 + 1
= square root of 35

Combination and Permutation

Permutation means when the order is important. You use this formula: nPr=n!/(n-r)! Combination means when the order is not important. You use this formula:Cr=n!/(n-r)!r! Example 1: 5people are standing in a line.  How many different outcomes are possible?   P(5, 5) = 5! = 5 × 4 × 3 × 2 × 1 = 120 Example 2:  In how many ways can you select a committee of 4 people from a group of 10 members? C(10, 4) = 210 Example 3: License plates for cars have to be unique. If a license plate contains 6 characters consisting of 2 letters followed by 4 digits  example:  QW2354,  A-how many different license are possible? B-how many different license are possible if letters were allowed to repeated but numbers are not allowed to be repeated? There are 26 × 26 × 10 × 9 × 8 × 7 =  3,407,040 different plates with no repeating numbers. Example 4: In how many ways can you select a committee of 3 people from a group of 12 members? C(12, 3) = 220

Function Notation

f(x) = x^2 + 5x - 2 is an example of a function. If it says "Evaluate f(2)", you can assume x = 2 and plug in 2 for all x's in the function and simplify. Here are some other things you can do with functions:

(f+g)(x) = f(x) + g(x)
(f-g)(x) = f(x) - g(x)
(f*g)(x) = f(x) * g(x)
(f/g)(x) = f(x) / g(x)
(f○g)(x) = f(g(x)) or plug in the equation g(x) into all x's in f(x)
(g○f)(x) = g(f(x)) or plug in the equation f(x) into all x's in g(x)


Ex. 1

f(x) = 3x+3
g(x) = x-8

Evaluate f(2)
f(2) = 3(2)+3
= 6+3
= 9

Evaluate g(3)
g(3) = (3)-8
= -5


Ex. 2
f(x) = 2x-8
g(x) = x^2 +4

Find (f+g)(x)
2x-8 + (x^2 + 4)
x^2 + 2x -4

Find (f-g)(x)
2x-8 - (x^2 + 4)
2x - 8 - x^2 - 4
-x^2 + 2x - 12

Find (f*g)(x)
(2x-8)(x^2+4)
2x^3+8x-8x^2-32
2x^3-8x^2+8x-32

Find (f/g)(x)
(2x-8)/(x^2+4)

Find (f○g)(x)
2(x^2+4) - 8
2x^2+8-8
2x^2

Find (g○f)(x)
(2x-8)^2 +4
4x^2-32x+64+4
4x^2-32x+68


Ex. 3
f(x) = 2x^2 -1
g(x) = x+1

Find (f○g)(2)
2(2+1)^2 -1
2(3)^2 - 1
2(9) - 1
18 - 1
17

Find (g○f)(2)
(2(2)^2 -1) +1
2(4) -1 +1
8 -1 +1
8

Find (g*f)(1)
(1+1)(2(1)^2-1)
(2)(2-1)
(2)(1)
2

Find (f+g)(8)
2(8)^2 - 1 + 8 + 1
2(64)-1+8+1
128+8
136

Find (f-g)(6)
2(6)^2 -1 -(6+1)
2(36) - 1 - 7
72 -8
64

Ch. 6

Find the equation of a circle having endpoints (3,7) and (5,-2).
center:  (3-5/2, 7-2/2)=(-1,5/2)
D:  square root of (-5-3)^2+(-2-7)^2
=square root of 64+81
=square root of 145
=12.04
=12.04/2=6.02
(x-h)^2+(y-k)^2=r^2
(x+1)^2+(y-5/2)^2=36.24

In order to find the equation for a circle, you have to find the center and the radius.  To do this, we use the midpoint formula, and the distance formula.  The midpoint formula is (x1+x2/2,y1+y2/2).  We are given tow different points, so we use those two points and plug them into to formula I just gave you.  Once you do that, you get (-1,5/2).  Now you have to use the distance formula to find the radius.  The distance formula is the square root of (x2-x1)^2+(y2-y1)^2.  You again use the two points you were given in the problem to solve for the distance.  Once you solve this, you get 12.04.  12.04 is the diameter, and we need the radius to find the equation of a circle, so all you do now is divide 12.04 by 2 and we have 6.02 now.  Now we just plug the answers we found into the circle formula.  The circle formula is (x-h)^2+(y-k)^2=r^2.  You have to square your radius you found and that will give you 36.24.  So your answer is (x+1)^2+(y-5/2)^2=36.24.

limits

**If you have a fraction with two polynomials divided, use the rules:
1. If the highest exponent of the top is equal to the highest exponent of the bottom, then you put the leading coeff over the leading coeff.

2. If the highest exponent of the top is greater than the highest exponent of the bottom, then you use plus or minus infinity (plug in to see if you get positive or negative)

3. If the highest exponent of the top is less than the highest exponent of the bottom, then it is equal to zero.

**If it doesn't follow the rules,
1. Plug into y=

2. Use the 2nd funtion, then table on your calculator.
3. Plug in 10, 100, 1000, 10000, 100000....

(In the table, if e is negative, then it is equal to zero. If e is positve, then it is equal to infinity.)

Until you see a pattern.

**If it is a geometric(a number raised to n) and |r| <1, then limit=0
If it is greater than 1, then it is infinity.

Example 1:
Find the given limit.
lim (n^2 -n)/n^3
= 0

lim [3n^2/n]
= infinity

lim (7n^2+5n)/(10n^2)
= 7/10

Area of a non-right triangle

To find the area of a non-right triangle, use this formula:
Area=1/2(leg)(other leg)sin(angle between)
So,

EXAMPLE
________________________________________

    78 degrees
    /\
9 /  \  12
  /__\              
                         
In triangle ABC, a = 12, B = 78 degrees, and c = 9. Find area of the triangle.
Area=1/2(12)(9)sin(78)
Area=15.82

Chapter 4

Domain and Range:-Domain- the interval of x values where the graph exist
-Range- the interval of y values whre the graph exist
-Zeros, x-intercept, root - set = 0, slove for x
-To be a function the graph must pass the vertical line test
-If given points the domain is the list of all x-values and range is of all y-values in { }

To find Domain and Range:
-Polynomials: domain and always (negative infinity, positive infinity)
-Square root: 1. Set inside root equal to 0 and solve
                      2. Set up intervals
                      3. Plug in if you get a negative x interval
                      4. Write answer in interval notation- use brackets by the number Range: [ a, infinity)
-Square root: Domain: 1. Set inside equal to 0 and solve
                                     2. Put answer in [-   ]
                      Range: [0, square root of number] if a=0
                                  [0 plus or minus a, square root of a number + a] if a is positive
                                  [0 minus a, square root of a number minus a] if a is neagtive
-Fraction: Domain: 1. Factor top and bottom
                                2. Cancell if possible and marke # if cancelled
                                3. Set bottom equal to 0 and slove for x value
                                4. Write in interval notation stopping at numbers found in 2 and 3
-Absolute Value: Domina: (- infinity, positive infinity)
                             Range: [0 + a, infinity) if v is ponted up and a is positive
                                         [0 - a, infinity) if v is pointed up and a is negative
                                         (- infinity, 0 - a] if v is ponted down and a is negative
                                         (- infinty, 0 +a] uf v is pointed down and a is positive


Example 1: Find the Domain and Range of the following: f(x) = 5x^3 + 4x^2 + 3x - 6

 Domain: (- infinity, infinity)
 Range: (- infinity, infinity)

Funtion? Yes


--Daniellee

Circles

Standard form (x – h)^2 + (y – k )^2 = r^2

Where (h,k) is the center and r is the radius

You can find the radius using the distance formula with the center

and a point on the outside or you can half the diameter found by

using 2 points on the outside through the center

Equations must be put in standard form by completing the square

a. Divide by leading coefficient, move all numbers to the right

b. Half the middle term, then square it and add to both sides

c. Factor

To graph in your calculator you must solve for y and put +

and – in formula in y =

To find the intersection of a line and circle solve the line for y

Plug into the circle equation for y

Solve the quadratic

Ex 1. Find the center and radius of (x – 3)^2 + (y + 7)^2 = 19

Center: (3, 7) Radius: sqrt19

Ex 2. Given C(-1, 7) and the radius is 7 units, find the equation

of the circle in standard form.

(x + 1)^2 + (y – 7)^2 = 49

Ex 3. Find the center and radius of x^2 – 2x + y^2 – 6y = 9

-2/2 = (-1)^2 = 1 -6/2 = (-3)^2 = 9 9 + 1 + 9 = 19

(x – 1)^2 + (y – 3)^2 = 19

Center: (1, 3) Radius: sqrt19

Conics

Conics are a type of graph. They can be graphed by calculator or hand.  Each conic has a certain equation and a way it looks when graphed. This is about circles.

Standard form: (x-h)^2 = (y-k)^2 = r^2

-You can find the radius using the distance formula with the center and a point on the outside or you can half the diameter foud by using 2 points on the outside through the center.

-Equations must be put in standard forn by completing the square. These are the steps:
1.Divide by leading coefficient, move all #s to right
2.Half the middle term, them saure it and add to both sides
3.Factor everything

-To graph in your calculator you must solve for y and put a + e - formula in the Y= slot in calculator.

-To graph without calculor you  must find center points and raidus. Graph as told.

EX1.  Write an equation for the line with a center of (2,8) and radius of 4.
plug into formula (x-h)^2 + (y-k)^2 = r^2
(x-2)^2 + (y-8)^2 = 16

EX 2. Find the center and radius of (x-3)^2 + (y +5)^2 = 9
center = (3, -5)
radius + sqaure root 9 = 3

Circle Stuff

The equation of a circle and the properties of the circle. The equation used is the standard equation that has the form: (x - h)2 + (y - k)2 = r2 where h and k are the x- and y-coordinates of the center of the circle and r is the radius. The exploration is carried out by changing the parameters h, k and r included in the above 1. h and k to zero and parameter r to 1. Check that the circle shown has the center at (0,0) and radius equal to 1. 3. set r to zero and parameters h and k to different values, the graph of the circle is a point, Explain.(Hint:Solve the equation (x - h)2 + (y - k)2 = 0 4 - Keep r equal to 1 and shift the circle by changing h and k. Check that the center of the circle is at (h , k). 5 - Keep h and k constant and change r. Check that the circle has radius r. 6 - Set h, k and r to 1. The circle has one point of intersection with the x-axis and one point of intersection with the y-axis. These are called the x and y intercepts. Find these points analytically using the equation of the circle. (x - h)2 + (y - k)2 = r2 (Hint: To find the x-intercepts set y = 0 in the equation and solve for x. To find the y-intercepts set x = 0 in the equation and solve for y.) 7- Set r to 2 and h to a certain value. Change k from -1.8 to 1.8 (|h| less than r). How many x-intercepts are there? Set k to 2 (the radius), How many x-intercepts are there? Set k to -2, how many x-intercepts are there? Set k to values greater than 2 (the radius), how many x-intercepts are there? Set k to values smaller than -2, how many x-intercepts are there? Explain analytically. 8- Try the same exploration as in 7 above with the y-intercepts by changing the value of h. 9- Exercise: Find (analytically) values of h, k and r such that the circle associated with these values has no x or y-intercepts. Check your answer graphically. From the website: http://www.analyzemath.com/CircleEq/CircleEq.html