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