Amy's Reflection #27

i hope everyone is enjoying their time off..well anyway, im just gonna post a review of something so we don't forget what we learned in the past...

1.) area of a non right triangle = 1/2 (leg)(leg)SIN(angle b/w)

Example: non-right triangle: HIJ (left to right)H = 65 degrees, j = 2, i = 6. Find the area.

A = (1/2)(2)(6)sin(65)

A = 5.438

2.) Law of Sines(used to non-right triangles):

Sin A/a = Sin B/b= Sin C/c

Example: you have a triangle with the sides 4 and 5 & you also have an angle of 30 degrees.

A = 1/2 (4) (5) Sin 30 degrees

A = 10 Sin 30 degrees which is aproximately = 5

3.) Law of Cosines (used when you can't use Law of Sines):

(opposite leg)^2 = (adjacent leg)^2 + (other adjacent leg)^2 - 2(adjacent leg) (adjacent leg) cos (angle between)

Example: you have a triangle with the sides of 5, 6, and 7. find the angle between 5 and 6.

7^2=6^2+5^2-2(5)(6)

cos a7^2-6^2-5^2= 2(5)(6)

cos acos a= 7^2-6^2-5^2 / -2(6)(5)

a= cos-1 ((7^2-^6^2-5^2)/(-2(5)(6))

a= 78.463 degrees

4.) For any line : m = tan (alpha)
**m = slope , (alpha) = angle of inclination

5.) For a conic: tan 2 (alpha) = B/A-C
**if A=C then pie/4 (always)
**A = coefficient of x^2, B = coefficient of xy, C = coefficient of y^2

Examples:

1. Find the angle of inclination of x^2 - 2xy + 3y^2 = 1.

tan 2 (alpha) = B/A-C

A = 1 , B = -2 , C = 3

tan 2 (alpha) = -2/1 -3 = 1

tan 2 (alpha) = 1

2A = tan^-1 (1)

2 (alpha) = 45 , 225

alpha = 45/2 , 225/2

alpha = 22.5 , 112.5

2. x^2 + y^2 - 3xy + 4x - sqrt.

x = 1alpha = 1 (because A = 1 & C = 1 so A = C)

hope that refreshened your minds...

Taylor reflection for 14 February 2010

I apologize for my reflection being so late i was in vegas and there was barely any cell phone service much less internet service

so ill post my study sheet for the test

Arithmetic= tn= t1 + (n-1) d
Geometric= tn= t1 * r ^ n-1

tn= actual number
n= address of a number in a series

Sn Arithmetic= Sn= n (t1 + tn) /2
Sn Geometric= Sn= t1 (1 - r)^n /1-r

LIMIT -> 1- infinity
polynomial equations uses rules always
rules: t= top b=bottom
t=b: coefficients so the coefficient of the top hightest exponent over the coefficient of the bottom highest exponent
t>b: infinity
t
all other limit problems plug in 100, 1000, 10000 for n in the calculator and record results for each then determine what number the results are headed toward

sum of series only used when r<1>1 then no solution b/c it diverges)
T2 over T1 to get r


Sigma has three parts

a top number
a middle number
a bottom number

top is called limit of summation
middle is called the summand
bottom is called index

top is the address of the last number in given series
middle is the result of the tn formula
bottom is what number you start counting at

if the equation is arithmetic then the bottom number will be 1
if the equation is geometric then the bottom number will be 0


when asked to evaluate for a sigma problem you plug in the numbers including and inbetween the bottom and the top numbers
so if the bottom number is one and the top number is 5 then you would plug in 1, 2, 3, 4, 5 for the variable of the middle equation
and add the results of each plug in together to get final answer


when asked to express then you draw the sigma sign and fill in the top middle and bottom parts



i think i really understand this chapter
ill post again with questions from another chapter later
hope this helps
if anyone needs any explaning or further help dont be afraid to ask!

Amy's Reflection #26

ok i'm just gonna do a whole bunch of examples from chapter 13 to help jog y'all's memory, kk??

Examples:

1. Find the 32nd term in the sequence: 1,4,7,10...
*figure out whether this sequence is arithmetic or geometric.
*arithmetic:because you're adding 3 each time.
*use the arithmetic formula: tn=t1+(n-1)d
*you are looking for "n" in the formula. So you plug in 32 wherever "n" is in the formula.
*So you get t32=1+(32-1)(3)
*that simplifies to =1+(31)(3)
*So t32=94

2. lim (n infinity) sin (1/n)
sin(1/100) = .010
sin(1/1000) = .0010
sin(1/10000) = .00010

lim (n infinity) n+5/n = 1, because the degree is the same, so the coefficients equal 1

3. where the values of x converge
1+(x-2)+(x-2)^2+(x-2)^3+
r=x-2
x-2<1
-1<1
1<3

4.In the arithmetic sequence:3,5,7,9-- find the 28th term.
t28=3+(27)(2)
t28=3+54
t28=57

5. In the geometric sequence: 2,4,8,16-- find the 10th term
t10= 2*2^9
t10= 2*512
t10= 1024

6. Find the sum of the first ten terms of the series:
2 - 6 + 18 - 54 +...
s10 = 2(1 - (-3)^10) / 1 - (-3)
s10 = -29, 524

*(2 being the first number in the problem, -3 being what you multiply each number by
to get the next term)

7. Find the sum of the first 25 terms of the arithmetic series:
11 + 14 + 17 + 20 +...
tn = 11 +(25 - 1)3
tn = 83
s25 = 25(11 +83) / 2
s25 = 1175

*(11 being the first number in the sequence, 3 being the number you add, Plug 25 into the
n-1 formula because your looking for the 25th term)

And for those who have trouble with the problems the involve sigma:

alrighty so say you have this problem...

write the series expanded form.

the limits of summation are 4 and k=1. the index is k. and the summand is 5k.

to expand it, your answer would be 5+10+15+20.

*you have to have 4 numbers in the series becaue thats the number that is the limit of summation. your summand is 5k so you would multiply 1*5, 2*5, 3*5, 4*5 and your answer is 5+10+15+20.

hoped that help...

Stephanie's Reflection

tn-t1+(n-1)d (arithmetic)
sn=(n(t1+tn))/2

• n is term number
• t1 is first term
• d is what you add
• tn is term number in the sequence

tn=t'∙r^(n-1) (geometric)
sn=(t1(1-r^n))/1-r

• r is what you multiply by

lim/n infinity

• if the degree of the top is equal to the degree of the bottom then the answer is the coefficient
• if the degree of the top is greater than the degree of the bottom the answer is infinity
• if the degree of the top is less than the degree of the bottom the answer is 0
• if the rules don't apply, use your calculator

for geometric sequences if the absolute value of r is less than 1 then it goes to 0

the sum of infinite series can only be found when a geometric sequence where the absolute value of r is less than 1

s=t1/1-r

• if the absolute value of r is not less than 1, the series diverges (doesn't approach a number)
• if the absolute value of r is less than 1, the series approaches a number

Stephen's Reflection

Ok so we still on ch 13. This chapter is kinda sorta easy for right now because we basically use the same formulas which are arithmetic and geometric. Im going to explain the formulas for series.

Arithmetic series: Sn=(n(t1+tn))/2

Geometric series: Sn=(t1(1-r^n))/1-r

Ex: Find the sum of the first 25 terms of the series...11+14+17+20+...

first you find tn which is t25 and use the arithmetic formula which is t1+(n-1)d so it will be 11+(24)(3)=83.

Then you plug in the arithmetic series formula: Sn=25(11+83)/2=1175 and that is your answer.

The only thing i have problems with is what do i do with n when they dont tell me how many terms there are?

Stephen

Dustin's Reflection

Didn't learn too much new stuff this week. I'm ready to start reading flatland being that it will help my grade. I missed the whole lesson on limits and didn't know what to do for them on the quiz. I could really use some help on them if possible. I'm just gonna help with some identification because B-rob said that alot of ppl missed those problems on the test.

Say the sequence is: 1,3,6,10,15,21,...

You can see that you are adding 2, then 3,...

Just because you are adding does not make it arithmetic though. In order for it to be arithmetic, the sequence must add or subtract the same number each time. Being that this sequence adds a different number each time, this sequence is not classified as arithmetic or geometric.

Same rules apply for geometric. If it does not multiply by the same number each time, it is not geometric. NEITHER IS ALWAYS A CHOICE.