It's Today!

For those of you who've forgotten, I'll be in my classroom today, Wednesday, December 30, from 1:00 until 3:00.

See you then.

More Probability

Getting things organized, and some probability stuff.

Today, we spent the first 10 minutes normally spent on a topic of Mr. Max choice, getting our old tests handed back. That's always a highlight of the class.

Then, Mr. Max gave us some things to think about:

1. Accelerated Math is now "out of" approximately 54 objectives. Where are you?

2. Pre-Test for upcoming Thursday test was handed out last week, have you done all questions?

3. NO CALCULATOR for thursday's test.

Here's what you should know how to do for Thursdays test:

Trig 1 (circular functions)

Transformations

Trig 11 (identities)

Logarithms/Exponential Functions

Perms & Coms

Conics

Now, here's what's left to learn:
1) Probability

2) Geometric Sequences

Here's what we learned today: Probablity

We first learned about sample spacing, which is basically creating a picture about the situation your given. Using ordered pairs, or a chart, or a T chart.

Then we learned about the two different event types, Simple and Compound.

A simple event is when the sample space can not get any simpler then what it is.

A compound event is two or more simple events are considered at once.

Then there's the 2 laws of probablity, first being the addition law, "or". Basically if your trying to find the probability of A or B, you will have to add. The formula is a little more complicated, but its on our formula sheets.

The second law is the multiplication law, "and". When your trying to find the probability of A and B, you will have to multiply. This formula is a little harder to understand, with the whole P(BA) part to it.
We basically spent the rest of the class doing examples.

So, do not forget!!!

Thursday = Test. (Forget your calculator at home kids, we don't need 'em.)

January 14th, 12:30 pm = Pilot exam (10 days after we get back from Holidays.. oh boy. Merry Christmas to us!)

Thats about it! Good luck everyone! Study it up!

Sketching Standard Form Hyperbolas (Conics)

Sorry for the lateness of this posting. I had a blonde moment. The video was showing up as text but I guess when I publish it, it shows up as normal. Anyway....



Today we learned how to graph hyperbolas.....YAAAAAAAY!, Lots of fun.......but nowhere as much fun as trig, because it's awesome.


The first slide is just showing you the equation of a hyperbola. A great way to look at it is that either the x coordinate or y coordinate is negative. Also, the coordinate that is positive is the way that the hyperbola opens. So, if you have +x and -y, the hyperbola is going to open left-right because your x coordinate is positive. Furthermore, your 'a' value always goes with your positive value (x or y); However, unlike circles, your 'a' value (transverse axis=2a) is NOT always bigger than your 'b' value (conjugate axis=2b).

Moving on, the next slide asks you to sketch the hyperbola.
The first thing that I do is find the centre of the hyperbola. This is the same as finding the centre for a line. It's just the opposite sign of what's inside the brackets. In this case it is (-2,1).
Next, I find the values for 'a' and 'b'. For this, you just have to square root the denominator to get your values. Now, while keeping in mind which way the hyperbola opens, you can trace your transverse and conjugate axes. I recommend drawing a box for your axes, it really helps. From there you can find your vertecies which are the endpoints for transverse axis. The vertecies are your starting points for your hyperbola.
Now, we have to find the equation of the asymptotes to make our sketch complete and where to draw our hyperbola. This is just finding the equation of a line: y=mx+b, going back to grade 10 if you remembered which I didn't. So, you know your x and y values, they're just the values of the centre (x=-2,y=1). You know the m (slope), it's (by the way that I look at it) the amount that you move from the centre to the corner of the box. In this case, you're moving two up and three over making the slope 2/3 and -2/3. The part that you don't know is b (different from b at the start), but since you know everything else, it is just a matter of putting everything in and solving. After you find your b values, you can draw your asymptotes equations (with dashed lines) and then draw your hyperbola, remembering that your hyperbola follows very close to your asymptotes.

The next slide is just another example.

Last, you are given the graph and it is just a matter of finding the equation.
What I do first is find out whether your x or y is negative by looking at which way the hyperbola opens.
Next, I find the centre of the graph; In this case it is (-4,1),making it(x+4)-(y-1).
Now, you have to find the value of 'a' and 'b'. Without aid of the box or it telling you what your endpoints of the conjugate axes are, you can't find your 'b' value. Since you know the vertecies of the graph, you can determine the value of 'a', but you have to put it in terms of a^2 as opposed to 2a. In this case, you're given a box so you can find the 'b' vale.
Last, you have to find the equation of the asymptotes which is the same as before since you're given the box.

Hoped this helped you out if you were having troubles with this.

Conics- major / minor axis ( from formulas) - Transverse / conjugate axis

For ellipses, you have two axis, a major and a minor . the major axis is always the longer of the two ( see video, starts on ellipses).





For Hyperbolas, you also have two axis, a transverse axis and a conjugate axis. ( see video, after ellipses)





Transverse axis


-connects verticies


-not always longer than conjugate axis


- length is alwas 2a units (a+a)


- there for length of conjugate is always 2b units (b+b)

More Conics

Today in class we talked more about conics. Last class we found out how to change from general form to standard form. This class it was the complete opposite, we learned how to go from standard form back to general form. The reason we convert from general form to standard form is to be able to graph. I didn't quite get the concept at the start of the conics lesson but I did some accelerated math and I am starting to understand it. Also the test has been changed to next Thursday.

More conics

Assignments: exercises 37, 38, 39. This was assigned previously, but after today's lesson, we can do more.

Mr Max expanded our knowledge on conics. The central idea of today's lesson is that we can change general form conics into standard form conics.


Conversion between general form conic and standard conics.

Ax^2 + Bxy + Cx^2 + Dx + Ey + F = 0


Remember how to complete the square?

Take: x^2 + 4x + 4

We can take a square and
(x+2)(x+2)
(x+2)^2

or:
How do we find x? Well, take b/2, and square the result. sqroot(3/2)







If we try to enter (X^2 + y^2 + 6x - 8y) = 11 into graphimatica directly, it fails to map the equation because it cannot parse the data. So, we use this form after we complete the square

(X^2 + y^2 + 6x - 8y) = 11
(x^2+6x+9)(y^2-8y+16) = 11 + 9 + 16
(x+3)^2 + (y - 4)^2 = 36

Using the general form:

(x-h)^2+(y-k)^2= r^2
(h,k) = center; r = radius
we enter (x+3)^2+(y-4)^2= 36
and get:








c)4x^2 - y^2 - 8x - 4y - 16 = 0
4x^2-8x -y-4y = 16 + 4 -4
4(x^2-2x+1) -1(y^2+4y+4) = 16 + 4 - 4
(4(x-1)^2 / 16) - ((y+2)^2 / 16) = (16 /16)
(x-1)^2/(2^2) - (y+2)^2/(4^2) = 1
(x-h)^2 / a^2 - (y+2)^2/b^2 = 1








The final is only ~27 days away. I'm going to start studying today! Yes, that sounds crazy, but the exam is difficult, and this class can be interpreted as being preparation for the exam.

Work Time

Next Test: Thursday, December 10

Although today is Mr. Maks' birthday (Happy Birthday!), he ended up giving us the gift of having the class to do work. Personally, I really enjoyed this to have time to get some work done and give myself some time to process this Conics stuff and allow it to stick.
I'd suggest going through the practice test this weekend at some point so you'll know what to expect and feel more confident when the test comes around. Also, probably check out those Exercises Courtney posted just below mine.
Along with our assignments, I find one of the links we have on our blog really helpful, The Online Version of This Course has a lot of really great examples and practice questions.
Link straight to the Conics section:
http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20-%20Conics.html
Everyone just keep on working hard, you're doing great! Have an awesome weekend!

"When we do the best that we can, we never know what miracle is wrought in our life..."
-Hellen Keller

Conic Sections and Other Cool Mathematicalness

That was the title of our class today.
Today was our introduction to conics and we learned there are 4 sections to conics.

1. circle
2. ellipse
3. hyperbola
4. parabola

This is a conic: x^2 +/- y^2
General form of a conic is: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
*A, B, C, D, E and F are elements of the set of reals. B cannot equal 0
Conics are always squared

A circle happens when the A value and the C value are the same.

(x^2 + y^2) = 1 is a small circle
(x^2 + y^2) = 7 is a bigger circle...etc

Ex. Given the general form of the conic equation,

a) Identify the conic
b) State A, C, D, E, F

given... x^2 + y^2 - 8 = 0
a) So this is a circle because both co-efficients are 1.
b) A = 1, C = 1, D = 0, E = 0, F = -8

given... 2x^2 + 2y^2 + 4x -2y -32 = 0
a) This is a circle, because A and C values are the same
b) A = 2, C = 2, D = 4, E = -2, F = -32

An ellipse happens when A and C values are the same sign, but A cannot equal C. An ellipse is a squashed circle, an ellipse is also a subset of a circle.

Ex. Given the general form of the conic equation,
a) Identify the conic.
b) State A, C, D, E, F

given...(1) x^2 + 49y^2 - 49 = 0
a) Ellipse because both A and C are positive, but the values are different.
b) A = 1, C = 49, D = 0, E = 0, F = -49

given... 4x^2 + 9y^2 - 3x + 2y +0
a) Ellipse because both A and C are positive, but the values are different.
b) A = 4, C = 9, D = -3, E = 2, F = 0

A hyperbola happens if A and C have opposite signs.
*If the x is positive, than the hyperbola will open horizontally, but if the x is negative the hyperbola will open vertically.

Ex. Given the general form of the conic equation,
a) Identify the conic
b) State A, C, D, E, F

given... 9x^2 - 4y^2 - 36 = 0
a) Hyperbola because the A value is positive, and the C value is negative.
b) A = 9, C = -4, D = 0, E = 0, F = -36

given... -3x^2 + 3y^2 +2x - 12y + 2 = 0
a) Hyperbola because A is negative and C is positive.
b) A = -3, C = 3, D = 2, E = -12, F = 2

If one of A or C is the value of 0, than you have a parabola.

Ex. Given the general form of the conic equation,
a) Identify the conic
b) State A, C, D, E, F

given... y^2 - 4x = 0
a) Parabola because A value is 0
b) A = 0, C = 1, D = -4, E = 0, F = 0

given... 3x^2 - 2x + 5y - 3 = 0
a) Parabola because A ihas a value bigger than 0, and C's value is 0.
b) A = 3, C = 0, D = -2, E = 5, F = -3

The exercises we were assigned are:
Exercise 36 #1-7
Exercise 37 #1-11
Exercise 38 #1-9

Dont panic because there is no way you wil be able to do all these questions with just the information we learned on conics yesterday. You need more information.

Communicating using technology, Japanese Vending Machines....and some more Binomial Theorem

We started the class with a discussion about how beneficial (or not) a ban on cell phones would be in schools, and how some people have lost the connection with people around them, by only talking to there "friends" on their phone or computer. Just something to think about...Back to math class we did mental math today. We did more examples of using the Binomial Theorem and had a matherific time! Remember that smoking and meatbelts are stupid.

Assignment is Exercise #35 Questions 3,4,5,6,7

Binomial Theorem

Binomial theorem is defined as (a+b)^N where N equals a natural number.

See these terms??? (look up), well, look at the cool patterns within them (look down)

Here is an example showing how to use the binomial theorem:

Here is an example showing how to find a certain term within an expansion:

Here is another example showing how to put the binomial theorem to use:

This nifty picture down here is Pascal's Triangle. What you'll notice is that if you add any two numbers beside eachother, their sum is the number found directly below it. Example: 6+15=21

Homework:

Exercise 32, Questions 1-10

Exercise 33, Questions 1-10

Exercise 34, Questions 1-10

Below is a calendar of our next few weeks of school before the end of the semester.

-November 29th to December 1st we will be learning BINOMIAL THEOREM
-December 2nd to December 8th we will be learning CONICS
- December 8th and 9th we will be given time to review
-December 10th will most likely be our test
-December 11th to December 17th we will learning PROBABILITY
-December 18th we are off for Christmas Holidays!

-January 4th to January 11th we will be learning SEQUENCES

-January 12th to the 19th we will be doing review, but remember PILOT EXAM IS JANUARY 14th, and our FINAL EXAM IS JANUARY 20th

KRISTIN HANSFORDS TEST QUESTION

Question: Simplify: (log210)(log48-log3)

Solution: (log10/log2)(log16)
=(1/log2)(log2^4)
=4log2/log2
=4

Test Question

1st question (2 marks)

Convert expression into a single expression logarithm:
½ln(x) - 2 ln(x-1) - 1/3 ln (x²+1)
=ln √(x) - ln(x-1)² - ln ³√(x²+1)
= ln (√x)/ [ (x-1)² * ³√(x²+1) ] <--- Log Law #2.
surprisingly, thats all you have to do for that question (simplify)

2nd question (1 mark)

Express log 6 1/216= -3 in exponential form
6^-3= 1/216

3rd question (1 mark)

Evaluate ln ( 1/ 4√e)
*multiple choice here guys..
a) ¼
b) -¼
c) -4
d) 4

Thursday Test Question

This is my test question:
Solve for x: ln6+ln(x−2)=−2
(Express your answer correct to 3 decimal places.)
This question is worth 3 marks as well
Here is the method I used to get the answer:

ln[6(x-2)]=-2
ln(6x-12)=-2 (1 mark here)
(e^-2)=6x-12
6x=(e^-2)+12
x=(e^-2)+12/6 (1 mark here)
x=2.022 556
x=2.023 (1 mark here)

(that's where the marking guide said the marks should be awarded)

I would suggest to everyone that you go over all the test questions mentioned on here. This way you know what you will have to answer and how to do it. Good Luck to everyone!

My Text Question

http://www.edu.gov.mb.ca/k12/assess/archives/pc_mg_june_06.pdf

page 58.

(3 marks)

Transformations Questions

Questions: #13 - 2 marks (1 per letter)
#14 - 3 marks (1 per letter)
http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20Standards%20Test%20-%20Transformations%20QUESTIONS.pdf
Answers Link:
http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20Standards%20Test%20-%20Transformations%20ANSWERS.pdf

Test Questions

Here are my questions for our test tomorrow:


1) Point P is a point on the terminal arm of an angle Θ in standard position. If cscΘ<0 and sin theta=-15/17, deterine the value of the other five trigometric functions.

2)P(Θ) is a point on the unit circle. If secΘ<0 and tan(theta)<0, and the coordinates for P are (x, 2root6/7)

a. Find the value of x
b.State the coordinates of P(Θ+pi)

Solutions:

1)step 1: Determine quadrat-(quadrat III)
step2: Find sinΘ=1-(-8/17)squared=-15/17
Step3: Determine other 4 trig. values
sec=-17/8
csc=-17/15
tan=15/8
cot=8/15

2) a.
step1: determine quadrat-(quadrat 2)
step 2:1-(2√6/7)²=-5/7
b.step1: determine quadrat-(quadrat 4)
step 2: determine integer values for x and y: (5/7, -2root6)

My question.

These questions are pulled from the Mickelson book.The solutions can be found on pages 198 and 199.

Page: 179

Number:16)

Letter: c)

Evaluate(worth one mark):

10! 4! / 8! 6!

= (1*2*3*4*5*6*7*8*9*10)(1*2*3*4) /

(1*2*3*4*5*6*7*8) (1*2*3*4*5*6)

= 9 * 10 /

5 * 6

90 / 30

= 3

Second question (page 180, number 23) (worth two marks):

How many different groups of four letters can be made from the letters A, B, C, D, E, and F if the letters can only be used once?

P(6, 4) = 6! / (6 - 4)! = 6! / 2! =

1 * 2 * 3 * 4 * 5 * 6 /

1 * 2

= 3 * 4* 5 * 6

= 360

We can create 360 groups of four letters from A, B, C, D, E, and F if the letters can only be used once.

TEST QUESTION!

HERE WE GO GUYS...QUESTION ON THE FUNDAMENTAL COUNTING PRINCIPLE (FCP)

This is out of the Mickelson (hmm i always thought it was nickleson) book. pg 176.

There are 4 roads between cities A and B, and 3 roads between cities B and C.
a) How many ways can a person travel from A to C through city B??
(4)(3) = 12
b) How many ways can a person make a round trip from A to C and back to A, through city B?
(4)(3)(3)(4) = 144
c) How many ways can a person make a round trip from A to C and back to A through city B, without using any road twice?
(4)(3)(2)(3) = 72

Alright, hope that helped and you all do great

Brett's Test Question

Question:

1. a) How many 6 letter arrangements can be made using the letters M A T T E R ? Give your answer as a whole number. (1 mark)

b) How many 6 letter arrangements end with T T? Give your answer as a whole number. (1 mark)

c) How many 6 letter arrangements can be made if the letters M T T R must be together? Give your answer as a whole number. (2 marks)

Solutions:

1. a) 6!/2!= 360


b) _ _ _ _ T T
4!= 24


c) (3!) (4!/2!) = 72

**3 groups, 4 consonants, 2 t's


GOOD LUCK THURSDAY!

My test question!

I just got this from accelerated math, so i'll write it out for you as opposed to posting a link!
(Worth 2 marks)

Write as a logarithm of a single expression:
2log5x + 3log5(x+6)

Answer:
= log5x^2 + log5(x+6)^3
= log5x^2(x+6)^3

Taaa Daaaa!

Test Question for Nov 26 2009

http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20Standards%20Test%20-%20Transformations%20QUESTIONS.pdf

Question #2
a) Worth 2 marks
b) Worth 2 marks
c) Worth 2 marks
Question #11 Part A, B, C and D


Answers can be found here...

http://www.bmlc.ca/Math40S/Pre-Calculus%20Math%2040s%20Standards%20Test%20-%20Transformations%20ANSWERS.pdf

my test questions

HEYYYYYYYY YOUUUUUUU GUYYYYYYYY'S
If you would like the advanced edge on tomorrows upcoming test, here it is.
I Picked two questions, one from exercise 3 and one from exercise 5.
from exercise 3, I picked question 1, part e), which was
find the exact value of tan(7pi/4), show work, ie. unit circle

from exercise 5, question 1
Solve for thaeta where the domain is the set of all reals.
sin thaeta=1/2
answer is pi/6+2kpi where k is an integer
and
5 pi/6 + 2kpi where k is an integer

dont forget to show how you got those answers. ie unit circle

In class today Mr. Max gave us more examples on Perms and coms. The first example that we were given showed us how to create possible cases for one question, but after creating the possible cases you combine them together to get the final answer.
Next, we were given the term and definition: Sample Space- some display, rendition, showing all probability outcomes. "has everything" Scrabble was used as the example to explain sample space, which is shown below.
Test has now been moved to Thursday everyone, and there is tutoring today after school with Mr.Max. The Test questions should be posted on to the blog by wither today or tomorrow, where we will be able to see the problems and solutions. Good luck everyone!

Tuesday November 24, 2009- More Perms and Coms!

First of all, TEST MOVEMENT! Instead of Wednesday, we're moving this to Thursday.
What happens if the orderings of perms/coms are circular? i.e) people around a table; bracelets; necklaces; basically things that require objects in sequence on a round form of something.

ex) How many ways can you sit 4 people around a table?
*moving to the cafeteria to do some shananigans around the tables.
-How many things are actually moving?
ANSWER- 3 things are actually moving, which makes the equation 3!
Final answer- 3!= 6 ways
FORMULA: (n-1)!, where n= number of things to order in a circular way.


ex) Regarding this necklace, how many ways can we make this thing unique?
Thinking the same as the previous example? If so, that is incorrect. The 3D object can be flipped around, as shown in the picture. FORMULA: (n-1)!/2















ex) 5 people around a circular table
a) if person A and person B have to sit together?
{(n-1)!} *2 <--- because they can switch around (A-B, B-A) b) if person A and person B can not sit together? FORMULA: (All ways)-(undesired ways)= Answer.
4! - 3!2! = 12 ways

c) person A and person B together; person C and person D together.
2! (from the n-1 ! formula); 2! ways to organize A/B; 2! ways to organize C/D
2! * 2! *2!= 8 ways.

Ordering identical things-
ex) "How many different 'words' are possible with these letters?"
Z O O M
*thinking the answer is 4! ? 24 ways? Wrong, again!
MOZO* is the same as MO*ZO..
FORMULA: total number of ways / "total number of identicals"! = desired ways
ex) S T A T I S T I C S
10 !/ {(3!)(3!)(2!) = 50400 UNIQUE words

Friday Nov 20. Work Period.

Friday we had a work period to catch up on Accelerated Math, Cumulative Exercises and to work on our 2 questions of our choice picked from Old exams, Cummulitave Excercises, Accel Math, ect to be submit for the test on Wednesday.

It would be wise to be into the Probility on Acell Math and have the following Cummulative Exercises completed prior to Wednesdays test...

**Up to and including**

Ex 1 #1-12
Ex 2 # 1-12
Ex 3 # 1-7
Ex 4 #1-8
Ex 5 #1-12
Ex 6 #1-16
Ex 7 # 1-10
Ex 8 #1-10
Ex 9 # 1-14
Ex 10 # 1-6
Ex 11 # 1-12
Ex 12 # 1-10
Ex 13 #1-10
Ex 14 #1-11
Ex 15 # 1-14
Ex 16 # 1, 2, 4, 6-13
Ex 17 # 1-12
Ex 18 # 1-18
Ex 19 # 1-10
Ex 20 # 1-13
Ex 21 # 1-12
Ex 22 #1-10
Ex 23 # 1-12
Ex 24 # 1-10
Ex 25 # 1-15
Ex 26 # 1-16
Ex 27 # 1-10
Ex 28 # 1-10
Ex 29 # 1-10

Deeper Thinking...

Yesterday in class Mr. Max had us do a different style of assignment. We were to read an article online (link is below in his post) and write our own blog post based on our thoughts on the topic. It was about an agreement made between 2 students and a teacher about having a No Homework agreement where the 2 students would get, well, no homework. Ever.

So, your homework is to write a blog post on what you think about having no homework. Ironic?

We spent the rest of the block working on accelerated math. Try to get that done guys, I know I myself am super far behind and its just taking me ages to get caught up, but once I complete an objective I really do feel like I know what I'm doing for the most part.

REMINDER: Test is next Wednesday, so lets get our study on guys!

Pilot Exam

I forgot to post this when I scribed for the class on Monday. For anyone who missed Monday's class, we were told that there is going to be a pilot exam (practice exam) for our Grade 12 Pre-Calculus exam after holiday break. This takes place at 12:30 on January 14th. I think this is a great opportunity to get a feel for what the real exam is like as well as to find find out where we need to improve before the actual exam. It is going to take place in Mr. Maksymchuk's room.

Perms and Coms

Yesterdays class, we learned more about perms and coms. Permulations are the number of possible orderings of a set, provided that the order of set elements matters. An example would be 281-0563 218-0563 560-3812 or books on a shelf, locker combos. The BEST EXAMPLE EVER of a perm, is create 5 letter words from the alphabet no repeats. So you pick any 5 letters, and then you would times 26(because there is 26 letters in the alphabet) then 25 because you used a letter, then 24 b/c you used another letter...etc. you get 7893 600. At that point we learned the formula : 26!/21!=n!/(n-r)! n being the number of items to select from and r being selecting items r at 2 time. "n things, permulated 5 at a time". The second part of THE BEST EXAMPLE EVER is: pizzas. you have peperoni pinapple and ham, either order you put it in, its still the same pizza.

To Homework or Not? That is the question?

http://www.cbc.ca/canada/calgary/story/2009/11/18/calgary-homework-school-students.html

So what do you think? As young adults enrolled in the final year of public education, specifically in mathematics, which side of this 'argument' do you line up on?

Discussion prompts?

Would you like a no-homework agreement?
Would you benefit from a no-homework agreement?
Would you learn enough with no homework?
Do you have enough time in school to both learn and practice?
Is the subject area a consideration when discussing such agreements?

I really don't know the answers, but inspirationally, I know you do. I look forward to reading your responses. Thank you in advance for informing both public education practice generally and specific decisions made in room 204 at the SVRSS.

Logarithmic Word Problems

Today in class we did logarithmic word problems. For the most part, it is just review from Grade 11 Pre-Calc. The first two images are a little small because of the formatting.

The first thing we did dealt with was radioactive decay and half lives. The first question had to do with finding the amount of substance left after a determined amount of time; In our case, it was 3 years. So, all you have to do is substitute your X value for 3 and from there it becomes calculator work.

The next question involves a little more work and use of the logarithmic laws. We were asked to find the half life of the substance. What is asked here is what amount of time does it take for the substance to decay to half its original mass. So, we're solving for X. We know that the original mass was 80g so the mass at its half life would be 40g. Our Y value is now 40. Next, you multiply everything by ln. This is usually beneficial because out original question has the variable e in it so there is a good chance that we can get rid of it by using ln. Remember, the ln(80e^-0.2x) is actually ln80+ln(e)^-0.2x since it is a law. Next, we bring down the x value (another law) so we can factor it and get x by itself. From here, it becomes calculator work.

Out next section had to do with compound interest. Our first question had to do with finding the value of the investment after 1 year. All you have to do is substitute 1 in for the t value. From there it becomes calculator work.

The next question had to do with finding the value of the investment after 10 years. Again, just substitute 10 in for the t value and take it from there.

The last question had to do with finding the interest after 10 years. All you have to do for this is find the value of the investment after 10 years and subtract it from the original investment ($5000)

This formula shows what the interest would be if it was compounded all the time as opposed to just monthly or annually. How this formula was derived is that when the interest reaches a very high number, it gets very close to the value e. So the formula (1+r/n)^n simply becomes e^r. The reason why banks don't use this way of compounding interest is because it means more money for the consumer. The use of this formula is strait forward; You just substitute in your values and solve with your calculator.

The next section had to do with population increase. However, this formula can be used for anything that has to do with exponential growth. In this case, it ts used to show growth rates of bacteria. Also for this one, you just substitute in your values and solve.

Again, we are asked to find the half life of a substance. This question is just like the last one other than it is a different formula. You use natural logarithms to get rid of your e value, use the logarithmic laws, get x by itself, and solve with your calculator.

We were assigned 1-16 on exercise 26 and 1-10 on exercise 27.

I think we're assigned up to objective 31 on our accelerated math to be done before our test next Wednesday.

Thursday, November 12, 2009

On Thursday we had a substitute teacher, so we didn't learn anything new. We worked on assigned work such as accelerated math and our student exercises. I was really having trouble with this unit until I found a website that explains the concept of logarithms and it gives you several example questions. Also if you are having trouble with things from the last units you can find all the previous topics on the website and they are explained fully. The website is http://www.brightstorm.com/. When you get to the site you just click on the math link then it will give you a list of topics, select pre-calculus, then a list of all the units we took or will be taking this year pops up. Then select a video.

"Success is a peace of mind which is a direct result of self-satisfaction in knowing you did your best to become the best you are capable of becoming."- John Wooden

natural logarithms(-notations-equations-and applications)

REMEMBER! loge = ln





OK, for this first slide he used 3 logarithmic identities to form the single logarithm
1. ln(x-1) + 3ln(x+3) becoming the numerator(x-1)(x+3)^3 from
the identity log e (MN)= log e M + log e N

2.from (ln(x-1)+3ln(x+3))-1/2ln(x^2+2) he got the denominator Square Root of (x^2+2) which is the same as(x^2+2)^1/2 from the identity log e (M/N) = log e M - log e N

3. and the third identity is log e (M^N) = N log e M which you can see done with both the numerator and denominator




the only thing i think i need to explain with this one is that if both sides of an equation have loge
in front we know we can take it away, correct? so i guess the opposite can be done as well where we add in loge, which is seen done in the second line (the red ink).


(remember! loge = ln)!



*lne=1


this slide uses the third identity mentioned in the first slide, and takes away ln from both sides(as mentioned in slide two), other than that i think it is self explanatory.




in this slide he adds ln to both slides making lne on the left side which previously stated equals one(see lime green * in above slides explination).

and again he used the third identity mentioned in slide 1.

Hope this helped you in anyway, shape, or for possible


Mr. Maksymchuk gave use another one of his shirt ideas this one was based on the Pink Floyd album cover the dark side of the moon with the rainbow and prisms, but his shirt would say Pink Freud. I didn't get the reference but maybe you did.


I later googled it and it was referanced to Sigmund Freud who was a phsycoanalysis who was best know for his theories of the unconcious mind and the defence mechanism of repression.


after all that i still didn't completely get it, oh well, my loss I guess

Approximating "e" and The Natural Logarithms (ln)

Homework: Exercise 25 #1-15
Also, remember to vote for which day you think our next test should be!

Mr. Maks told us that the general idea of the lesson was that: loge=ln, in the case that e~2.71828...

We used the formulas: f(n) = 1+1/n , g(n)= (1+1/n)^n (for values n E I)
and Mr. Maks showed us, using Excel, the comparison between the two formulas using numbers between 1 and 1000.

We were also taught about Natural Exponential Functions and what they look like.
The graph of f(x) = e^x has these properties: (Important to know and understand)

• D= R (reals)
• R= (y>0)
• increases on its entire interval (moves upwards to the right)
• concave up
• one to one (** means it has an inverse**) ; if e^x1 = e^x2 (therefore, x1 = x2)
• 0<> 1 for x> 0

This relates back to the important equations Mr. Maks gave us yesterday of:
e^1nx=x and f(f^-1(x))=x (try them out on your calculator, they actually work!)

We also sketched some Exponential Graphs on Graphmatica
a)y= e^x-2
Compared to y=e^x, this graph is shifted 2 units down.

b)y=-e^-x
Compared to y=e^x, this graph is flipped over the y-axis, then the x- axis.

c)y= abs(e^x-1)
Compared to y=e^x, this graph is moved 1 unit down, then taking the absolute value of it causes the negative part of the graph to be bounced up and over onto the positive side of the x-axis.

If you compare graphs a, b, and c to the original equation (y=e^x) on Graphmatica, being able to see the actual changes really help. Also, if you know the approximate look of the original equation, you can just apply what we learned in our transformation unit and visualize where to move the graph by how much and so on. (It's pretty neat how all math builds on each other for the most part!)

Also, like Mr. Maks told us there's really great explinations and examples of logs in the Mickelson book starting on page 37.