Due to the activities taking place today, attendance was optional. Therefore, we had a class to work on Accelerated Math and ask Mr. Maksymchuk questions. I found it to be a pretty productive class and I hope everyone else did too. Being the kind teacher that he is, Mr. Maksymchuk hasn't assigned us any homework over Spring Break. Recall though, that he assigned all of Exercise 23 yesterday and it would probably be a good thing to do some of the review questions of earlier exercises.
Here's a link to a slide show of some really cool math tricks that will make everyone think you're a genius! (As if they didn't already know that.)
http://www.slideshare.net/simonesu/amazing-math-trick
Good luck to Cramer's team in Team Stuff Day!
Have a great Spring Break everyone!
Bethany
Thursday, March 27, 2008
Tuesday, March 25, 2008
Logarithm Laws
Greeting peers today in Mr. Max's dictatorship, I mean class, he started by trying to recruit more students for the national math competition on April 15.
After this largely unsuccessful attempt we dived head long into logarithm laws which are largely based on rules for exponents that we already should know.
Exercise 22 1-14 was assigned and don't forget about the 19 accelerated math objectives due on thursday
http://www.swop.net/
After this largely unsuccessful attempt we dived head long into logarithm laws which are largely based on rules for exponents that we already should know.
Exercise 22 1-14 was assigned and don't forget about the 19 accelerated math objectives due on thursday
http://www.swop.net/
adding to Amy's post
So as you all know, today Mr. Max taught us about logs!
He began with explaining to us what a log was, and it is really just another type of notation that can be used to express things such as really big numbers and it is the inverse of an exponential function.
He also explained what the parts of a logarithmic function were.
In the function:
x = b^y --> x is the argument, b is the base, and y is the argument.
In the second screen shot, he then went on to show us how to write exponential funtions in logarithmic notation and vice versa.
In the third screen shot he showed us a logarithmic function that does not exist. This is because if you try to sub in a +2 for the x value you wont arrive at a -4 and if you sub in a -2 for x, you will end up with a +1/2
On the fouth and fifth slides he graphed examples of logarithmic functions, and on the following slide he gave us a list of 8 common log properties.
Exercise 20 #'s 1-12 and Exercise 21 #'s 1-8 are due for tomorrow!
Hope these notes were helpful and have a great day!
Anna
He began with explaining to us what a log was, and it is really just another type of notation that can be used to express things such as really big numbers and it is the inverse of an exponential function.
He also explained what the parts of a logarithmic function were.
In the function:
x = b^y --> x is the argument, b is the base, and y is the argument.
In the second screen shot, he then went on to show us how to write exponential funtions in logarithmic notation and vice versa.
In the third screen shot he showed us a logarithmic function that does not exist. This is because if you try to sub in a +2 for the x value you wont arrive at a -4 and if you sub in a -2 for x, you will end up with a +1/2
On the fouth and fifth slides he graphed examples of logarithmic functions, and on the following slide he gave us a list of 8 common log properties.
Exercise 20 #'s 1-12 and Exercise 21 #'s 1-8 are due for tomorrow!
Hope these notes were helpful and have a great day!
Anna
Monday, March 24, 2008
LOGs
Hello everyone. Today Mr. Max taught us about LOG's. We defined them and wrote some notes about the properties of LOG's. Here are the screenshots from today:
Well, since there will be more people than just me putting stuff up today, I'll keep it short and sweet. You can hear what they have to say. Later!
Seems like it won't be too terribly hard. He also reminded us that doing the cumulative excercises really does help.
He also gave us a game plan for the week:
Monday(today)-define LOG's
Tuesday-Law's of LOG's
Wednesday-Solving LOG's with theorems
Thursday-Natural LOG's("i")
Some notes about Thursday:
-Mr.Max is planning on teaching a lesson, so if you haven't signed up for "Team Stuff Day", DONT!
- 19 Objectives are due on Thursday at 3:40! Don't leave it till the last minute(like I tend to do...)
Well, since there will be more people than just me putting stuff up today, I'll keep it short and sweet. You can hear what they have to say. Later!
Amy
Thursday, March 20, 2008
Exponential Functions
Hey class,
So today Mr. Max taught us a few things on Exponential Functions, beginning with identifying some of the parts of a function and some useful terms that we should iNvEsTiGaTe:
*this picture above shows what an increasing or decreasing graph would look like
*He also explained to us what a discontinuous function would look like
and here he explained the one to one property, this is a property in which there can only be one corresponding domain value per output value
So today Mr. Max taught us a few things on Exponential Functions, beginning with identifying some of the parts of a function and some useful terms that we should iNvEsTiGaTe:
*this picture above shows what an increasing or decreasing graph would look like
*He also explained to us what a discontinuous function would look like
and here he explained the one to one property, this is a property in which there can only be one corresponding domain value per output value
So i was checking Edline last night and when i saw my math mark i had a mild heart attack. Turns out I forgot to post those examples and i am so sorry since maybe if i had it would have helped someone with the test we wrote.. my apologies. I'm hoping that i can still get a mark for this but even if i don't i thought i'd do it for the benefit of my classmates.
So here she is:
1. Find exact value of sin75.
sin(30+45) = (sin30*cos45)+(cos30*sin45)
= [(1/2)*((sqrt.2)/2))]+[((sqrt.3)/2)*((sqrt.2)/2)]
= ((sqrt.2)/4)+((sqrt.6)/4)
= ((sqrt.2)+(sqrt.6))/4
2. Simplify sin5xsin3x- cos5xcos3x.
= -(cos5xcos3x+sin5xsin3x)
= - cos(5x+3x)
= -cos8x
So here she is:
1. Find exact value of sin75.
sin(30+45) = (sin30*cos45)+(cos30*sin45)
= [(1/2)*((sqrt.2)/2))]+[((sqrt.3)/2)*((sqrt.2)/2)]
= ((sqrt.2)/4)+((sqrt.6)/4)
= ((sqrt.2)+(sqrt.6))/4
2. Simplify sin5xsin3x- cos5xcos3x.
= -(cos5xcos3x+sin5xsin3x)
= - cos(5x+3x)
= -cos8x
Better Late Than Never
Well, I finally figured out how to do these examples! I may not get marks for them, but I guess the main thing is that I know how to do them.
Example One: Determine the exact value of sin135
sin(90+45)=sin90 cos45 + cos90 sin45
= (pi/2)(sqrt.2/2) + (0)(sqrt.2/2)
=(sqrt.2)(pi)/4
Example Two: Simplify: sin6xsin4x - cos6xcos4x
= -(cos6xcos4x-sin6xsin4x)
= -cos(6x+ 4x)
= -cos10x
Amy
Example One: Determine the exact value of sin135
sin(90+45)=sin90 cos45 + cos90 sin45
= (pi/2)(sqrt.2/2) + (0)(sqrt.2/2)
=(sqrt.2)(pi)/4
Example Two: Simplify: sin6xsin4x - cos6xcos4x
= -(cos6xcos4x-sin6xsin4x)
= -cos(6x+ 4x)
= -cos10x
Amy
Wednesday, March 19, 2008
Wednesday, March19
Today was test day. I found it to be quite complicated.
rinkworks.com/graffiti/g/math.png
That is a true mental breakdown.(I couldn't get the picture to appear clearer)
www.saynotocrack.com/img/geometry/question3.gif
This one also made me laugh.
I do not know what to write about to benefit the class mathematically so here are some random funny things because humour is beneficial in its own special way.
rinkworks.com/graffiti/g/math.pngThat is a true mental breakdown.(I couldn't get the picture to appear clearer)
www.saynotocrack.com/img/geometry/question3.gif
This one also made me laugh.
And now some jokes.
Teacher: What is 2k + k? Student: 3000!
Q: What do you get if you divide the cirucmference of a jack-o-lantern by its diameter? A: Pumpkin Pi!
Q: Why do you rarely find mathematicians spending time at the beach? A: Because they have sine and cosine to get a tan and don't need the sun!
Q: Why do mathematicians, after a dinner at a Chinese restaurant, always insist on taking the leftovers home? A: Because they know the Chinese remainder theorem!
Teacher: "Who can tell me what 7 times 6 is?" Student: "It's 42!" Teacher: "Very good! - And who can tell me what 6 times 7 is?" Same student: "It's 24!"
A math student is pestered by a classmate who wants to copy his homework assignment. The student hesitates, not only because he thinks it's wrong, but also because he doesn't want to be sanctioned for aiding and abetting. His classmate calms him down: "Nobody will be able to trace my homework to you: I'll be changing the names of all the constants and variables: a to b, x to y, and so on." Not quite convinced, but eager to be left alone, the student hands his completed assignment to the classmate for copying. After the deadline, the student asks: "Did you really change the names of all the variables?" "Sure!" the classmate replies. "When you called a function f, I called it g; when you called a variable x, I renamed it to y; and when you were writing about the log of x+1, I called it the timber of x+1..."
Q: What is the most erotic number? A: 2110593! Q: Why? A: When 2 are 1 and don't pay at10tion, they'll know within 5 weeks whether or not, after 9 months, they'll be 3...
Trigonometry for farmers: swine and coswine...
Q: How does a mathematician induce good behavior in her children? A: `I've told you n times, I've told you n+1 times...'
And finally for gregs benefit, the rock lobster video.
I hope you enjoyed my random post.
Tuesday, March 18, 2008
identities sum and differences
Ok now being only a couple days late for this due to some technacal difficulties with logging on to blogger for some reason i can now conribute my wise learnings, I keep trying to tell myself this anyways, in the form of two examples.
So hear is exhibit A using the identity Cos(a+b):
= Cos(A)* Cos(B) -Sin(A)*Sin(B)
= Cos(5x)*Cos(2x)-Sin(5x)*Sin(2x)
= Cos(5x+2x)
= Cos(7x)
And now for exhibit B which is like A in reverse and more specific:
Cos(75 degrees)
= Cos(45 degrees+ 30 degrees)
= Cos(45)*Cos(30)-Sin(45)*Sin(30)
= (sqr(2)/2) * (sqr(3)/2) - (Sqr(2)/2) * 1/2
= (sqr(6)/4) - (sqr(2)/4)
= (sqr(6)-sqr(2))/4
Well hope this helps someone
So hear is exhibit A using the identity Cos(a+b):
= Cos(A)* Cos(B) -Sin(A)*Sin(B)
= Cos(5x)*Cos(2x)-Sin(5x)*Sin(2x)
= Cos(5x+2x)
= Cos(7x)
And now for exhibit B which is like A in reverse and more specific:
Cos(75 degrees)
= Cos(45 degrees+ 30 degrees)
= Cos(45)*Cos(30)-Sin(45)*Sin(30)
= (sqr(2)/2) * (sqr(3)/2) - (Sqr(2)/2) * 1/2
= (sqr(6)/4) - (sqr(2)/4)
= (sqr(6)-sqr(2))/4
Well hope this helps someone
Aloha, everyone. Today Mr. Max assigned Excercise #17, 1-12; and Excercise #18, 1-16. We get all of today's class to work on them and study for tomorrow's test.
He also told us that the test will be the same as the pre-test, but with different numbers. Since the whole pre-test idea seems to be liked by the whole class, we will continue to get pre-tests before each of our real bi-weekly tests.
One last thing-The latest due date for Accelerated Math is March 27th with 19 objectives.
Well good luck on the test tomorrow everyone!
Amy
from my.opera.com/offspring/blog/show.dml/359734
He also told us that the test will be the same as the pre-test, but with different numbers. Since the whole pre-test idea seems to be liked by the whole class, we will continue to get pre-tests before each of our real bi-weekly tests.
One last thing-The latest due date for Accelerated Math is March 27th with 19 objectives.
Well good luck on the test tomorrow everyone!
Amy
from my.opera.com/offspring/blog/show.dml/359734
Double Angle Identities
Hello class,
Sorry this post is a little late.. I never had time yesterday and today it took me awhile to upload images due to the sweet sun computers.
Anywho, today Mr. Maksymchuk continued on his journey to explain identities. Today we did some more work with double angle identities. As shown below, Mr. Max took all the equations given to us in our exam formula sheet and showed how they derived the formula. I found this part very helpful and makes learning identities easier to understand!
Another things that I found useful in today's lesson is to always keep in mind the key triangles that we learned at the start of the year. Also take not of a 30, 60, 90 triangle with lengths of 3, 4, and a hypotenuse of 5. Knowing these triangles will come in very handy.
In this picture below, they wanted to know the value of 2(theta). So Max simply input the values for theta into another formula from our exam formula sheet, and solved for cosine and sine. Always note the signs you apply to the final answer so you make sure you have it in the right quadrant.
Below is just another example of solving double angles using exact values. Remember to FACTOR out like terms!!!
Well this is the end of my post. Any questions or comments on how I can improve PLEASE let me know. Thank you for your time.
Remember!!! Always keep the stick on the ice.
Love Cramer
Sorry this post is a little late.. I never had time yesterday and today it took me awhile to upload images due to the sweet sun computers.
Anywho, today Mr. Maksymchuk continued on his journey to explain identities. Today we did some more work with double angle identities. As shown below, Mr. Max took all the equations given to us in our exam formula sheet and showed how they derived the formula. I found this part very helpful and makes learning identities easier to understand!
Another things that I found useful in today's lesson is to always keep in mind the key triangles that we learned at the start of the year. Also take not of a 30, 60, 90 triangle with lengths of 3, 4, and a hypotenuse of 5. Knowing these triangles will come in very handy.
In this picture below, they wanted to know the value of 2(theta). So Max simply input the values for theta into another formula from our exam formula sheet, and solved for cosine and sine. Always note the signs you apply to the final answer so you make sure you have it in the right quadrant.
Below is just another example of solving double angles using exact values. Remember to FACTOR out like terms!!!
Well this is the end of my post. Any questions or comments on how I can improve PLEASE let me know. Thank you for your time.
Remember!!! Always keep the stick on the ice.
Love Cramer
Friday, March 14, 2008
Pi Day!!!
HAPPY PI DAY EVERYONE!!!
picture via the Hope University math newsletter Off on a Tangent
In case you weren't aware, it is also Albert Einstein's birthday. Also, a proper Pi Day celbration should not begin until it is 1:59:26 on March 14 (π = 3.1415926... = 3rd month, 14th day on 1:59:26).
Today's lecture was short as Mr. Maksymchuk decided that we should use the period to work on the pre-test and catch up with accelerated math. Employing the sum and difference identities that were taught to us yesterday, Maksymchuk showed us how to derive sleek-and-sexy numerical values for trigonmetric functions with"strange" angles, among other things:
Here he derived an equivalent expression for sec(θ-π/4):
Continuing with the sleek-and-sexy values:
And finally we were given our assignment:
Mr. Maksymchuk finished this around 12:30 p.m., giving us about 40 minutes to work on our pre-test and finish up our accelerated math.
Anyway, since this post was so short I think I will provide my fellow classmates with a puzzle they may or may not choose to figure out:
You have 9 balls - all of which are indistinguishable by sight - and a balance scale. Eight of the balls weigh exactly the same while the last one is either heavier or lighter than the other eight. This differential in weight is so small that you can not tell by holding the "odd" ball in your hand whether or not it weighs any differently than the other eight; that is, you will have to employ the balance scale to figure this out. What is the smallest amount of weighings that you can do to find the odd-ball-out? How did you carry these weighings out?
Feel free to post your solutions.
-IB
Idenities Example
I have decided to use an example of one Double Angle Identity, and on example of a Sum and Difference Identity..
1) Find the exact value of the following:
a. sin (π/12) =sin (π/3-π/4) = sin(π/3) cos(π/4) - cos(π/3) sin(π/4)
= (√3/2)(√2/2) - (1/2)(√2/2)
= (√6/4) - (√2/4)
final answer = (√6 - √4)/4
2) Solve for "x" where 0° ≤ x ≤ 360°
a. sin2(x) = sin(x)
1) Find the exact value of the following:
a. sin (π/12) =sin (π/3-π/4) = sin(π/3) cos(π/4) - cos(π/3) sin(π/4)
= (√3/2)(√2/2) - (1/2)(√2/2)
= (√6/4) - (√2/4)
final answer = (√6 - √4)/4
2) Solve for "x" where 0° ≤ x ≤ 360°
a. sin2(x) = sin(x)
2sin(x) cos(x) = sin(x)
2sin(x) cos(x) - sin(x) = 0
sin(x) (2cos(x) - 1) = 0
sin(x) = 0, therefore x = 0, 180, 360
2cos(x) - 1 = 0
cos (x) = 1/2, therefore x = 60, 300
Using identities (sum and difference examples)
Here are my two examples, i wasnt sure of what i was doing, but
βεθany was nice enough to help me out!!
Example 1:
Simplify: sin6x sin4x - cos6x cos4x
(for this you will have to use the identity) :
cos(α+β)= cosαcosβ - sinαsinβ
sin6x sin4x - cos6x cos4x =
-(cos6xcos4x - sin6xsin4x) = -cos(6x+4x) =
-cos10x
Example 2:
Determine the exact value of sin 75°
To begin, use the identity: sin(α+β) = sinαcosβ
sin(30+45) =sin 30cos 45 + cos 30 sin 45
= 1/2(√2/2) + √3/2(√2/2)
=√2/4 + √6/4
=√6+√2 /4
βεθany was nice enough to help me out!!
Example 1:
Simplify: sin6x sin4x - cos6x cos4x
(for this you will have to use the identity) :
cos(α+β)= cosαcosβ - sinαsinβ
sin6x sin4x - cos6x cos4x =
-(cos6xcos4x - sin6xsin4x) = -cos(6x+4x) =
-cos10x
Example 2:
Determine the exact value of sin 75°
To begin, use the identity: sin(α+β) = sinαcosβ
sin(30+45) =sin 30cos 45 + cos 30 sin 45
= 1/2(√2/2) + √3/2(√2/2)
=√2/4 + √6/4
=√6+√2 /4
Trigonometric Sum and Difference Example Post
1). Find sin (60º-45º)
Employing sin(a-b) = sin(a)cos(b) - sin(b)cos(a)
sin(60º-45º) = sin(60º)cos(45º) - sin(45º)cos(60º)
=(√(3)/2)(√(2)/2) - (√(2)/2)(1/2)
=(√6 - √2)/4
2). Find cos(-x)
Employing cos(2π-x) = cos(-x) and cos(a-b) = cos(a)cos(b) - sin(a)sin(b) we have:
cos(-x) = cos(2π-x)
= cos(2π)cos(x) - sin(2π)sin(x)
= (1)cos(x) - (0)sin(x)
= cos(x)
Voilà!
Thursday, March 13, 2008
Sum and Difference Identity Examples
Well, I basically just copied what the examples were in the green Mickelson book, but changed the numbers, but just working through the examples and my own really helped me understand them better. Here's what I came up with. By the way, I'm sure there's a way to use greek letters, but I couldn't find it so I'm substituting a for alpha and b for beta.
Example 1
Simplify: sin4xcos3x - cos4xsin3x
By using the identity sin(a - b) = sinacosb - cosbsina,
sin4xcos3x - cos4xsin3x = sin(4x - 3x)
= sin(x)
Example 2
Determine the exact value of cos(315). (That would be 315 degrees, but I don't know how to get a degrees sign on here either.)
By using the identity cos(a - b) = cosacosb + sinasinb,
cos(360 - 45) = cos360cos45 + sin360sin45
= 1(SQRT(2)/2) + 0(SQRT(2)/2)
= SQRT(2)/2
OR
By using the identity cos(a + b) = cosacosb - sinasinb,
cos(270 + 45) = cos270cos45 - sin270sin45
= 0(SQRT(2)/2) - (-1)(SQRT(2)/2)
= -(-1SQRT(2)/2)
= SQRT(2)/2
Hopefully that makes some sort of sense. Don't forget, tommorow's pi day!!
Example 1
Simplify: sin4xcos3x - cos4xsin3x
By using the identity sin(a - b) = sinacosb - cosbsina,
sin4xcos3x - cos4xsin3x = sin(4x - 3x)
= sin(x)
Example 2
Determine the exact value of cos(315). (That would be 315 degrees, but I don't know how to get a degrees sign on here either.)
By using the identity cos(a - b) = cosacosb + sinasinb,
cos(360 - 45) = cos360cos45 + sin360sin45
= 1(SQRT(2)/2) + 0(SQRT(2)/2)
= SQRT(2)/2
OR
By using the identity cos(a + b) = cosacosb - sinasinb,
cos(270 + 45) = cos270cos45 - sin270sin45
= 0(SQRT(2)/2) - (-1)(SQRT(2)/2)
= -(-1SQRT(2)/2)
= SQRT(2)/2
Hopefully that makes some sort of sense. Don't forget, tommorow's pi day!!
Today Mr, Maksymchuk gave us a lesson on sum and difference identities, he gave a the algebraic and calcuator methods for solving. We also got our tests back at the begining or clss, hope everyone did well.
The following slides are from todays lesson.....
Tomorrow be ready to hand in all accelerated math, due tomorrow at 3:40!
Wednesday, March 12, 2008
Wednesday, March 12
Today in class we continued to learn about triginomic identities and how to solve them. To prove an identity to be true, basically you must try to make the left hand side of the equation equal to the right side. Eight basic techniques to try and solve or simplify an equation to prove its identity are shown below (a-h).
If the left hand side (LHS) equals the right hand side (RHS) then the identity is proved to be true. The correct terminology is shown in the slide below
In order to master the art of proving identities you must practice practice practice! To some, (like myself) it can be very difficult and frustrating which is why it is important to keep up with your exercises and practice.
It is important to know that to verify and to prove are similar but not the same. When you are verifying something you finding an identity for a specific case but it does not prove the identity for the whole equation.
Similarily to graphing, verifying cannot be used to prove an identity but you can use it to disprove.
Together we went through a few different problems for proving identities which are shown in the above slides.
Even though today was our deadline for accelerated math, Mr. Max was ever so kind to give us until friday to get done our first nine objectives. So don't forget to get those done by 3:40 friday and for tomorrow have questions 1-9 of exercise 15 to complete.
Thats all for now, so remember, keep your eye on the ball!
By the way happy birthday Cindy Lou!!! Hope you have a good one!
Picture courtesy of "http://www.freerangechick.com/images/bdayCard/BBigBirthdayCake.jpg"
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