AP Calculus AB (2006-07)

So Long ...

2007-06-29T09:38:00.000-05:00

And so we begin where we left off ... don't let the sky be your limit. ;-)

I'm so glad we've had this time together,

Just to have a laugh or learn some math,

Seems we've just got started and before you know it,

Comes the time we have to say, "So Long!"

So long everybody! Watch this space for pointers to new blogs for each of my classes.

Farewell, Auf Wiedersehen, Adieu, and all those good bye things. ;-)

My very best wishes!

2007-06-19T08:50:00.000-05:00

I am thinking that these days each of you, your family, and your friends are looking to your future. I'd like to do that too and wish for you all that Emerson describes--

Lani

Passing it forward!

2007-06-10T09:43:00.000-05:00

Some 44 years ago (when I was about your age) my father who used to travel a lot, penned a letter to me and enclosed a short article clipped from a magazine. In that letter, he wished for me a life of purpose and joy. The enclosed clipping described such a life and quoted George Bernard Shaw whose words you can read above.

I carried that article with me for years; unfortunately, somewhere in the many moves, it was lost. But not the thought and the power of those words; they have become a part of who I am. I know that my career as a teacher, and my mentoring are but "drops in the ocean" in this often violent, globalized world of ours but they are a significant part of my life which has been filled with joy, happiness and love. I'm sure that my father's writing and sharing those thoughts adds to their meaning for me, especially now that Alzheimer's prevents him from recalling what occurred. His belief in me, in mankind in general,and his wish that I might find a life of purpose and joy shaped my world.

I'm passing that forward --my belief in you, my belief in mankind, and my wish that you find real purpose and joy in life!

You're in The News

2007-06-08T16:26:00.000-05:00

I just stumbled across a couple of links where folks are talking about your work here on the blog and linking to it as an example of high quality blogging in mathematics ... thought you'd like to know ...

Website of the Week (05-28-07)

Wikis and blogs in senior secondary classrooms

I know I've mentioned this before but it's worth mentioning again ... I've very proud of you. All of you.

Cheers!
Mr. K.

You Should Know ...

2007-05-23T17:52:00.001-05:00

As we wind down towards the end of the year and you finish your Take Home Exam, complete your Developing Expert Voices projects and look back on all you've learned using flickr, wikis and your blog (including various entries into The Hall of Fame) you may recall my telling you the world is watching and talking about you. Well, they really are. Look at this blog post by Dr. Paul Martin, an English professor at the University of Vermont. (Scroll down to Best Practices.)

Any time you want to see whose been talking about and linking to you can do that by clicking here.

Photo Source: Security

Happy

2007-05-21T22:34:00.000-05:00

I am so happy to be done my DEV PROJECT and so proud of it too.. Anyway, hope you guys like it. http://liiinger.blogspot.com/

Today's Slides: May 16

2007-05-16T14:15:00.001-05:00

Here they are ...

To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Food For Thought

2007-05-13T23:08:00.000-05:00

Watch this. It's only about a minute long. You really need to see this ... really ...

Research Assistants

2007-05-13T22:11:00.000-05:00

How are Research Assistants Chosen?
Choosing to be a Research Assistant is entirely voluntary ... but if you do it you can get a bonus mark on the unit test for each unit you undertake to act as Research Assistant. A maximum of one bonus mark is available for each student that publishes as Research Assistant.

IMPORTANT: There can only be one Research Assistant for each scribe post.

What Does a Research Assistant Do?
The Research Assistant must search the internet to find a minimum of two links in support of one of our daily scribes.

(a) One link must be to a site that reviews the content we have been studying in class as described in the scribe post for which you are acting as Research Assistant.

(b) One link must be to a site that has one or more interactive quizzes (self correcting) that deals with the specific content covered by the scribe post for which you are acting as Research Assistant.

(c) The links you find cannot have been published by any previous Research Assistant.

(d) Tag your post with i) your name, ii) the unit ii) Research Assistant.

How Do I Know What Previous Research Assistants Have Done?
Below is a list of the units and Research Assistants to date. This list will be updated as each Research Assistant posts the results of their research. Each Research Assistant's name is linked to the post they made with the links they found. This post is your "one stop shop" to find out what has gone before and learn/review when you need it.

The Research Assistants List

Conics
MrSiwWy

Photo source: HoP Mad Science Fair: Jon and Natalie

2007-05-12T23:18:00.000-05:00

heres a funny video =) about calculus

Today's Slides: May 8

2007-05-08T16:00:00.000-05:00

Scribe Post

2007-05-07T23:01:00.000-05:00

Hi, this is Christian! In our last class, the lovely Mrs. Stawski came to help us with prep stuff for our upcoming exam. It's on Wednesday everyone!!!

After that, we had to do questions. As soon as we go over those questions, and we get the answers (we haven't done either), I'll do a follow up on this scribe post. For now, hang in there!

And one more thing!!

2007-05-07T19:02:00.000-05:00

On the day that I was to take my "big" test, I was just about to leave for the testing center when I asked my husband to wish me luck. "No," he said, "I won't do that." I was crestfallen. I felt like I needed one last boost before the "big" one.

Then he continued, "You don't need luck. You're smart. You're prepared. You're good. I believe in you. Go out there and just do it!!!! I'll be here when you get back."

It's time for me to pass that forward to you! You don't need luck. You're smart. You're good. You're well prepared (thanks to your hard work and Mr. K). I believe in you. Go out there and just do it!!!!

From ZZZZZ's to A's

2007-05-07T18:55:00.000-05:00

The bottom line: Teens need 9.25 hours of sleep per night

from Do Teens Get Enough Sleep?

In experiments done at Harvard Medical School and Trent University in Canada, students go through a battery of tests and then sleep various lengths of time to determine how sleep affects learning. What these tests show is that the brain consolidates and practices what is learned during the day after the students (or adults, for that matter) go to sleep. Parents always intuitively knew that sleep helped learning, but few knew that learning actually continues to take place while a person is asleep. That means sleep after a lesson is learned is as important as getting a good night's rest before a test or exam.
from Adolescents and Sleep

At the risk of sounding "mom-ish", have you taken this into consideration in your preparation for your upcoming test? I know you mentioned it in a scribe as being discussed in class!

Asking only because, when I was sleep deprived, I know I wasn't fully aware of how much more difficult problem solving and remembering was. I never fully realized how sleep deprivation changed my abilities and me until after I started getting adequate sleep.

Another factor in your preparation to be your very best for your test??

Visualize Success on Your Test!!

2007-05-03T09:55:00.000-05:00

SCribe post

2007-05-02T18:53:00.000-05:00

The Calculus exam is in seven days and seven nights.

Things to remember:
- Bring food
- Sleep early the night before
- Bring water
- Stretch when it is break during the exam

The wiki assignment is due at midnight tonight. Remember to log in with your first name on the wiki. Constructive assignment starts tomorrow.

AP MINI EXAM ANSWERS
1) A

2) B

3) D

4) D

Free Response question
A) Using a mdpt sum we get 43.1 multiply it by six which is equal to 258.6

B) Yes there is a a value where r'(t) = 0 because if you look at the table, there is a maximum value between 9 and 15 hours.

The next scribe is christian

Scribe Post

2007-05-01T23:02:00.000-05:00

Here are the answers for the exam review quiz. Just click on the images to make them bigger.

Developing Expert Voices Rubric v1.1

2007-05-01T00:30:00.000-05:00

Here is the updated version 1.1 of our rubric ...

Teaching mathematical concepts is the main focus of this project; so we can teach other people and learn at the same time.

Achievement Descriptors
Instead of levels 1-4 (lowest to highest) we use these descriptors. They better describe what this project is all about.

Novice: A person who is new to the circumstances, work, etc., in which he or she is placed; a beginner.
Apprentice: One who works for an expert for instruction or to learn a skill or trade.
Journeyperson: Any experienced, competent but routine worker or performer.
Expert: One who possesses special skill or knowledge; trained by practice; skillful and skilled.

Achievement	Mathematical Challenge (25%)	Annotation (40%)	Solutions (15%)	Presentation (20%)
Novice	Problems illustrate only an introductory knowledge of the subject. They may be unsolvable or the solutions to the problems are obvious and/or easy to find. They do not demonstrate mastery of the subject matter.	Explanation does not "flow," may not be in sequential order and does not adequately explain the problem(s). May also have improper mathematical notation.	One or more solutions contain several errors with insufficient detail to understand what's going on.	Presentation may or may not include visual or other digital enhancements. Overall, a rather uninspired presentation. Doesn't really stand out. It is clear that the student has invested little effort into planning their presentation.
Apprentice	Problems are routine, requiring only modest effort or knowledge. The scope of the problems does not demonstrate the breadth of knowledge the student should have acquired at this stage of their learning.	Explanation may "flow" well but only vaguely explains one or more problems. Some parts of one or more solutions are difficult to follow. May include improper use of mathematical notation.	One or more solutions have a few errors but are understandable.	The presentation style is attractive but doesn't enhance the content; more flashy than functional. It is clear that the student has invested some effort into planning their presentation.
Journeyperson	Problems showcase the writer's skill in solving routine mathematical problems. They span an appropriate breadth of material. One or more problems may require careful thought such as consideration of a special case or combine concepts from more than one unit but not necessarily.	Explanation "flows" well and explains the problems step by step. Solution is broken down well and explained in a way that makes it easy to follow. May have minor use of improper mathematical notation. May point out other ways of solving one or more problems as well.	All solutions are correct and easy to understand. Very few or no minor errors.	The presentation may use multiple media tools. The presentation style is attractive and maintains interest. Some of the underlying message may be lost by some aspects that are more flashy than functional. It is clear that the student has given some forethought and planning to their presentation.
Expert	Problems span more than one unit worth of material. All problems are non-routine. Every problem includes content from at least two different units. Problems created demonstrate mastery of the subject matter. Showcases the writer's skill in solving challenging mathematical problems.	Explanation "flows" well, explains the problems thoroughly and points out other ways of solving at least two of them.	All solutions correct, understandable and highly detailed. No errors.	The presentation displays use of multiple media tools. The presentation style grabs the viewer's or reader's attention and compliments the content in a way that aids understanding and maintains interest. An "eye opening" display from which it is evident the student invested significant effort.

Creativity (up to 5% bonus)
The maximum possible mark for this assignment is 105%. You can earn up to 5% bonus marks for being creative in the way you approach this assignment. This is not a rigidly defined category and is open to interpretation. You can earn this bonus if your work can be described in one or more of these ways:

unique and creative way of sharing student's expertise, not something you'd usually think of;

work as a whole makes unexpected connections to real world applications;

original and expressive;

imaginative;

fresh and unusual;

a truly original approach; presentation method is unique, presented in a way no one would expect, e.g. song, movie, etc.

Today's Slides: April 30

2007-04-30T16:29:00.001-05:00

A thought--

2007-04-27T13:30:00.000-05:00

Exam Prep Wiki Solution Manual

2007-04-26T11:33:00.000-05:00

As we gear up for the exam two short weeks away I want to push your thinking to yet a higher level. To a level commensurate with what will be required of you on the AP Exam. Towards that end here is your final major assignment (other than the take home exam in May) for the course.

You will collectively create a solutions manual for this course, one question each. Actually, there are two parts to this assignment: A Significant Contribution and a Constructive Modification. Here's what you have to do:

Part 1: A Significant Contribution
to be completed no later than May 2, 2007.
(1) Go to your Solutions Manual wiki (also linked in the side bar right under the Scribe List) and pick one question in the Side Bar list.

(2) Solve the problem completely. Show all work and annotate it as if you were explaining it to a classmate who was having trouble with it. (Similar to the way you write up your scribe posts.)

Part 2: A Constructive Modification
to be completed no later than May 6, 2007.
(1) Scan through the solved problems your classmates have written up. Find one with an error and fix it or correct a diagram or improve on what they wrote if it is unclear in any way or ... there's more about this on the front page of the wiki.

IMPORTANT
Make certain you always log in to the wiki using your first name. If you don't I won't know what work you have done and will be unable to assign you the marks you earned. Not logging in with your name is the same thing as handing in a paper assignment without putting your name on it. Also, if you've forgotten the password email me and I'll email it back to you.

Photo source: no solution

BOB

2007-04-25T20:37:00.000-05:00

The course has concluded... This unit has been very interesting. I remember when Mr.K said we will learn how CSI people do their investigations. I saw it in the homework assignment. For me, Newton's Law of Cooling is a very interesting lesson. We learned how to calculate how long it would take for a certain object to cool down. I've learned a lot for the past couple of months in this course, even though the past things are just starting to make sense (since I'm just taking PreCal 40S). I still have to worry about my DEV in calculus. Some people say it's not fair for me to do 8 questions for DEV. I'm not usually the kind of person that "complains" about what my teachers give. Yeah, its a lot of work, but I'm up for anything the teacher throws at me.

I hope everyone did well on the test! =D

BOB

2007-04-25T20:10:00.000-05:00

WOW it's the last unit test. I'm amazed by how quickly the time passed... anyway, I think I'm ok for this unit for the most part. We'll see how the test goes though. I'm more concerned about the DEV project right now though so I don't really have time to study. I guess it's a good lesson though, I should have thought more carefully about the due date I picked. I really wanted to hand it in earlier so my life wouldn't be so stressful after the AP exams, but now it's really REALLY hectic :S Right now I'm starting to feel like I don't even care anymore. One of my teachers said that it's because we're so close to the end we can see the light at the end of the tunnel, and we're just looking towards that and hoping to get there on momentum alone... and another thing I want to mention, all the people who write math textbooks and come up with problems, my respect for them has increased by about a billion percent. Coming up with questions is a lot harder than I thought it would be. Even if you can solve a certain type of question, creating one in a way that works seems to me like pulling a rabbit out of a hat. an enormous rabbit. :)

BOB

2007-04-24T23:37:00.000-05:00

Differential Equations is the final unit. It was short and I guess a bit difficult in the beginning, but what chapter isn't. During the beginnning it's mostly mechanical. Near the end there's more thinking, especially for the word problems. The day we did all those word problems really helped me out. Practice makes perfect. Those problems were practically the same, only the story was different. The assembly today really got to me especially about the attendance. When one of the guidance councillors discussed about the statistic or math problem he took about relations between attendance and marks it was kind of interesting for me. Even more when it was only for DMCI students not national or overall. We completed this unit, but I still have to study the previous units to get ready for that crazy exam coming up. Work hard everyone =)

Blogon the Blog #9

2007-04-24T13:47:00.000-05:00

This unit in Calculus is fun to me. Even though I did pretty horrible on the pre-test, I think I have the right process of thinking to go about the problems. Had I remembered the antiderivative of e^-x was -e^-x, and not the same positive number, I'd have a perfect paper. Well near perfect paper, because I forgot to cube a number in the last part of the long answer question. But I'm pretty happy because it was just a mechanical error, or spelling error as Mr.K would like to call it. Going into this test I feel pretty confident in myself. Like I said this unit is pretty enjoyable. Once you grasp the idea of it, everything seems to flow easily. Good luck everyone.

BOB

2007-04-24T01:43:00.000-05:00

omg... i almost for about BOB.. ok .. this unit is finally our last...but still we have the exam we have to study for... i really need to study for the exam.. because yes.. i don't want to waste this school year that I did calculus, all for nothing. =) test is tomm.. eventually it was going to come sonner or later.. I feel as if I didn't study as hard as I usually do... I guess i just thought in my head, "last chapter, 4 chapters, won't be as hard." Now as I study, I realized that its almost equally diffilcult/easy as the other units...It's really late, I should sleep now.. btw...I'm glad i didn't forget about bobbing. =D

BOB

2007-04-24T01:03:00.000-05:00

Believe it or not, we've covered everything in the course!! I'm honestly proud of everyone!! First, Calculus isn't a class that everyone in high school takes. Second, even for the ones who can take it, it's not an easy course to get through. It takes patience and hard work. I'm lucky to be among people who actually care about their education and care about their peers. Thanks you guys for a great year.

The AP Exam is 15 days away. Anyone getting panic attacks yet? Well no one should, haha, we all deserve peace of mind. It boils down to choice, don't you think? This exam doesn't affect our marks after all, so why should we worry about it? Well, this is how I justified the put off homework and lack of prep for quizzes and tests. It wasn't a wise choice.

The reason why I took Calculus was to prepare for University. I've made up my mind even before taking the course that I'd take Calculus in University again. Like Mr. K said, we won't write an exam like this in our lives again. If we get through the AP Exam, it'd help us with our post-high-school education, and that's worth something. Why would anyone take this course in the first place if s/he is not willing to work for it?

We are all aware that we're busy and there are so much things going on in our lives. Honestly, my life doesn't revolve around Calculus. Throughout my high school career, I've worked my butt off. The only time I had a spare, in my almost 4 years at Daniel, is this semester, and even then I'm still having a hard time juggling everything. I'm sure all of us experience this. But we're almost there! Never give up, yeah? It would've been so easy to just drop this course altogether. But we're still here. What might be possible strategies to get through the course? Personally, I like working with other people. Might we do those study groups? Should we have "classes" where we're just having a conversation about Calculus topics, very casual, non-threatening? I think those work.

I'm starting to go through the Scribe Posts throughout the year. Haha, good times. Just before I leave this off, a reminder to everyone that we shouldn't get stressed out. Haha, someone once said, "one has achieved success, he who has lived well, laughed often and loved much". That, for me, is success. Recipe for success is passion + hardwork... then the money starts pouring in.. hurray!

Good night everyone.

BOB

2007-04-23T23:49:00.000-05:00

Wow our last unit. This unit is alright. I think that the hardest part is figuring out what to do in the beginning. I think I did alright on the pre-test. I keep on forgetting the "C" and it's annoying! That is where I messed up on the last question on the pre-test. it's something that should have been avoided. I really like having the smart board in class. I can go back to check what we did in class. It's a big help. It's too bad we didn't have it at the beginning of the year. Although I find that it takes a little bit longer.

I liked questions like these:

But I'm not so hot with question like these:

because it is so easy for me to mess up on the question. It's just figuring out how to start.

My bio teacher has this thing he calls his students. He calls us all "Dr. Yeah Buts." They think of reasons not to do something like calculus homework. I don't want to be a "Dr. Yeah but," but sadly I am. I think its a very very bad habit to get into since we'll be in university next year. If I can't handle a busy workload in high school, well imagine university.... bahh!

I've started reviewing for the exam last week. It's frustrating because... well it's frustrating. I'm very glad I've stuck with this class. There were countless times where I wanted to drop this class, but I chose not to... plus my mom's a bit scary =S ... in a good way though =P Mom's are good at making you feel guilty....

bob

2007-04-23T23:44:00.000-05:00

Well yeah this is theunit is over. well i guess this unit is pretty short compared to others and should be simple. i know that i shouldn't have missed those classes i have missed in the past few weeks and i am holding myself totally accountable. i guess its just becuase ive just been so busy with everything going on in my life that i just needed a break from everything and to just relax. anyways well the test is coming and i guess i can say that i am somewhat ready. ive just been ready the text book today and i think i understand most of it. well good luck to me cause i know ill need it.

BOB

2007-04-23T23:42:00.000-05:00

We're done, done, done!!!! So the last unit was short... and that should have made it easier to learn, easier to understand, easier to do homework, etc, etc. Truth is, it hasn't gotten any easier for me. This is not the place to complain and I only have myself to blame. I haven't attempted to do all of the homework, or completely read the blog for this unit. I'm pretty clueless. I know if I did the homework, I would have a better understanding and idea of how to go about solving the problems. The exam is in two weeks. My project is due before that. We may be done the unit, but I am far from where I want to be in this class. I'm glad we're done the unit, I'm just not looking forward to the test =/

BOB

2007-04-23T22:51:00.000-05:00

Hm the last unit! Our exam is coming up...and half of us will be in Banff this week. Chapter 9 was short and mostly mechanical. I think the hardest part is finding what to do first. After you know what to do, you can do the rest easily. The Pre-test wasn't that difficult but I still have some things to relearn. The only thing I'm worried about is Euler's method but I'll read about it in the textbook. I also have to practice those problems on those worksheets on Mr.K's post.

Scribe Post

2007-04-23T22:04:00.000-05:00

Hello guys, it's Christian. 16 days left 'til the exam!!

Today we had the pre-test in class. The solutions can be found in the day's slides. The questions we worked on today were applications of ideas we learned this chapter. Not surprisingly, they were pretty much the same questions we've done before. That's all for me..

The next scribe is Suzanne.

Today's Slides: April 20

2007-04-20T13:14:00.001-05:00

I was expecting you all to show up at lunch hour ... only Christian came ... and that was an accident!

19 Days Left until the exam.

Anyway here are the slides I filled out for you ... read them ... understand them ... ask questions if you don't ...

To see a larger image of the slides go here. When you get there you'll see a button in the bottom right-hand corner that says [full]. Click it and the slides will display in full screen mode.

Scribe

2007-04-19T22:55:00.000-05:00

Well we were supposed to have the pre-test today, but as a class we were not that ready. So instead we went into three different groups and worked on word problems. All the problems were practically the same, the steps were: 1) finding two points on the parent, 2)seperating varibles, 3) integrating, and 4) creating an equation. However we still had some difficulties or maybe it was just an off day. I don't know. But the more practice the better.

Remember everyone to keep up with the assigned homework. The AP exam is just around the corner. I also have to catch up and review. There's a homework sheet below for more practice.

The next scribe is Christian.

Today's Slides and Homework: April 18

2007-04-18T14:09:00.000-05:00

Scribe Post

2007-04-16T15:49:00.000-05:00

Ahoy everyone! This is Jann, and I'll be your scribe for today.

We have finally finished the whole course. Today, we looked at some word problems on Differential Equations.

Here we go... (Note: I will be using Mr. K's slides... )

In order to solve this problem, we need to find the parent function. dy/dx is some constant multiplied by the parent function. We know that the temperature, T, varies with respect to time. (Basically, the temperature of the roast decreases over time) We need to find 2 points on the parent function...

At time, t=0, the initial temperature of the roast, which is 68 degrees F. Therefore, we have our first point, (0,68). For our second point, at time, t=2 hours, the temperature of the roast is down to 40 degrees F. This is our second point, (2,40).
Then, after acquiring the 2 points in the parent function, we integrate the formula. The Red Line shows that the calculus part of the problem ends. The following line is all Pre Calculus stuff. If we can recall our natural law from Grade 12 PreCal, "kt + C" is an exponent. Therefore, it is an exponent of the base "e". "e^C1" can be substituted into "C" because it is a constant. Then, we isolate the "T" in order to obtain the Parent Function.

Now that we have the structure of the parent function, we can solve for the missing terms such as "C" and "k".

Now that we have the parent function. We are ready to solve the problem...Then, we looked at one more example...

That's all we did today in class. We were suppose to look at another example, but we ran out of time. Good day everyone! XD

Oh yeah... next scribe is Crystal.

Today's Slides: April 16

2007-04-16T12:13:00.000-05:00

Scribe Post

2007-04-15T19:05:00.000-05:00

So during Calculus class today we opened up the class reviewing Newton's method, which is used for approximating a nearby value using the slope of a tangent line on a differiential equation. e.g. approximating √(37) from using tangent line at √(36)

Then after that we programed our calculator with Euler's Method, which is just a mechanical process and is explained in the slides below. When you have a function that tells you the rate of change, we'd use this method to solve the differentiable equation. And a diffentiable equation is when one of the variables is a derivative. These problems can also be called initial value problems.

There's not much to explain for what we did in class, as you can see below in the slides.

Homework is chapter 9.3. Next class we'll tackle the newton's law of cooling question.

Next scribe will be Jann.

Perseverance

2007-04-13T12:15:00.000-05:00

Do you think? Where does calculus fit into this?

Today's Slides: April 12

2007-04-12T13:29:00.000-05:00

Scribe Post

2007-04-11T18:03:00.000-05:00

THIS IS LINDSAY =)

In class, we learned how to draw slope fields.

Let's review from our previous calculus class:

The order of a differential equation is the highest-order derivative.

y'=y is a first order differential equation.
y"=-y is a second order differential equation.

The solution to a differential equation is any function that fits the eqatuion.

For example, to find the differential equation to y'=y you find that you have to find a function that is its own derivative. We know that y=e^x is a solution.

When we think about derivatives, we think about slopes! For every input x, f''(x) is the slope of f(x) at (x,y). This is a first order differential equation: when it describes the slopes at a specific point.

Example: y'=y is a differential equation. At any point (x,y), the slope of the solution curve is the same as the y coordinate. If we use the point (1,2), the slope of the curve is 2. Whenever the y coordinate is y=3, the slope will be 3.

We were given a handout with some examples. We did this in class and the solutions are in the slide post.

1) y'=y; you would use y=e^x because it is its own derivative.

2) y'=2x; the parent function is x²

3) y'=-x/y;
x² + y² = k
2x + 2yy' = 0
y'= -x/y

*parent function is a circle

4) y'=x+y

For homework, we need to create a new program to create slope fields. You can find how to create it in 9.2 or you can ask a calculus buddy to send it to your calculator! Also do all the odd questions!

Euler's Method Solutions and Answers

2007-04-10T10:00:00.000-05:00

Here is a copy of the assignment you had today showing the answers and solutions. The second page below shows two spreadsheets that you may find helpful.

Today's Slides: April 5

2007-04-05T17:06:00.000-05:00

First Things First? Again??

2007-04-04T09:00:00.000-05:00

Linger mentioned in a comment:

"These challenges that we're faced with everyday only make us stronger and better people. I know that there's been times when I wanted to give up... but then I think about how far I've come. My classmates are great and so is Mr. K... and yes, it gets frustrating sometimes, but it's always worth it at the end of the day. I think that these challenges are just oppurtunities for us to learn more about ourselves and each other, so whatever comes our way, we can and will defeat it =)"

So very good to remember now that May 9 is not far away!

I’m hoping that in my sharing one of my experiences, you can find some bit to help guarantee your success that day! In early 2003, I too faced an important test date and I too appreciated the opportunity and felt I had to find a way to defeat it as Linger says. Achieving this goal was too important to me and my test if I passed would lead to a national teacher certification (only 40% of test takers passed). I was really trying to be superwoman—just as I imagine many of you work at being superteens. I thought I was prioritizing but ooohh---a to-do list with 20 items all the time!!! Urgent “stuff” kept happening and I was always responding to that. Does that sound familiar?

I was overwhelmed!!!! I sat down and broke my test review into manageable chunks (I had 2 months until the test) and put them into my planner. I did that first because the goal to pass the test was so important. And I planned to turn off instant message, not answer the phone, or have the TV on during the review sessions. I had tried before but was always interrupted by the phone or my students on instant message with questions about our studies.

Then I looked at my other “stuff”, and categorized it: vital, important, or nice. Then I took the vital “stuff” and categorized them again: vital, important, or nice. And I let go the nice. So my house wasn’t very clean during the process, and we didn’t have gourmet dinners. But the laundry was done and we had quickie suppers. My students’ work was graded but I didn’t plan any big field trips or projects during that time. I set aside a time every second night to evaluate their work. As I look back now, I prioritized, and then prioritized again. I did a mental daily check of my goals and made every effort not to be dragged down by urgent if it didn’t help me achieve my goal. Of course, I had to be flexible at times. I couldn’t always follow the plan exactly. But since I knew where I was going and I had planned for time to get there, my review was accomplished by “the day”.

I truly believe, that with good preparation and putting first things first, you'll too feel that great rush of a job well done, and a goal achieved when you learn your scores. I share these experiences, knowing that you are planning and reviewing, but wondering is there one little piece here you could use to help you on your way? Or can you point us to some tips that are really helping you manage to put first things first?

2007-04-03T23:37:00.000-05:00

Hey math buddies...

This is not math related, but I need your help. In my jounalism class and also for the school paper, I'm writing a story on downloading music and I need your opinions! ... I also don't have much time :

Most people don't think of downloading music as stealing. Do you? I've certainly never thought of it that way. I just thought of it as a way of listening to music. If I really like the song then go out and buy it. You don't want to be stuck with an album that sucks and your stuck with it. I mean how much is a CD, like $15 to $20. Most people think "why spend the money when you can get it for free?" Therefore, saving that money for something else. I mean what about libraries. Is there a difference between borrowing a book and reading it for free and listening to a song for free?

There are copyright laws that need to be respected but most people don't think about it. In the states you here about people, even as young as 12, being sued by RIAA for downloading. I mean imagine being a 12 year old and recieving a letter saying your being sued. Does the same thin happen in Canada.

I think the music industry needs to discover a new approach. Internet is obviously here to stay and we have access to sooo much, including free music, to millions of people.

You're opinions will be greatly appreciated, it wouldn't be fair to include only my opinion since this topic has so many other views.

By the way, have you guys checked out the new school paper?? =D

I don't know what to label this... BOB? Yes I'll call it BOB

Developing Expert Voices - Invitation to the Blog

2007-03-28T14:35:00.000-05:00

As promised, I built a unique blog for your Developing Expert Voices projects. Email me and I will send you an invitation to the new blog. When you publish your projects on the DEV blog tag them with the name of your class (AP Calculus) and the topics covered by the problems you created.

You can take a sneak peak at the new blog here.

Click that picture ... there's a great article at the other end of that link.

BOB

2007-03-26T22:55:00.000-05:00

Hi! My BOB's a bit late, because I haven't done the test yet. It's 11pm of Monday, March 26th right now, and I just finished a couple of calculus questions. That tells you how boring my life is when it's the first day of Spring Break, and of all things, cALcuLus is in my head. Well, it's because of the anxiety some people have expressed on Friday by the people who have taken the test. They said it was very difficult and they had a hard time doing it. I thought I had the main ideas down, but according to a few classmates, that wouldn't be enough.

This is my gameplan:

I'll try to come up with a step-by-step method in solving these problems. These are integral questions still right, but see, I still have problems with integrals. I'm not super duper comfortable with them yet. If there's a step-by-step method with the problems, I think it'll help me. Sometime when we come back to school, I'll probably stop by Mr. K's room to ask for help. Til then!

Developing Expert Voices Rubric out of beta v.1.0

2007-03-23T16:35:00.001-05:00

This is the final version of the document we built together.

Developing Expert Voices Rubric

The teaching of mathematical concepts is the main focus of this project; so we can teach other people and learn at the same time.

Acheivement Descriptors
Instead of levels 1-4 (lowest to highest) we use these descriptors. They better describe what this project is all about.
Novice: a person who is new to the circumstances, work, etc., in which he or she is placed; a beginner.
Apprentice: to bind to or place with a master craftsman, or the like, for instruction in a trade.
Journeyperson: any experienced, competent but routine worker or performer.
Expert: possessing special skill or knowledge; trained by practice; skillful or skilled.

Acheivement	Mathematical Challenge (25%)	Solutions (55%)	Presentation (20%)
Novice	Problems illustrate only an introductory knowledge of the subject. They may be unsolvable or the solutions to the problems are obvious and/or easy to find. They do not demonstrate mastery of the subject matter.	One or more solutions contain several errors with insufficient detail to understand what's going on. Explanation does not "flow," may not be in sequential order and does not adequately explain the problem(s). May also have improper mathematical notation.	Presentation may or may not include visual or other digital enhancements. Overall, a rather uninspired presentation. Doesn't really stand out. It is clear that the student has invested little effort into planning their presentation.
Apprentice	Problems are routine, requiring only modest effort or knowledge. The scope of the problems does not demonstrate the breadth of knowledge the student should have acquired at this stage of their learning.	One or more solutions have a few errors but are understandable. Explanation may "flow" well but only vaguely explains one or more problems. Some parts of one or more solutions are difficult to follow. May include improper use of mathematical notation.	The presentation style is attractive but doesn't enhance the content; more flashy than functional. It is clear that the student has invested some effort into planning their presentation.
Journeyperson	Not all the problems are "routine" in nature. They span an appropriate breadth of material. At least one problem requires careful thought such as consideration of a special case or combines concepts from more than one unit. Showcases the writer's skill in solving routine mathematical problems.	All solutions are correct and easy to understand. Very few or no minor errors. Explanation "flows" well and explains the problems step by step. Solution is broken down well and explained in a way that makes it easy to follow. May have minor use of improper mathematical notation. May point out other ways of solving one or more problems as well.	The presentation may use multiple media tools. The presentation style is attractive and maintains interest. Some of the underlying message may be lost by some aspects that are more flashy than functional. It is clear that the student has given some forethought and planning to their presentation.
Expert	Problems span more than one unit worth of material. All problems are non-routine. Every problem includes content from at least two different units. Problems created demonstrate mastery of the subject matter. Showcases the writer's skill in solving challenging mathematical problems.	All solutions correct, understandable and highly detailed. No errors. Explanation "flows" well, explains the problems thoroughly and points out other ways of solving at least two of them.	The presentation displays use of multiple media tools. The presentation style grabs the viewer's or reader's attention and compliments the content in a way aids understanding and maintains interest. An "eye opening" display from which it is evident that the student invested significant effort.

Creativity (up to 5% bonus)

The maximum possible mark for this assignment is 105%. You can earn up to 5% bonus marks for being creative in the way you approach this assignment. This is not a rigidly defined category and is open to interpretation. You can earn this bonus if your work can be described in one or more of these ways:

unique and creative way of sharing student's expertise, not something you'd usually think of;
work as a whole makes unexpected connections to real world applications;
original and expressive;
imaginative;
fresh and unusual;
a truly original approach; presentation method is unique, presented in a way no one would expect, e.g. song, movie, etc.

BOB

2007-03-23T09:00:00.000-05:00

This unit was about finding the definite integral. I understood most of it in the beginning, but then I started getting confused near the end. I should have went to class more, but that was my fault. I crammed all night and attempted some of the questions. But I'm still not confident in how well I'll do the test this morning. Well hopefully the lunch classes will help me for the big exam. Good luck everyone!

BOB

2007-03-22T22:09:00.000-05:00

This unit was pretty quick. I understood most of what we were learning in class. The only thing I have trouble with if figuring out the image with the shells and all. The picture is what I get mixed up on, then my answer's wrong. In class today, I understood it, so I was happy. I liked this unit. I don't really like the word problems, again because of picturing the problem and the graph. I don't like the density problems I get confused easily. I didn't do very well on the pre-test. It was a lot of guessing. When Mr. K explained it all, it seem so simple. I usually think that it's more complicated than it really is. So I end up doing all this funky complicated stuff. This week was very, very busy. I had four other tests (this one's the last =D!), essays, and articles to do. So, that didn't really help. I'm soooo glad Spring Break is coming up!!

BOB

2007-03-22T20:48:00.000-05:00

Right now I'm working on the supplementary problems for this chapter, and so far so good. I had some trouble on the pretest yesterday, I think mainly because I didn't practice the different types of problems as much as I should have, so I forgot how to solve them after a few days. But yesterday and today has refreshed the basics of this chapter a little, so I think I can manage tomorrow. I was a bit confused by the density problems, and haven't really tried any outside of class yet, so that's my plan for the immediate future. And I want to work on the average value and mean value stuff too. (Though we didn't do much with the mean value thing, so I don't know if it'll even be on the test.) I have three tests tomorrow though and I still have to study for one of them... but hey, get through tomorrow and it's spring break :) Good luck everyone! Have a nice spring break, preferably homework-free.. preferably somewhere with sandy beaches and no snow.

BOB the Snail

2007-03-22T20:25:00.000-05:00

Hi everyone! Our unit on Integral Applications is done and we're going to have a test. This unit has been really short. I found this unit easier to understand than the unit before this. The applications are fairly straight forward. We just need to practice these problems. Why BOB the Snail? A snail is a very slow creature. It's having difficulty in crawling from one place to another; but it never gives up. Like us, we should never give up on our studies. Even though I feel like falling behind, I still think about "going for it".

Good luck on the test everyone! XD

BOB

2007-03-22T19:48:00.000-05:00

BOB,

To be completely honest, I do not understand this unit. I hear everything that's being said in class, and during that time, I think I understand. When I leave the class, WHOOSH, it's all gone. Why? Maybe it's because in calculus class, you have to think about it, so you understand it. It's fresh in my mind. When I leave the room, I walk into a world where you have to think about many other things and all that calculus knowledge is pushed back into the dark corner of a filing cabinet...which MAY or MAY NOT come back. This unit is supposed to be the fun part of calculus but I'm really not enjoying it. It could be because I don't understand it, I can't picture what the graphs look like. The paper towel and washer analogies are helpful, but to me, it doesn't have a purpose.

I wish, like many of my fellow classmates, that we had more time to study. It's late in the year, we're tired. We need to recharge our batteries. Tomorrow is the test and honest to goodness, I am more afraid of this test than the AP exam. I really am. There are bits and pieces that I do understand, but I just can't use them to get the whole picture.

Good luck to my classmates. After this test, we could relax, Spring Break is so near, I can feel it. I wish we had more time to study x 2. Tests are supposed to show how well you understand a topic...but if we really don't understand, I don't think it's beneficial to have one. If we don't understand this unit, we will struggle in the next unit. Spring break, spring break, spring break.

Ohhh my BOB

2007-03-22T19:04:00.000-05:00

Dear Bob,

As you may already know, we've been having lunch hour classes, because we're behind in the curriculum. The exam is in the beginning of May, which is really just around the corner. Hmm.. you see, I took AP Calc for the experience and as a challenge. And so far, it really has been a challenge.

This unit was really easy to follow and understand in the beginning. But as we progressed, I had diffuculty in picturing what the arbitrary slice would look like when taken from the solid. They come in disks, washers, and prisms depending where you rotate them. To find the volume, you need to find the volume of one slice by finding its area. The only thing that varies is the radius, and I think its true, how knowing what the radius represents is the tricky part of it. Once you have the integral, the rest is just a matter of punching into the calculator. I don't know why, but I understand things when we do it in class... when it came to the pretest, I thought too hard about it. It was simpler than I thought. It sometimes frustrates me, because I should know better... I do know better.

I wish I had more time to study. I have another test to study for, a pile of homework to do, a book to finish reading, and other things to take care of. Nonetheless, don't we all wish we had more time? If we could rewind and do it better, put more effort, DO not some, but ALL of the homework, would we not? Yes, the test is tomorrow.. but that is the beauty of AP Calc.. because now, if and when we take it in University, we can remember what it felt like when we weren't prepared and we can laugh... because by then, we'll have learned our lesson and know better.

Biiig siiiigh.. the test is tomorrow!!! But you know what that means.. SPRING BREAK is on its way.. and it's getting nice out! WOOHOO... one more unit and we're done, done, done. When you think of giving up, just remember Sisyphus. Good luck to all and to all a good night =)

BOB #8

2007-03-22T15:03:00.000-05:00

This unit started off easy and towards the end it got harder. I think that's the way all the units were. At the beginning I did the homework and then stopped because I thought I already knew what I was doing. When the pretest came, it hit me that I didn't quite know what I was doing and right now I feel like I need to do the last chapters so that for the test tom. I feel confident enough to complete the questions. I think its going to be tough for me but I'm still going to try. I need to do more questions about when the function spins around the y-axis and at different values. I'm not quite sure about those. Good luck to everyone and be happy b/c remember it's spring break next week! =)

Today's Slides, Review and Worksheet: March 22

2007-03-22T13:29:00.000-05:00

BOB

2007-03-21T23:33:00.000-05:00

well, im not sure this unit is kind of weird. i am having a hard time seeing how a function would look like if we spin it around an axis. on the multiple choice, i noticed they were questions similar to what we did in the beginning of the unit. For example, the first question that dealt with wrapping the function around y= -1. i actually found the rate multiple question easier (last question on first page). must be careful with those itty bitty pieces.

Blogon the Blog #8

2007-03-21T20:50:00.000-05:00

The pre-test was hard. This unit is pretty tough! But once you get around how to find the integral, which pretty much is the hard part, then it becomes mechanical from there. I think my biggest problem is deciding whether or not the solid volume produce spinning a function around the y or x axis is going to be a cylindrical shells, or round discs.. when the y or x axis you're spinning it about is translated units. But I think by doing more problems like these, I could surely get it. I've been focusing my attention to another subject that is of much greater value to me and resulted in less time for calculus. After spring break, everything should be fine.

Today's Slides: March 21

2007-03-21T14:04:00.000-05:00

Scribe Post 90

2007-03-20T20:41:00.000-05:00

First off, i would like to remind everyone to go and add to the marking rubric for our project. Rubric is due this Friday.

1) Oil is leaking at a rate of R(t) = 2000e^-0.2t
, where t is measured in hours, how much oil has spilled in 10 hours?

We want to end up in gallons, so we multiply gallons per hour (which we are given) by some change in time. We want to find the total sum over the ten hours.
*When we are given a rate, such as the one above, we are trying to find a unit related to the rate. I.e. (gallons/hour)x(hour) = gallons.*

A) Density of an oil slick on a body of water is defined by

Suppose the oil slick is extends from 0 to 1000 m. Determine the mass of the slick.
We have Kg/m² and we want to get kg, so we have to get rid of the m²

We cut a piece from the oil slick. If we drill down, the density of the oil slick is the same throughout.

Here we have cut a piece out of the oil slick. The ends of the strips (triangles) become smaller. So we take the triangle and move it to one side to form a rectangle. The area of the rectangle is what we need to find. Going back to our equation we can now get rid of m²

With our our information and our missing piece found, we can create an integral to solve for the mass of the oil slick. We take the limit of n as it approaches infinite and it becomes smaller. We can then take the sum, where n equals one. p(a) is our given equation so we multiply that by the rectangle piece we cut out, Am². Then we get the integral from 0 to 1000 meters. 100/(1+r²) is kg/m² multiplied by 2pi r dr m², which will give us the mass of the oil slick 1000 m out. 2pi r is multiplied by dr because dr becomes infinitely smaller.
*check the slide for march 19 to view the solution to the integral*

B)What is the smallest radius that contains 75% of the oil slicks mass?
With our solution from A, all we do is take seventy percent of the value we found. We can then pull out 200pi. From here we just solve for r, by expanding the bracket notation first.

For the last part of class, we talked about how our physics formula can be derived. We also did a quick question before the bell rang. We were given an amount of cars per kilometer. instead of trying to cancel out a unit area we had to cancel a length to get the number of cars.

URGH, i got a cold.... next scribe is Suzanne... pre test tmrw?

Today's Slides and Homework: March 20

2007-03-20T13:28:00.000-05:00

A Second Wind?

2007-03-20T10:12:00.000-05:00

"Most people never run far enough on their first wind to find out they've got a second. Give your dreams all you've got and you'll be amazed at the energy that comes out of you." --William James

I wonder, would this apply to calculus too?

Today's Slides: March 19

2007-03-19T16:25:00.000-05:00

SCRIBE POST: DAY 89

2007-03-16T16:34:00.000-05:00

The plan was to have this up by Friday. Sorry guys. Anyway, I haven't been scribe for quite some time, so it took me awhile to get back into it...

We started class by talking about our due dates for our DEVELOPING EXPERT VOICES assignment. Remember guys: you can always move your due date earlier than expected. We also talked about the rubric for this assignment. Remember that changes must be made no later than Friday, March 23rd.

Let's get the math started...
To solve for a you simply plug in 1 for k. To solve for b, c, and d, you must integrate it from 1 to the number of items. We start with 1, because this is where the domain of the function begins. If you plugged 0 into the function, you would get 23 minutes. This doesn't make sense; it shouldn't take 23 minutes to make nothing. Once you know the integral, the rest is just punching it into the calculator. The answers can be found on the bottom of this page. http://3.bp.blogspot.com/_tlF3qs3v9cw/RfmQ9vFtnUI/AAAAAAAAAIA/ymR08nIwBLc/s1600-h/AppsOfTheIntegralExerc.jpg Note: This is a good exercise to practise for the unit test.

With the help of Fooplot http://www.fooplot.com/, we were able to see what the graph looked like. Points of intersection tell you where to integrate.

1. To find the area of S, we simply find the integral from 0 to 4. Because the function f(x) is on the bottom, we subtract it from g(x).

2. To find the volume of S, we first find the volume of one piece. We take an arbitrary slice, and find its area. In this case, the shape of the slice is a semi-circle, where the radius is the top function, g(x), subtract the bottom function, f(x). Once you have the area, you must write the integral. You then apply this integral to find the volume of the whole thing.

3. Because we S is rotating, you can imagine a solid being formed with a hole in the middle. To find the area, we must take the area of the smaller circle and subract it from the area of the bigger circle. Questions a, b, and c are all the same question; the only difference between them is how the radii are formed.

a)

b)

d) In this case, the solid is rotating around the y-axis, therefore the shape of the arbitrary slice is no longer a circle. It becomes a prism. The width of the prism is the circumference. (remember: the paper towel) Its thickness is dx, because we want the slice to be infinitesmally small. Its height is simply the top function, g(x), subtract the bottom function, f(x).

So that brings us to the end. If you're still having trouble, the slides from Thursdays class http://www.slideshare.net/gofull/30241/1 are up.. thank goodness for our super-di-duper smartboard!!! HMWK for the weekend is EXC 8.5 # 1,3,5,11,15. And by the way, MARK is the scribe for Monday! Enjoy the rest of your weekend guys =)

Today's Slides and Homework: March 15

2007-03-15T13:21:00.000-05:00

Pi Day

2007-03-14T12:49:00.000-05:00

Happy π Day!

Developing Expert Voices

2007-03-14T09:03:00.001-05:00

The Assignment
Think back on all the things you have learned so far this semester and create (not copy) four problems that are representative of what you have learned. Provide annotated solutions to the problems; they should be annotated well enough for an interested learner to understand and learn from you. Your problems should demonstrate the upper limit of your understanding of the concepts. (I expect more complex problems from a student with a sophisticated understanding than from a student with just a basic grasp of concepts.) You must also include a brief summary reflection (250 words max) on this process and also a comment on what you have learned so far.

Timeline
You will choose your own due date based on your personal schedule and working habits. The absolute final deadline is May 31, 2007. You shouldn't really choose this date. On the sidebar of the blog is our class Google Calendar. You will choose your deadline and we will add it to the calendar in class. Once the deadline is chosen it is final. You may make it earlier but not later.

Format
Your work must be published as an online presentation. You may do so in any format that you wish using any digital tool(s) that you wish. It may be as simple as an extended scribe post, it may be a video uploaded to YouTube or Google Video, it may be a SlideShare or BubbleShare presentation or even a podcast. The sky is the limit with this. You can find a list of free online tools you can use here (a wiki put together by Mr. Harbeck and myself specifically for this purpose). Feel free to mix and match the tools to create something original if you like.

Summary
So, when you are done your presentation should contain:
(a) 4 problems you created. Concepts included should span the content of at least one full unit. The idea is for this to be a mathematical sampler of your expertise in mathematics.

(b) Each problem must include a solution with a detailed annotation. The annotation should be written so that an interested learner can learn from you. This is where you take on the role of teacher.

(c) At the end write a brief reflection that includes comments on:

• Why did you choose the concepts you did to create your problem set?
• How do these problems provide an overview of your best mathematical understanding of what you have learned so far?
• Did you learn anything from this assignment? Was it educationally valuable to you? (Be honest with this. If you got nothing out of this assignment then say that, but be specific about what you didn't like and offer a suggestion to improve it in the future.)

Experts always look back at where they have been to improve in the future.

(d) Your presentation must be published online in any format of your choosing on the Developing Expert Voices blog. url: tba.
Experts are recognized not just for what they know but for how they demonstrate their expertise in a public forum.

Levels of Achievement
Instead of levels 1-4 (lowest to highest) we will use these descriptors. They better describe what this project is all about.

Novice: A person who is new to the circumstances, work, etc., in which he or she is placed.

Apprentice: To work for an expert to learn a skill or trade.

Journeyperson: Any experienced, competent but routine worker or performer.

Expert: Possessing special skill or knowledge; trained by practice; skillful and skilled.

Happy Pi Day!

2007-03-13T18:34:00.000-05:00

Click here to view the card Lani has sent you!

Scribe Post

2007-03-13T16:10:00.000-05:00

Good day ya'll! This is Jann and I'll be your scribe for today.

Today, we learned how to find the Average Value for Functions.
We started by getting the average value of a set of numbers.

8 +5 + 7 + 7 + 3 = 6
5

Here's the definition of the Average value of a Function:
"Let 'f' be a function which is continuous on the closed interval [a,b]. The average value of 'f' from x=a to x=b is the integral...

Q: Where did this formula come from?

A: Good question! If we examine a graph of any function, we can say that "a" is the lower limit of the graph and "b" is the upper limit of the graph.

This is the formula used to find the width of the intervals."x" is the width of the intervals. If we recall, we divide the graph of a curve into rectangular divisions to find the area under the curve. "x" varies because the more the intervals, the width of the intervals will decrease. "n" is the number of intervals used to divide the curve.

If we try to solve for "n", we need to rearrange the formula.

According to our Average Value rules, we need to add all the outputs and divide it by the number of intervals used. Therefore... We substituted the formula for n in the denominator. The formula boxed in red is the Riemann Sum. The "E" symbolized the summation of all the outputs in the function, "f". "x" is a constant since the width of an interval never changes. The number of intervals will go to infinity to get the area under the curve. The limit and the summation are symbolized by the integral from [a,b].

Example:

Find the average value of the given function on the given interval.

f(x) = 1 - 2x [0,3]

Using the average value formula for functions, we can get the average value...

Then, we talked about the Mean Value Theorem for Integrals.

If we recall, the Mean Value Theorem for Derivatives state that: Let f be continuous on a closed interval, [a,b]. There lies a value "c" that is equal to the rate of change of the curve.

The Mean Value Theorem for Integrals state that: Let f be continuous on a closed interval [a,b]. Then there exists "c" in the closed interval such that...

With all that said and done, we practiced 1 problem.

Find the i) average value of "f" on the given interval, ii) "c" such that the average of "f" is equal to the f(c), and iii) sketch the graph of f and a rectangle whose area is the same as the area under "f".

f(x) = 2x [0,3]
I think that's all we did in class. Next scribe is... err.... I'm not sure who to pick. XD I'll just pick during the next class.

Today's Slides and Homework: March 13

2007-03-13T11:22:00.000-05:00

Scribe Post

2007-03-13T00:20:00.000-05:00

HI Everyone!! Here's the post for friday's class:

On friday, we started with a non-calculator quiz then did we did the following lesson:

Note: If pics hard to see, just click on pic so its bigger.

_____________________________________

____________________________________

Homework: Ex. 8.3 odd #'s

Next Scribe is going to be Jann =D

Today's Slides and Homework: March 12

2007-03-12T13:31:00.000-05:00

Today's Slides: March 9

2007-03-09T11:44:00.000-06:00

Scribe Post

2007-03-09T00:59:00.000-06:00

Finding the Volume of a Solid
(note: The pics aren't as clear, so please just click on them for a better view)

Imagine a rectangular, three-dimensional solid. To take the volume of this figure, we multiply its thickness, height and width together.

V = (thickness)(length)(width)

Another way to find V is if we divide the solid into even figures, find the volume of one of the smaller pieces, and multiply it by how many smaller pieces there are. In other words:

Volume has three dimensions, and Area has two. By integrating, we are essentially finding the thickness of an object. Going back to the equation V = (thickness)(length)(width), one dimension is already accounted for. We only need to find two more, and this is the area of the cross section.

In general, any function rotated around the x-axis will have circular cross sections, and its volume can be found by:

A(x) is the function representing the changing areas. The area for all the cross sections is represented by A = πr², but r varies, so we need a function that represents r.

Let’s take a look at an example:

Imagine a line segment from the point (0, 4) to (8, 0). Revolve this line on the x-axis, and you’d get a cone, as seen in the diagram below. To find the volume, let’s follow the steps we learned in class:

(1) Find the function that represents the radius.
(2) Replace “r” with this function in the “volume equation” up there.
(3) Integrate over the interval.

If we compare this to what we get if we use the geometric equation we’ve used in elementary (V = 1/3πr²h), we get the same answer!!

So there you go folks. I think the three steps are doable. Homework is Exercise 8.2 ODDS. Scribe is JANN!

Today's Slides: March 7

2007-03-07T16:45:00.000-06:00

Scribe Post

2007-03-06T21:02:00.000-06:00

Hi guys :) Yesterday we started unit 8.

First off we started with a few problems which you probably recognized from past units.

Find the area bound by the two curves:

To solve these, you must look at the graphs, and determine which function is above and which is below. Then you take the integral of the top function minus the bottom function. To find the interval, make the two functions equal each other and solve for x.

After that we discussed a problem where we are given a velocity function and asked to find the displacement, also known as net distance, and total distance travelled. The net distance is found by subtracting the negative integrals from the positive integrals. The total distance travelled is found by taking the integral of the absolute value of v(t).

That's all you need to know to do excercise 8.1.

Then we started to discuss how a 3 dimensional object is created by 'spinning' the function around the x axis (or y axis, but we didn't get into that just yet). We discussed how you can take many 'slices' of the solid, so thin that the width approaches zero. I guess we'll be expanding on this in class tomorrow.

The rest of the assigned homework includes the odd questions from 1-7 of exercise 8.2.

The scribe tomorrow shall be...... Danny.

More about ants ...

2007-03-01T08:37:00.000-06:00

I just stumbled upon this. After Lani's recent post I thought it might be interesting to all of you ...

To view the slides in full screen mode go here and click on the [full] button in the bottom right hand corner of the slideshow.

BOB

2007-02-28T22:28:00.000-06:00

Hi! I like differentiation rules because they're straightforward. If two terms are multipled together, we use the product rule. If they're divided, we use the quotient rule. The reason why I think we're anxious about antidifferentiation rules is because there are no guarantees that we're even on the right track. There are guidelines that we've discussed over and over again, but as we all know, sometimes we still end up being on the wrong road. As with everyone in the class, I don't feel the best about this unit either. I DO believe though that we'll get through this, and we'll turn out alright. Remember that we took this class, among other things, to get exposed with the material? to be prepared for university? Well, I say let's work our butts out right now too, cause if we don't, we're just wasting our time.

BOB

2007-02-28T19:09:00.000-06:00

I am not even going to lie and say I understand even the simplest part of calculus . . . it's some hard stuff. I really don't understand much of the material. I just get confused with anti derivative and derivative and the chain rule, integration by parts. There's too much going on in one problem. My brain's exhausted just from thinking about The composite of functions and the product of functions. I really wish I understood calculus but I don't think that math is one of my strengths as some people are good at some things other people are not. I'm just trying to hang on to this class and hopefully come out of it with some knowledge and hopefully it will help me later on in life and if it doesn't at least I can say I tried. So what am mostly confused about is the general stuff like how to use integration by parts otherwise I really don't know.
I am going to go and try to attempt the chapter 7 review one more time and hopefully I'll get some of my confusion cleared up.

Scribe

2007-02-27T20:20:00.000-06:00

What did we do today?

We discussed how the pre-test went with the substitute and the multiple choice questions were pretty straight forward and no one really had any problems. What did leave us all puzzled was the long question in the back, the "open response".

The question was . . .

(1) Let f be a differentiable function such that f " is continuous and f and f ' have the values given in the table below.

(now I wished I had the smart board so I could draw the box like Mr. K had done in class..ah the wonders of technology, I honestly can't keep up with it.)

Don't forget the mnemonic " LIATE ".

L ogarithmic
I nverse trignometric
A lgebraic
T rignometric
E xponential

So an example problem that we did that even got Mr. K, who's a math gyro (uh I don't know if I am using that word right) thinking was . . .

So to review . . .

+ When the integral is a composite of functions, you are better off using substitution.
+ when the integral is a product of functions, you are better off using integration by parts.

Last but not least next scribe is Suzanne.

BOB

2007-02-27T13:34:00.000-06:00

Everything was quite clear when we started this chapter. That is, until we got to 7.6 everything I thought I knew completely left me. I'm not sure if I'm ready for this test. The pre-test was going great until the open response question. My confidence just left me. For the next couple of days, I'll have to do some MAJOR reviewing.

Ants? Geese? Calculus?

2007-02-27T13:21:00.000-06:00

Concerned after reading your BOB's and wondering if you've forgotten these:

The excitement of an early Hall of Fame scribe

How much you supported each other

Linger’s "Won’t let you Fall"

"Lessons from the Geese"

Yes, it's second semester. Yes, the schedule is hard. And yes, there's so much to do--

Didn't you have something pretty special? Is it lost? Can it be found?

Today's Slides: February 27

2007-02-27T12:48:00.000-06:00

bob

2007-02-27T08:43:00.001-06:00

all i can say is that this unit is very hard. it made it more difficult because we only had class every other day so it was hard to remember all the things that was going on. also it was hard to do all the homework since there was alot of other homework from other classes. but i know that it is all my fault lol, i cannot blame anyone. next time i will be sure to do all my homework. p.s. i don't think i am ready for this test lol

bob

2007-02-27T08:43:00.000-06:00

BOB #7

2007-02-26T22:23:00.000-06:00

Well, everyone else more or less summed up my situation. I felt that I had a fairly good grasp of the stuff we covered at the beginning of this unit when last semester came to an end. And this semester, what with only having class every second day and missing a number of classes due to tours, illness, etc. I have forgotten what I knew. I think if I had done the test back when we were on exercise 7.5, I would have a better chance than doing it now. It's just that too much time has passed, it's like coming back from summer vacation... The pre-test was difficult for me, I used my calculator for all the multiple choice so I think they're right, but though I had several ideas for answering the long answer, I don't have much confidence in what we finally decided to hand in. Having class every second day makes me procrastinate more than usual. But however this unit goes, I am going to get back to my usual (relatively) good habits for the next unit. I promise!

BOB

2007-02-26T22:14:00.000-06:00

I am honestly disappointed in myself. I know I could do so much better. But it's all my fault that I've done so bad in this unit. I barely attempted my homework... and I feel so awful ( that's currently changing) I know I'm going to do bad in my test. I think I'm in some sort of denial. I say I want to do a good job, but then why do I keep digging this hole for myself that I can bury myself with later. If I really really wanted to do a good job then why am I not doing a good job. It's weird for me. I have that turning feeling in my gut. That's guilt. I haven't felt like this before about a class. I usually do much better. I think I've just really hit my turning point. I am, FOR SURE!!!, going to do my homework. I'm making my mom check my homework from now on! It's kind of sad that it had to come to that, but if it's what I have to do to get better than it'll be worth it. I think it's kinda late for that but at least I'm going to keep trying. We still have like 2 units left or something. Like everyone else is saying, having calculus every second day is bad! I think that it's good for me to be brutally honest because it's helps me accept that I'm not doing the best that I could and I want to change that, honestly.

Blogon the Blog #7

2007-02-26T21:45:00.000-06:00

Ooo, I almost forgot to blog! Like others I'm having difficulties with this unit. Friday's pre-test seemed fairly easy for the first part... well the multiple choice anyway. But everything was pretty much solved using a calculator, which probably isn't a good thing in the end. For the long answer part, I was a bit sketchy about. Although I'm pretty sure we'd use integration by parts to solve for the definite intergral, but still remain second guessing myself. I think I'm pretty good with my arc functions and finding the intergral that involves the usuage of them. If there is any problem solving questions available, I think doing those will sure help us understand more. And I agree with others that having calculus class every second day is different... but we just have to adjust. It's a bit troubling for me because some of us have physics every other day we dont have calculus, but I'm taking it for the first time. But, in the end there should be no excuses. I really just start to get cracka-lakin down to business!

BOB

2007-02-26T21:34:00.000-06:00

Wow, this chapter was and still is really difficult. During the class discussions and Mr. Kuropatwa's explaining I have a good grasp on the work and what's going on. However when I get home and attempt some of the questions on the assigned exercise it takes me a really long time to solve it, if I even succeed in finding the right answer. It's like the light burned out or something. I again have a lot of reviewing to do. Those antiderivitives just do not want to be found. Well I guess I'll have to read some of the text and my notes to do well on this test. A review next class would be helpful though :) I'll probably try some more of those supplementary problems. Good luck to everyone and study hard.

BOB #7

2007-02-26T18:45:00.000-06:00

This chapter on different ways of finding antiderivatives is truly a chapter that I found quite difficult. I think its the amount of methods that I got confused with. I wasn't sure whether to use the substitution method or the integration by parts method. I still understand how to use the substitution method but when I went to look back at my notes, I totally forgot how to use the integrartion by parts method. I agree with people in the class who believe that having class each odd day is a disadvantage. It's a disadvantage because for me, I end up forgetting most of the things I've learned in class. I guess that's why I do need to study each day so that when the next class comes I don't forget any lessons that I've learned from the previous class. My goal for the next chapters is to do the homework right when it is assigned and to study the lessons each day instead of leaving it to the end. Test is tomorrow and I have to admit that I'm not fully ready. Right now I feel like I really really need to study. I hope everyone studies too and does well. good-luck =)

BOB

2007-02-26T15:42:00.000-06:00

Hi everyone! We have finished Chapter 7 and we are now getting ready for a test. For me, this unit was very hard. Finding the antiderivative is harder than getting the derivative because there are no rules to find an antiderivative. From what I observed, Mr. K explains everything well. When it comes to doing the homework exercises, it's very hard to answer. I guess we just need more practice. I'm scared for the test because Mr. K gave us a pre-test without him. I hope we go over the answers before having the test. But for tonight, I need to go over some stuff and make sure I know how to solve antiderivatives. Good luck everyone!

BOB

2007-02-25T18:53:00.000-06:00

Hey hey hey... another unit down, another BOB to put up. I must say, having calculus every other day is no fun at all. I feel like there's more pressure to do the homework and to be honest, I still haven't done it all. I don't know what it is... maybe it's what Mr. K is saying, how we're tired and we don't care as much. I mean I want to do good in this class, I'm just getting super lazy. I was lost with the open response question on the pre-test. I dont think this unit was super duper difficult. I just think that I haven't been giving it a hundred and ten percent. I pay attention in class and I understand... but since we only see each other every other day and sometimes longer, the thought leaves me after I walk out of class.. and I've been putting off the homework because of all the other classes and other activities going on. I still need to review my notes and finish my homework to prepare for the test. Having calculus every other day scares me, because everything is happening so quickly. It's almost March... which reminds me, aren't we suppose to be writing that exam this month?!? Mr. K?!? Our Panda Project is far from being done, and my group still hasn't started on our video project.. I don't think anyone has. There's just always something going on. I haven't had a free lunch period in a long long time. Even with my spares, I feel like I'm running out of time. Ohhh, it's all about this semester. There's only two more units left. We're so close guys... and I want to get it over with but then again I don't. Can we please have a review day next class?!? I feel like I'm not ready for the test... && Good luck guys =)

2007-02-25T16:06:00.000-06:00

Well this class was pretty funny we were all really lost and confused with the pre test open answer question. Instead of forming three people groups like we usually do, we formed one big class group and tried to help each other out.

Chapter 7 pre-Test: Antidiffernetiation

Multiple Choice
The multiple choice were pretty much calculator work. I will just list the answers our group got.

#1) If the definite integral 1S3 (x²-1)dx is approximately using the Midpoint Rule with n=4, the error is:
(B) 1/24
#2) If -1Sk (3x²-2x)dx=6, then k=?
(D) 2

#3) If 0Sk (sec²x)/(1+tanx)dx=ln2, and k>0, then the value of k is:
(B) 3.14/4

#4) Let R be the region in the first quadrant enclosed by the graph of y=xcosx, x=0, y=0, and x=k for 0<(3.14/2). The area of R, in terms of k is:
(A) ksink+cosk-1

Open Response
This is where all of us (as a class) just became clueless. We looked at it for a long time and read it over again, but we were not 100% or even 90% sure what to do or where to even begin. We're not sure what that -2 is all about and if it's supposed to be in that missing spot in the table.

#1) Let f be a differentiable function such that f " is continous and f and f ' have the values given in the table below.

Use the information in the table to evaluate:
a) 0S2 xf ' (x²)dx
15.333...

b)1S3 xf ' (x)dx

This was the answer our group got (aka manny figured it out :) ).

Well that's it for that class. As a class we are really confused about this section, we feel we're not ready for that test on tuesday. We just need a huge review Mr K., a HUGE review.

Oh the scribe for next class is ANH :)

BOB

2007-02-18T21:20:00.000-06:00

Well, i know it's early , but im not going to delay this. It has been tough these last two weeks back
, since we only have class like every second day. =T tomorrow we have a Jr High tour, so that means our test might be on Friday? or Next Tuesday? I'll admit i am finding it difficult to even review and practice for unit seven. The test dates keep moving farther away and i do my other subjects first. I am going to try my best to get some study time down for this unit. Everything is confusing... good luck everybody =).

scribe post

2007-02-18T18:36:00.000-06:00

hey everyone as you all can see its charlene here and yes i am scribe for today. today we started off with a quiz just like every single class from now on. so the quis consisted of four questions and we got like eight minutes to do it. we did go over some of the questions on the quiz so here they are...

A) given f(3)=8 and f'(3)=4 find f(3.02)

y - y1 = m ( x - x1)
y - 8 = -4 (x - 3)
y = -4x + 20

y(3.02) = -4 ( 3.02) + 20
Y(3.02) = -12.08 + 20
Y(3.02) = 7.92

B) given y=1 and xy² - 2y 3 4y³ = 6

solve for x
x - 2 + 4 = 6
x = 4

differentiate
y² + 2xyy' - 2y' + 12y²y' = 0
y' ( 2xy -2 + 12y² ) + -y²
y' = -y^{2<.sup> / ( 2xy - 2 + 12y²}

y'(1 , 4) = -1 / 18

after reviewing some of the questions on the quiz mr.k gave us a table to fill out that consisted of the trapazoid sum, reiman sum, and the simpson sum

i wasn't allowed to upload the table so im just gonna type it on

N ..Tn ....... error ..... Mn ....... error ..... Sn .....error
4 ...0.2656. 0.0156 ... 0.2422 ..-0.0078 .. 0.25 .. 0
8 .. 0.2539 .0.0039 ....0.2480 ..-0.0200 ...0.25 ...0
16 .0.2510 .0.0010 ....0.2495 ..-0.0005... 0.25 ...0

Tn = (B + B)H / 2

Mn = REIMANN SUM
*use 0.5 as the x-choice

Sn = T + 2M / 3
* WE DO NOT HAVE TO KNOW THE SIMPSON SUM FOR ANY TESTS OR EXAM

ERROR = approximation - actual

unforntunately i do not understand the table as much as i should so i will just wait till monday to explain more clearly what is going on in the table. sry guys

by the way the next scribe is CRYSTAL

Scribe Post: Day 82

2007-02-14T21:36:00.000-06:00

It's Danny! The day started off with a multiple choice quiz which we will be having every class in preparation for the final exam.

After that Mr. K gave us three integrals to antidifferentiate. When solving these three questions you must consider these two derivative rules:

Now onto the questions and solutions:

QUESTIONS:

SOLUTIONS:

After that Mr. K talked about "Trapezoid Sums" and "Midpoint Sums". I'll sum it all up in four simple diagrams.

For Trapezoid Sums when the curve is concave down, the sum would be an under estimate. When the curve is concave up, the sum would be an over estimate.

For Midpoint Sums, if you rotate it and make it a line tangent to the curve you'll see that when the curve is concave down, the sum would be an over estimate. When the curve is concave up, the sum would be an under estimate.

Well that's all I was able to muster up for this class, as you all know Charlene was the next scribe, her post is already up. Enjoy!

I also have a very important announcement to make. Even though I was convinced to stay in the course by Mr. K himself I truly believe now that it's best if I don't continue on. I experienced something in our last class that couldn't be a better reason for my leaving. I was having trouble with completing a chart, not only because I was unaware of the fact I was doing my task improperly, but also because there was something in there that I SHOULD'VE KNOWN from past pre-calculus courses that I completely forgotten. That explains to me that I haven't done my past math courses to the best of my ability, therefore hindering my learning experiences in my present math courses. I've already begun trying to fix up my past by taking my PRECAL 40S course again. This decision hopefully will allow me to take Calculus again in University, and this time understand things completely with the mind set of thinking I know material I should know and that'll make things easy for me. One last thing, I know Mr. K you tried to keep me in your class and you succeed. However, this time I've decided and no matter how disappointed you may be, I feel this is the right thing for me to do in my life and my soon to be career in education. Please I'm not asking you to agree with me on my decision, but I'd like to know you're always there for me when I need you. I'm terribly sorry Mr. K, but I'm not going to ask for your forgiveness on my decision.

Scribe Post: 81

2007-02-07T18:33:00.000-06:00

Hello everyone! Today, Mr. K talked about Gr.9's from other schools who are going to take a tour of Daniel Mac and how we're going to be teaching them how to do several games. We decided on 5 games but one of the games is yet to be changed.

Here is the draft of whose in what group and whose in what game:

1)Game: Magic
Group:
-Jann
-Mark (a maybe)

2)Game: River
Group:
-Lindsay
-Danny
-Christian

3)Game: Hanoi
Group:
-Suzanne
-Manny
-Katrin

4)Game: Painted Faces **Has yet to be decided on**
Group:
-Linger
-Danny
-Mark

5)Game: Tangrams
Group:
-Charlene
-Anh
-Ashlynn

At the end of Class Mr. K talked about the rules of the Calculus Exam.

**keep in mind that nobody is NOT allowed to say a word about the exam EVER
**remember in the exam it's better to skip and not answer a multiple choice question than to just totally guess on a multiple choice question. ***you will NOT lose any marks if you skip a multiple choice question***...but if you totaly guess on a m.c.q. and you get it wrong, marks WILL be deducted.***

NOTE:
-starting next class we will be doing timed multiple choice quizes at the beginning of class to prepare us for the exam! DO NOT BE LATE FOR CLASS b/c we're going to be timed!

Homework: read the next exercise: 7.6 (it will help you for the next class)

NEXT SCRIBE: DANNY (for friday's class)

Scribe Post

2007-02-05T22:47:00.000-06:00

Hey everyone! I tried making my scribe as a stop-action animation but I couldn't get it to work right so I guess I'll make it the old way...

With this definite integral, we can find its antiderivative and find the area.

Let's take a look at the graph...

Solving problems like this are not always this easy as this one. We can't always find the exact value of the definite integral.

Take a look at this problem and its graph.

We could use Riemann sums to find the area under the curve.

____________________________________________________________________

These are the differentiation and integration formulas. We use these to help us determine the problems below.

Determine:

1)
Use substitution. (I FORGOT TO PUT dx INTO THE PICTURE)
u= x ³
du= 3x²
1/3du= x² dx

If your wondering why u = x³ its because

(x³)² = x⁶. So u is u². We are trying to make the original function look like d/dx Arctan(x) integration formula.

Substitute ....................................................Resubstitute

You don't need " + C " in the intermediate steps because the C are not the same while your substituting.

(There's supposed to be an equal (=) sign in between the pictures)
_____________________________________________________________

2)
If the 8 was a 9 in the polynomial we could complete the square.
So we do this,
6x-x² -8 Factor out the negative.
-[(x² -6x+8 +1) -1]
-[(x² -6x+9) -1]
-[(x-3)² -1]
1 - (x-3)² We can replace this into the integral.

Next we use substitution.

Let u = x+3

du = dx

Then RE-substitute..

And we're done

Homework: finish up arctrig exercises (7.5)

Wednesday's scribe will be Katrin.

Scribe Post

2007-02-04T22:06:00.000-06:00

THIS IS LINDSAY! On the last day of semester 1's Calculus class, which was Thursday for those who do not remember, Mr. K had to supervise the Pre-Cal Provincial Exam, so we were on our own! We were given two worksheets to do and we worked on it as a group. We only finished one problem though, but that's okay. TEAMWORK!!

Wrong Way Joe!

Joe Physics was driving down the highway on this past Memorial Day weekend. Joe's car ride was getting boring so he decided to calculate his total distance from his house from t=1 to t=3. Normally he could have used his odometer, but he drove backwards on part of the trip. Joe figured that his velocity as a function of time could be given by v(t)=3t^4cos(t). If at t=1 his displacement is 5 what is the total distance traveled by Joe from t=1 to t=3 using integration by parts?

This is what we came up with as a class. XD

On Friday we went to the University of Manitoba for a workshop on "Personal Learning Environments: A Better Learning Landscape?" It was quite fun. =) Driving down the highway with the Calculus class. I thought it was a day well spent. Bravo to Mr.K for his awesome improvisational skills.

The related rates photo is due on, I believe, Tuesday.

I hope everyone did well on their exams. BACK TO CLASS TOMORROW! Next scribe is ASHLYNN!

Scribe Post: Day 78

2007-01-23T15:43:00.000-06:00

Hi everyone! This is Jann and I'll be your scribe for today.

Today, we practiced more antiderivative questions using "Integration by Parts".
Here we go...

1. Find the antiderivatives of the following:
Q: How can we decide which function should be "u" (aka "f") and which one should be "dv" (aka "g' ")??

A: Good question! Today, we learned a new "rule". It's the...
This rule is used to decide which of the two functions will be "f" and "g' ". This rule is followed from top to bottom (or left to right if you wrote it in that type of manner). For example, if an "algebraic" and a "trigonometric" is present in the problem, then "f"= "algebraic" and "g' "= "trigonometric". According to the rule, "A" comes before "T".

Moving on...
Q: How come we did the "Integration by Parts" twice?

A: We did the technique twice because we dont know the antiderivative of "xcosx". We have to use the technique again to find the antiderivative of this expression.

Then, we answered some trig problems...
Note: "Arc(cos or sin)" is the same as "sin or cos"^-1.

The answer to letter "c" is undefined because the value is greater than 1. The answer to letter "d" is -3/10 because cos and arccos cancel each other out.

Finally we looked at this graph...
This is the graph of y= Sine. It is fixed in the domain, (-pi/2, pi/2). By doing this, the graph of sine is now a "One-to-one" function. It means that each input has 1 and only 1 output.

Refresher:

1. Vertical Line Test- this test is used to determine if the graph is a function or a relation. If the line test touches the graph more than twice (1 input has 2 or more outputs), then it is a relation.

2. Horizontal Line Test- this test is used to determine if the inverse of a function is a function or a relation. If the line touches the graph of the parent function more than twice, then the inverse of the graph is a relation.

I hope I covered most of the things we did for this class. XD

Next scribe will be Danny.