Luis DP FP Rough Sketch

1. Jim and Bob are racing from the origin to the point (5, 10). Let t be the number of seconds after the start of the race. Jim's position at any time t is given by the parametric equations x = t, y = 2t. Bob's position at any time t is given by the parametric equations x = 5t, y = 10t. Who will win the race? How long does it take each competitor to finish the race?
2. The following plane curve is a circle: x = 2 cos(t), y = 2 sin(t), 0≤t < 2π. Is its orientation clockwise or couterclockwise? What happens when you reverse the parametric equations, so that x = 2 sin(t), y = 2 cos(t)?
3. Given a point in rectangular coordinates (x, y), express it in polar coordinates (r, θ) two different ways such that 0≤θ < 2π: (x, y) = (- 7, - 7).
4. Given a point in polar coordinates (r, θ), express it in rectangular coordinates (x, y): (r, θ) = (1, 2π/3).

Rough Sketch of Problems

Well I've encountered a prblem. Once I did the equations in Word and pasted them here the equations get messed up and don't show properly. I've tried several things but nothing seems to work. Help would be appreciated, I've decided to post what happens just as proof that I did the work.

1. 12xsinxdx' type="#_x0000_t75">

2. 12x²cosπx' type="#_x0000_t75">

3. 1224ex' type="#_x0000_t75">

4. A car is traveling from Bogota to Cartagena. The car’s acceleration is given by:
12at= t²lnt' type="#_x0000_t75"> Were time is given in seconds and distance in m
Find the distance the car traveled 10 seconds.

Another thing I'm worried about is that my topic has no word problems in the book, making it really hard to come up with some on my own. I will try to come up with at least one more.


Just going to try bubble share to see how it will look like

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Final Project Rough Draft- Andrea

1. Sketch a derivative. Given a graph, sketch its instantaneous rate of change.
2. Analytical and Numerical approximation. Given an equation of the velocity travelled by an airplane as a function of time, find the instantaneous rate of change at t=a using analytical (use the definition of the derivative) and numerical analysis.
3. Solving difficult derivatives involving chain rule, ln, and trigonometric functions. Given a function that describes the position of a roller coaster as function of time, find the velocity at a specific interval of time.
4. Implicit Differentiation. At the instant that two cars, A travelling from west to east at 5 miles of the intersection point and B travelling from east to west at 7 miles of the intersection point a police man standing at this point measures the instantaneous rate of change when car A travels at 70mph and car B at 45mph. What will it be?

Final Project: Rough Sketch

My final project is about limits. This is a rough sketch of what my four problems would be.

1. A teenager that goes to the gym, is moving a box. The boy pushes the box hard, and it starts sliding with a friction force that oppose the motion created by the push. If it is 130 pounds, the friction is the same as the force the boy uses to push the box. If it goes over 130 pounds the friction force will equal 100 pounds. With this information sketch a graph of friction as a function of the boys applied force. A). Where is the graph discontinuous? B). Keeping this description, try to make the graph continuous.

2. The pupil of an alligator is given by the function F(x) mm. The variable x is the intensity of light in the pupil. If there is a given function f (x)= 195x-.6+75/4x-.6+ 25. A). Find the diameter of the pupil of the alligator with the maximum light.

3. After a boy was injected with a medicine, the docotr wanted to know the concentration of drug in the muscle according to the function of time f(t). Suppose that t is in hours and the function is f(t)=e-.07t - e-.94t. Find the limit as t approaches infinity.

4. Find the limit as x approaches infinity (e-6x cos 6x2)

As Anto I had various problems as well. When I tried to coyp the limit symbol into the post it showed me something very strange. Also the power values. If I can get any help I would apreciate it.

Test Case: Answer Tips

I'm trying a new feature on the AP blog only.

Double click on any word in the blog that is not a link and see what happens.

Let me know what you think.

I'll keep it if it works well and does not have any nasty side effects.

Bubble Won't Share

Bubbleshare is down so today's slides will be posted Tuesday morning.
Yes, I will be working while most of you are vegetating.

Anyone else working?

Final Project HELP!

Hi everyone,
I was writing the rough sketch of the problems for my final project and I realized I can't write integrals or square roots as you normally would write them on paper, but as you would type them in to the calculator, what makes them much harder to understand. If anyone knows how I can write my problems in a pretty way like we would see them on the mimio, please let me know.

Mentor Call

The 12A blog now has around 20 new posts. Please go there soon and offer your assistance.

Also, for those helping the 11th graders, your posts are excellent. Please share them with both sections. A simple cut, paste, and post will go a long way.

Thanks again for your wonderful work.

Any mathematical insight to Mac's optical illusions on the 11A blog?

Final Project due date and topic

I chose to do my project about Integration by Parts. My due date is going to be May 18, because like Luis said, its better to have the greatest amount of time possible in order to hand in a great project.

Friday's Slides 4/27/07

We started today by finishing the batter problem and determining the distance at which the batter can no longer track the incoming baseball. (Slide 1)

The we solved a few derivative and integral problems that featured inverse trigonometric functions. We eventually derived a formula for inverse secant. The absolute value is there because of the definition of y = sec^(-1)(x). The function is always increasing over its entire domain.

The last problem involved the derivative of the inverse cot function. We solved it by deriving a formula and then applying it to the specific case. Alternatively, we could have used the relationship cot^(-1)(x) = tan(-1)(1/x) and used the inverse tan derivative formula. You should try this.

You should also try deriving all of the derivative formulae for the inverse trig functions using the general relationship between the derivative of any inverse functions. Do you remember it?

d/dx(f^(-1)(x)) = 1 / f'(f^(-1)(x))


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Review sheet # 2 should be started (finished?) over the weekend.

Luis DP's FP Due Date

5/18. Not like a last-minute man. More of an all-minute man. You can never have all minutes unless you choose 5/18. It's how this minutism works. I'm giving the lesson much earlier though.

2=1

Ok, so, the following operations (if you can call it that way) were posted by someone on the MIT Facebook group. The idea was to show that 2 = 1 only that everyone else in the group posted a thousand reasons why the proof was incorrect that it made me feel bad for the person who made the post. Either way, I just wanted to post it here to see what you think about it and if you could come up with different reasons why this may be valid or invalid.

x=1
x=x
x^2=x
x^2-1=x-1
(x^2-1)/(x-1)= (x-1)/(x-1)
x+1=1
2=1

Wednesday's slides 4/25/07

Today we continued exploring the derivatives of inverse trig functions.

The 1st two slides show problems that require the use of the chain rule and the derivative of the inverse sine function.

We then returned to the derivation of the derivative formula for the inverse sine function. During the derivation we ignored the +/- square root when solving for cosy. We must ignore the negative root because the inverse sine function is increasing over the entire domain. A negative root would imply a negative slope to the curve and this is not the case over the domain [-1,1].

We then derived the formula for d/dx(inv cosx) and d/dx(inv tan x). (see the slides for the formulae)

Lastly we returned to the batter problem of yesterday and finished it. The batter would need to track at 65 radians/sec to keep his or her eye on the ball. Mr. A. reported that scientists estimate a limit of 3 radians/sec as the maximum rate at which humans can track a moving object. The last slide addresses this limit.



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Homework: Page 542 17-31 odd, 45

Keep working on your final project.

Lina's Final Project

I'm changing my project topic from "Extrema, Optimization and Concavity" to "Trigonometric Techniques of Integration", so if anyone wants my former topic, he or she may do his or her project on it. Also, sir, if I'm not mistaken, you said we were going to cover the new topic I chose after the AP test. Therefore, do I need to have my project done earlier or do I still have until May 18 to hand it in?

Tuesday's Slides 4/24/07

Today we revisited inverse trig functions. Of particular importance were the domain and range of the inverse sine and the inverse cosine functions. Sine and inverse sine are inverses only if we restrict the domain of sine to [-pi/2, pi/2]. Otherwise, the inverse sine function would fail the vertical line test. Having thus restricted the domain of sine, the range of inverse sine is also restricted to [-pi/2, pi/2]. The domain of the inverse sine function is simply the range of the sine: [-1,1].

Remember with inverse functions:

• D-->R
• R-->D

Likewise, the cosine & inverse cosine functions are inverses only over the interval [0, pi], therefore the range of the inverse cosine function is also [0, pi]. See slides 1 & 2 for a graphical interpretation.

Next we started a problem involving the derivative of an inverse trig function (see slide 3). We did not solve the problem as we do not yet know how to differentiate the inverse tangent function. We did derive the derivative formula for the inverse sine function (slide 4). The derivation process is a little hard to think of but is algebraically pretty straight forward. The key is rewriting the inverse trig equation as the corresponding trig equation.


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Tomorrow, we will derive the derivative formulas for the other inverse trig functions & then finish the batter problem.


Keep Bloggin'

Final Project Timeline

I hope that each of you have begun working on your final projects. As you prepare your instructional presentation, you are likely to find that many unexepected issues will arise. Please give yourself time to address these.

To help you, I have created the following timeline for your projects.

I am looking forward to seeing what form of technological wizardy some of you may choose to employ. If this is something that concerns you, please begin a conversation with Mr. Moyano soon. Many of the available tools are easily acessed but beginning users often experience a period of frustration as little glitches crop up. I am still stumbling through these little traps in managing the class blogs.

Good luck and be sure to let me know how things go.

Mentor Tip

You have probably already found that many of the students in the other classes were not very precise in the formulation of their questions. If you are having difficulty in finding a particular question to which you might respond, add a comment directing the student in question to be more specific and precise in communicating his or her needs. Try guiding them so that they might help you help them.

Blog Rubric

Below you will find the rubric by which you may measure your level of achievement in earning the blog grade that you desire.

Although it is not incorporated into the rubric, each level of performance requires that your posts and comments be respectful in tone, content, and language usage.

Click to see a larger image.







Please attach a comment if you any ideas that would improve the quality of the rubric.

Tiling

I took the following picture through the shop window of a business that is located just behind the physics lab.

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I have been passing the tiling for weeks (probably much longer) but I recently started thinking about it. I have some questions for you to consider. I do not know the answers but I am interested to hear some of your thoughts.

How was this tile pattern generated?

One obvious answer is that someone said something like, "Ok. In the first row I want 4 whites, 1 gray, 2 whites, 1 blue, 3 whites, 1 gray, and 1 white. For the second row, I want 1 gray, ..."? Do you think this likely? Explain.

If the pattern was not created tile by tile, how was it done? A likely answer was using a computer program. But what type of instructions would the programmer give?

The name of the pattern is "Degradado Azul." Starting at the top and going from one row to the next, the number of blue tiles sometimes increases and sometimes decreases. Overall, though, the rows start without having almost any blue tiles and then end up, in the bottom rows, with all or nearly all tiles being blue. The number of blue tiles in each row must be increasing even though it sometimes decreases.

Do you see any other patterns? Could there be some sequence formula that a computer could use to generate the number of blue tiles in each row? Given the number of tiles, in what positions should the blue tiles be placed? How would that be decided?

Do you see any other patterns?

Do other questions occur to you?

Gera 1st Assignment

1. The forgetting curve does not seem accurate. The curve is exagerating because I think the mayority of people in the world dont lose more than 50% of the information the next day they learned it. The forgetting curve is very general, because obviously there are people that lose 50% of the information just a few hours later. In the other hand, I agree with the curve that is created when you study each day. People in the world that are not good at math, but they practice it each day, would do a better job. Right now I dont have study habits, but I know a have to start working on them because I would need them for collage.
2. I am having problem with the word problems. Ive hated them all of my life, because most of the time I get stuck when I am trying to find the initial equation. This year I am having problem with the optimization problems, because it envolves word problems. I dont have any questions about this topic.

Assignment 1 by Luis

100% to 50% in 24 hrs?! Not so drastic actually. I’ve seen people drop from 100 to 0 as they walk out of the classroom. So that curve is way too general. I enjoy my learning fully. I have no regrets. However, I have a tip for you math lovers, even if you never struggle. Solve your mysteries in the shower. During your sleep. While you walk. Unleash the demon inside you. That demon that dies and lives for knowledge. Enjoy time where there is none. Life is too short to be passive. I wish I could follow my own tip fully.

I do have a question about the test: As one deals with an equation and performs integration, constants come up. As we have witnessed previously, a constant may vary throughout the process of simplifying such equation. And although the last step or result is what matters, the previous steps are false. Does the AP test require the test-takers to point out which constants are different from one another? It might sound silly and the AP probably doesn’t care about distinguishing constants, but I don’t think we want to give ourselves excuses to lose any point on that test.

Gusti: 1st Assignment

1. The forgetting curve is seems a real conclusion about how your brains work with the things we learn everyday. This demonstrates that homework really work and revising it every some time works a lot making your life easier in mid terms. My study habits are very weird. If I don't get out of my house in the evenings I would eat and sleep all day and would not do nothing of homework. But if I go out and do some of kite boarding or go out and do something in the night I would do some homework. The thing is that in the night I don't have time to do all my hw and by study habits go down. This shows why I forget things very fat because I dont study all days and in a good way making me forget things easier and faster.
2. I am having trouble in the topics of Extrema, Optimization, and Concavity. I don't know why really bust since y learned this i have had problems. I don't really have a question about this topic because I don't know it in general.

Another Blogging Tip

Give your response a meaningful and relevant title. That way, when the post appears in our blog archive (see sidebar), everyone will know the topic.

You can also attach a label in the bar below the post so future bloggers can more easily search for the post. For instance, I attached the label "tip". Later, when more tips have been posted we can find them all by searching for "tip".

See my earlier post that I labelled, "Forgetting Curve." Later, I can go back and find it easily.

Blogging Tip

When creating a new post, use the built in spell check.

We all make mistakes, but remember, what you write goes out to the entire world. You can make yourself look better with just a simple click.

Lina's Forgetting Curve

1. The curve does seem accurate if we think about it in a wide context. If I compare that curve to how I forget math concepts, forgetting 50% of the material in one day does seem extreme. However, think about other subjects or perhaps other people. If I compare the curve to how I forget the material covered in philosophy or sociales, forgetting 50% of the material in one day does seem accurate. Actually, there are times in which I’d probably forget more. Now try to put your shoes in those of people who you know struggle with math class. They probably do forget 50% of the material after one day. So the curve does make sense and seem accurate as an average of how people forget class material but maybe not for a particular case.

The curve created by reviewing, however, seems accurate for any case. If you don’t practice, you’re most likely to forget. How many times have we had to do some Pre-Calculus problem and struggled with it because we couldn’t remember the equation for a parabola? If we were to review last year’s material every month or so, we wouldn’t have that type of problem.

To be honest, I don’t have study habits and that’s probably one of the things I have to work on the most when I get to college.

2. The definition of a limit was very confusing, especially the thing we saw on the computer with epsilon. Mid-point Riemman sums were challenging at first since I missed a class to go to the Hay Festival with Ms. Monroy (big mistake) but it was all clear after some practice. Learning the derivatives and antiderivatives of trigonometric functions has and will always be a torture (the problem of not reviewing). I don't have any question at the moment (sorry).

Answer to Assignment #1

1. The numbers don't seem accurate, they actually seem a bit extreme, I mean it doesn't seem normal that you lose around 50% of the information you learn in a lecture by the next day. However, the new curve you can create by studying each day does seem accurate and not very complicated to do. To make math easier for me I should get in the habit of studying a little bit of older topics every day, to keep them fresh in my mind.

2. The two topics I had the most trouble with were finding the volume or areas of shapes because I have a hard time visualising things in my head. After practicing some problems it got easier but its still challenging for me. A question I have about the subject is what short cut or tool can you use to find the shape of the object your trying to find the volume of.

Welcome

You found it!!!

Welcome to our mutual venture into the land of math blogging. This is the place to come when you did not quite understand that last topic from class, or you were too shy to ask your question, or you want to share an interesting and useful math website or new problem-solving strategy, or maybe just to chat about your math struggles and/or successes.

As with everything, you will get out what you put in.

So let's get started. Here is your 1st assignment. The first 2 people that respond appropriately will receive a 105% for this assignment. (It will be your job to help others figure out how to get here and post.) Everyone must respond by midnight, Friday April 13.

Go here http://www.adm.uwaterloo.ca/infocs/study/curve.html and read about The Forgetting Curve.

1. Describe how this relates to your study habits. Do those numbers seem accurate? What specific changes could you make which would make learning math easier for you?

Blogging is just one way to revisit a new piece of information. Below each post you will see a section for comments. Use these comments to help and learn from your friends but also as way to review new info. Keep that curve high!

2. Think back over this past year and identify one or two particular calculus concepts that were, and maybe still are, particularly troubling for you. Ask 1 or 2 questions (on the blog) about these topics whose answers might help give you a better understanding. Remember, the final still awaits. This is your chance to get your questions heard.

Once you finish your assignment, go here http://oos.moxiecode.com/examples/cubeoban/ to play a fun and deceptively challenging game. Level 1 is automatic. Level 2 is a quick hello. It's not until level 3 that you will appreciate the game. Remember, this is for AFTER you finish your assignment!

Happy bloggin'.

Your second calculus joke.

Joke 2

Two mathematicians were having dinner in a restaurant, arguing about the average mathematical knowledge of the American public. One mathematician claimed that this average was woefully inadequate, the other maintained that it was surprisingly high.
"I'll tell you what," said the cynic. "Ask that waitress a simple math question. If she gets it right, I'll pick up dinner. If not, you do."
He then excused himself to visit the men's room, and the other called the waitress over.
"When my friend returns," he told her, "I'm going to ask you a question, and I want you to respond 'one third x cubed.' There's twenty bucks in it for you." She agreed.
The cynic returned from the bathroom and called the waitress over. "The food was wonderful, thank you," the mathematician started. "Incidentally, do you know what the integral of x squared is?"
The waitress looked pensive, almost pained. She looked around the room, at her feet, made gurgling noises, and finally said, "Um, one third x cubed?"
So the cynic paid the check. The waitress wheeled around, walked a few paces away, looked back at the two men, and muttered under her breath, "...plus a constant."


Dedicated to Antonio - or as I like to call him - Mr. C.

Final Project

The Assignment

Think back on all the things you have learned so far this year 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 11, 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 2 Canadian teachers specifically for this purpose). Feel free to mix and match the tools to create something original if you like.

Topics:
You will each choose one of the following topics to address:

• Limits
• Differentiation & the meaning of the derivative
• Implicit Differentiation & Related rates
• Extrema, Optimization, Concavity
• Integration, Areas
• The Fundamental Theorem of Calculus
• Differential Equations
• Integration & Volumes

A consideration of analytical, graphical, and numerical approaches, approximation versus calculation, and applications, where possible, would help you to plan a thorough explanation of your topic.

Choice of topics will be given in order of request. The first person to post gets 1st choice. The last person gets whatever is left.

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) Your presentation must be published online in any format of your choosing. Experts are recognized not just for what they know but for how they demonstrate their expertise in a public forum.

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

• Why did you choose the concepts (not the unit) you did to create your problem set?
• How do these problems provide an overview of your best mathematical understanding for your topic?
• How did the publishing requirement effect either your enjoyment of or ability to complete the task?
• 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.)
  
  
  
  Labels: final project