1, 2, 4, 8, 15, 30, 50, 100, ?
Sir, you never told me what number came next, hopefully you can share the secret.
I also want to take this opportunity to thank you for everything you've taught us these past two years. Due to the high intensity of our classess I know feel more prepared than ever for college and I know that I won't have any problems with calculus. Good luck on your next job,
anto
Wednesday, July 4, 2007
Monday, May 21, 2007
Final Proyect: Integration and Volumes
Here's my final proyect, check it out if you want to learn more about this topic
http://www.freewebs.com/riroman/
http://www.freewebs.com/riroman/
Friday, May 18, 2007
Final Project
Final project done. I guess this is it, besides the final. Good luck and enjoy my project.
http://www.freewebs.com/jcsp88/
http://www.freewebs.com/jcsp88/
Monday, May 14, 2007
Trigonometric Techniques of Integration
My final project was on Trigonometric Techniques of Integration and can be found in any of the following links. I apologize if you get any pop-ups. I understand they can be quite annoying. Drawings and graphs shouldn't take long to load. Either way, if you come up with any spelling mistake or other type of error while looking at it, please let me know so I can fix it. Enjoy!
Lina Garcia
www.trigint.741.com
trigint.741.com
Lina Garcia
www.trigint.741.com
trigint.741.com
Test
I was just testing to see if the built in blogger function microsoft word has could be used to publish equations. Well, it doesn't work
Friday, May 11, 2007
Luis DP's FP Solutions
1. Jim reaches the point (5, 10) after t = 5 seconds. Bob will reach the point (5, 10) after t = 1 second. Bob will win the race.
2. The orientation of the first curve is counterclockwise. When the functions for x and y are exchanged, the curve's orientation becomes clockwise.
3. (r, θ) = ( √98, 5 pi/4) (-√98, pi/4)
4. (x, y) = (-.5, (√3 )/2)
2. The orientation of the first curve is counterclockwise. When the functions for x and y are exchanged, the curve's orientation becomes clockwise.
3. (r, θ) = ( √98, 5 pi/4) (-√98, pi/4)
4. (x, y) = (-.5, (√3 )/2)
Problems Posting Answers
I'm having two major problems posting the answers. The first one is that they are to big to post on bubble share, so I don't know how to post them atm, the second one is that I'm having trouble working out one problem. I'll present the annotated answers on monday, in hopes of getting some partial credit, or maybe just some corrections on my answers.
Wednesday, May 9, 2007
MAC's Beauty Post
I recently received a forward called "La Simetria de los Numeros" and I was really amazed by it. Mathematics can sometimes be beautiful... I wanted to share this with all of you and that is why I'm making this post. Look at the patterns.
1.
1 x 8 + 1 = 9
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321
2.
1 x 9 + 2 = 11
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111
3.
9 x 9 + 7 = 88
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888
4.
1 x 1 = 1
4.
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=12345678987654321
Amazing... check the answers if you want, they are real. I like them all, but my favorites are numbers 1 and 4, which are yours?
Tuesday, May 8, 2007
AP TEST
The AP Exam is finally here. I just wanted to say, good luck to everyone that is taking the exam. Hope you all get a 5.
Friday, May 4, 2007
Final Draft: Problems
This are my final Problems. I change the very much from my first problems and I add a word problem. If a problem is not well elavorated or needs something please add a comment and tell me. Thanl You.
Differential Equation Problems
1. A new sport called kiteboarding gets to Cartagena. Initially 10 persons starts doing kiteboarding. After 2 years passed the number of persons that are now doing kite boarding is 120. Find the equation at any time. What would be the number of people doing kiteboarding in 50 years?
2. Solve the equation.
Differential Equation Problems
1. A new sport called kiteboarding gets to Cartagena. Initially 10 persons starts doing kiteboarding. After 2 years passed the number of persons that are now doing kite boarding is 120. Find the equation at any time. What would be the number of people doing kiteboarding in 50 years?
2. Solve the equation.
3. Solve the IVP.
4. Make a slope field with the next equation. 
Problems Final Draft
Well this is the final version of my problems. However I'm still open to suggestions on how to incorporate some word problems into the mix, since this compilation of problems seem quite plain and boring
Thursday, May 3, 2007
Tuesday, May 1, 2007
Monday's Slides 4/30/07
Today, we solved a few more inverse trig problems. A couple of interesting topics arose.
The first slide shows an integral that required an inverse secant and substitution. Don't forget to change the limits of integration when you make your substitution.
The second slide shows a derivative of an inverse cosecant function that also requires the chain rule. Anto's solution involved inserting an x^4 inside a square root. (To explain this step, he showed an example of the technique in the upper right corner.) Can you see another way forward that avoids inserting x^4 inside the radical?
The last slide shows a definite integral of a rational function with trigonometric expressions in both the numerator and denominator. Juan used the wise step of doing some algebra before doing the calculus. This step avoids using any inverse trig functions. The problem is that he forgot to extract both the negative and positive roots. It turns out we only needed the negative root because in the interval [3,4], cosine is negative.
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Then we looked at the answers to the multiple choice questions from review sheets 1 & 2.
Be prepared on Wednesday to ask questions on any problems for which you have confusion or doubt.
The first slide shows an integral that required an inverse secant and substitution. Don't forget to change the limits of integration when you make your substitution.
The second slide shows a derivative of an inverse cosecant function that also requires the chain rule. Anto's solution involved inserting an x^4 inside a square root. (To explain this step, he showed an example of the technique in the upper right corner.) Can you see another way forward that avoids inserting x^4 inside the radical?
The last slide shows a definite integral of a rational function with trigonometric expressions in both the numerator and denominator. Juan used the wise step of doing some algebra before doing the calculus. This step avoids using any inverse trig functions. The problem is that he forgot to extract both the negative and positive roots. It turns out we only needed the negative root because in the interval [3,4], cosine is negative.
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Then we looked at the answers to the multiple choice questions from review sheets 1 & 2.
Be prepared on Wednesday to ask questions on any problems for which you have confusion or doubt.
Finding squares of numbers ending in 5
I came across this website showing how to easily compute, in your head, the squares of numbers ending in 5. The formula is:
(n)* (n+1) + 25
Now, the trick is that the 25 isnt "added" it is just the last two digits of the number you find computing the first part of the formula.
Example
1. 85*85
In this case n = 8, therefore
8*9 + 25 = 7225
2. 135*135
In this case n = 13, so
13*14 + 25 = 18225
(n)* (n+1) + 25
Now, the trick is that the 25 isnt "added" it is just the last two digits of the number you find computing the first part of the formula.
Example
1. 85*85
In this case n = 8, therefore
8*9 + 25 = 7225
2. 135*135
In this case n = 13, so
13*14 + 25 = 18225
Monday, April 30, 2007
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).
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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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
This album is powered by BubbleShare - Add to my blog
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?
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.
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.
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?
Yes, I will be working while most of you are vegetating.
Anyone else working?
Sunday, April 29, 2007
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.
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.
Saturday, April 28, 2007
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?
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?
Friday, April 27, 2007
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.
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.
Wednesday, April 25, 2007
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
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.
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.
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