On limits and life

These last two weeks I've been studying limits in various ways, from various sources, and different media. I've used videos, sites, ebooks, paper books, ad apps. It is great to see how each reference approaches the topic differently, and how the central concepts emerge from the gathering of information.

I am starting to see why many people regard Calculus as beautiful. Being able to find the equation of a line that is tangent to a curve using limits is breathtaking. I am not kidding. There was something awe inspiring when out of some algebra and some elementary Calculus I arrived at a formula, that when graphed, touched a curve at exactly one point before continuing in its path. Can you see the significance of that? These two functions for an instant, touched, and then moved on.

To delve deeper into this existential stream of consciousness about Calculus, consider that a limit tells us about where a function is heading. It doesn't care about what happens when the function gets there. It cares more about what happens as it gets closer and closer to a given point. Therefore, in Calculus as in life, the most important thing is the journey rather than the destination.That gets me to think about the people we meet briefly once in our lives. Those chance encounters might not change our paths, but there is a cosmic record that they happened. At that moment when two people meet briefly, you can describe each of them as being together. In other words we could say that at 9:00pm on Sunday, May 19th, 2013, John met Mary and Mary met John. Therefore John and Mary became part, in that instant, of the greater formula of life as variables sharing the same time and space coordinates. And that point in time, much like a limit, gives us information about where each of them is and how they are behaving.

I am liking this calculus stuff more and more each day.

Let me know your thoughts.

 

Slight Detour: The story of a cube

As I was gathering resources and reference for Calculus. I ran by a page (that I cannot locate now), of a math tutor that made a small comment about how he also had a Rubik's cube solving page.

 

Flash back at least 11 years ago.

 

For a holiday (I can't remember which) my girlfriend of at least 8 years years (who is now my wife of 10 years) gave me a picture cube as a gift.

After briefly looking at all the pictures of our life together, in fast, deliberate motions, I scrambled the cube. The look of horror on my future's wife face is still etched in my mind.

 

I looked down at the cube and shared her concern. I had never solved a Rubik's Cube. We knew no one who had, therefore this cube of our pictures would never be rearranged again.

 

I gave it all I had, for hours and days and weeks I tried to solve it. But I could not. The cube sat in my car for at least two years. One day, after we got married, my wife found it and gave it a sad forlorn look. She took it and stored it with the rest of our pictures and heirlooms. I felt like the worst person on earth for being foolish enough to scramble that picture cube.

 

Flash forward to two weeks ago.

 

When I saw that page on how to solve a Rubik's Cube using algorithms. I knew what I had to do.

 

As fate had it, one day we were shopping at Party City and they were selling mini Rubik's cube for 89 cents. I grabbed one as casually as I could and bought it.

 

Like I said, I never found the first page I used, but afterwards I got these others and these were the ones that helped me the most. Beginner's Solution to the Rubik's Cube. Beginner's Rubik's Cube Solutions. In the beginning, as I read these I felt overwhelmed, almost as overwhelmed as I felt doing Calculus without precalculus back in January. They were talking about Faces, primes, clockwise and counter-clockwise moves. They also mentiones middle pieces, corner pieces, edge pieces. Then they would give me string of algorithms that looked like this: R2 U F B' R2 F' B U R2.

 

There was no way this could work, could it? My first attempt with my mini cube, after at least 3 hours, ended in utter failure. My wife, who thought I had gotten my self another hobby besides calculus was not impressed. I had kept my real intentions secret from her. And on top of everything, she thought I was just cheating using formulas to solve a cube. At the time, so did I. I felt like I was painting by numbers. All I had to do was follow those instructions to the letter and I would solve a cube.

 

By the time I had solved my mini cube, I felt otherwise. This was not painting by numbers. Solving a Rubik's cube with the beginner method was about recognizing patterns and executing moves to make the pieces go where you wanted them to...without messing with the other pieces you had already done.

 

After solving my mini cube, I did not feel ready enough to tackle our picture cube. So I looked for more practice. I got the Rubik's cube app for my iPhone and Ipad and bought a "magic cube" from China on EBay. My goal was to practice with these as much as I could.

 

As I waited for my Chinese cube to arrive, I practiced with the app. I found it hard to use at first because the controls were weird. But after a while, I was "fluent" in it.

Still remembering the combination of moves that I needed to make and when I was supposed to make them still took me a long time.

 

My first time solving a cube in the app took me 1 hour 49 minutes. And I had to refer back and forward to my notes. More than once my wife found scraps of paper with algorithms on them...I got some weird looks from her.

 

The great thing about having the app is that I could fire it up at a moments notice if I was waiting in line, in an elevator, or in my lunch break. I could even pause the game and pick it up later at night right before bed. I was getting good practice out of this. After 5 tries I could keep my time under an hour. After 10 tries I could solve it a shade under 15 minutes. In the last 5 tries I could solve a cube in about 6 minutes.

 

I now felt ready to solve our picture cube.

 

All last week I have been asking my wife if she remembered were we had put that picture cube. I still did not tell her what I wanted to do, but I knew she knew. What I bet she did not know was that I could solve it this time. After a few attempts at finding it, and making a mess of our closet, I got my hands on it. The cube felt weirdly solid in my hand. As if it had gathered the mass of 10+ years just waiting for this moment.

 

I looked at it and remembered how hard I had tried to solve it the last time. Yet, all that was solved was a single face. It showed a picture of a kiss I gave my future wife after I had caught the garter from my brother-in-laws wedding. I would propose to María soon after that. It seemed fitting that I would attempt solve this cube on the year of our 10th wedding anniversary.

 

However, after I look a the rest of the pictures, I panicked. I could not tell which pieces belonged together. Many of the pictures had parts that were almost the same color. This would not be as easy as I had hoped. María was walking around the house doing some stuff and would check in me from time to time. When she did I asked her questions that must not have been reassuring to her like: where do you think this piece goes? She would give me her best guess and walk on.

 

After about 20 minutes it all clicked and I was back on track. A minute later I had solved all the pieces. However, I was not done. This is a picture cube which has a key difference than a regular cube.

In a picture cube the center pieces need to be rotated to fit the rest of the pattern. If you don't, you will have a solved cube looking like my mini cube did: I bet Jessie would love to have her face back in its proper orientation. Before you say anything...yes, my mini cube was of Toy Story...moving on!

Having centers not rotated properly can be quite distressing to many people. Because you can actually see it happening while you are solving the cube. If you type, Rotate center pieces, in google, you will get dozens of hits on this problem and how to solve it. The one I found to help me most was this video. The process involved a very simple algorithm repeated 5 times to turn a center piece 90 degrees clockwise and another center piece counterclockwise.

To finish my cube I would have to do the algorithm twice. Since I had to do 4 moves to complete the instructions once. That meant I had to do 40 moves to rotate all center pieces to their proper place. If I make a single mistake, I could scramble the cube in such a way that all my effort would be wasted. And guess what, at move 38, I lost track of my next move. I had no idea what I had to do next and the cube looked horribly scrambled.

I was heartbroken. I knew I was really close, but if I made the wrong move It was over. I just sat there, looking at all the faces of the cube and see if I could get back on track. A week ago I would have been lost, but after all that practice, I could see the relationships between pieces clearer. I felt a rush of adrenaline as my brain registered that I had already solved the cube. It must be the same rush a chess player gets when he or she knows the game has already been won a few moves ahead. I took a deep breath and mave the last two turns.

And there it was...solved. After all those years.

I showed it to María and the look on her face was amazing. I sat her down and told her how this cube had been on my mind for many years. how I always wanted to solve it. To rearrange all those pictures of our lives together. It was symbolic for me. And then she told me something that melted my heart.

A Rubik's cube has 54 squares. 9 squares per face per 6 faces. María told me that When our picture cube arrived the first time her hands. She was disappointed. The sticker that had the pictures looked really fragile to her. She knew that unless she did something, it would deteriorate and fade really soon. So she cut 54 squares of clear adhesive paper and covered all of them, one by one. It must have taken her hours of painstaking and risky work to get that cube to me. No wonder she was so chocked when I just scambled it the first time, and so sad when she found it still unsolved.

Now that we had it back we put it in a place of honor. Our bedroom table. To be looked upon by us and our little girl for many years to come.

 

Pedagogical Quandary

It's been a while since I last wrote. I have been busy studying but not writing about it. There's no good reason for me not writing, even when I have made little progress. I guess if I wrote about what I've been doing it would be repetitive.

 

I am starting Calculus on my own with no formal teacher or course. I am relying on three textbooks and video lessons. Still after starting Calculus a couple of weeks ago, I have not gotten that far because all three books start with pre-calculus. The only resource I have that starts with Calculus are the video lessons but they don't give me any homework exercises to practice.

 

Therefore here is my quandary: should I skip the precalculus and go right to calculus since I have 10 weeks worth precalc? Or should I glance over what the books have to say about precalculus just in case they shed light into how they will cover calculus?

 

I have been inclined to do the former and check out the preparation chapters. That means that I have studied the precursors of limits (slopes of secant lines) at least in three different ways. At least, each time I get the introduction to limits I understand them better.

 

My other quandary is with practice exercises, I want to get as much homework as I can but all my books only have the answers for the odd numbered items. That took me back to school. I remember being assigned odd numbered items at home for practice and even numbered items for hand in assignments. It always made me anxious when I could not check wether or not I had done the exercises well.

 

In high school, Mr. Quintero, our notorious but brilliant math teacher changed all that. He had no problem assigning odd numbered exercises to hand in. He realized the back of the book only gave us the answers, so he would put all the weight in the process. That way you could either get the exercise right and check the answer, or get it wrong and work through it to find out what happened. I found this approach far more effective and instructive.

 

Without a teacher's help, all I have is the answers provided by the book to know if I am right. I guess that is a trade off I'll have to work with.

 

What do you think?

 

The two questions of Calculus

At last on day 113, I dive in into Calculus. What I write bellow is my interpretation of what I have read me researched. If you are using this post as reference, I suggest you double check my statements. I am by no means a reference source on calculus, just another student trying to learn it.

 

Calculus, from what I have learned far, seems to be the study of change. It deals mostly with two major subjects differentiation and integration. If fact I finds some sources that specifically reference differential calculus versus integral calculus.

 

If I use Professor Edward Burger's approach to explain calculus from one his Thinkwell videos, then calculus is the mathematical discipline created to answer two mayor questions: What is the instantaneous velocity of an object? And What is the area or volume of an exotically-shaped object? The first question is in the realm of diffential calculus and the second belongs to integral calculus.

 

All courses I am reviewing right now, start with differential calculus.

 

In differential calculus I read that the original question that started the discipline was finding the slope of tangent line of a curve. I found this surprising because I remember drawing or working with tangent lines and alopes back in intermediate school geometry. However, when I read deeper I was blown away with the reason the slope of a tangent was such a problem.

 

A tangent is a line that intersects an object, like a circle or a curve, at one (and only one) point. That's the rub. Way back in my geometry class, I learned that you needed at least two points to make a line. The difference between this two point will give us the rise and run of the line, which is its slope. So how can Impossibly find the slope of a tangent line to a curve if a tangent is only one point? The answer, learn calculus. The short answer, and the heart of all calculus it seems, is to find another point in the curve that is sooooooooooooooo close to the first point that the distance between them is infinitesimal and therefore negligible.

In The chart opposite, as point point B in the curve get closer and closer to point A that is also in the curve, the line that passes through them looks more and more like the tangent of the curve.

 

Now imagine the curve actually represents a car's velocity traveling on a straight line. Then point A is we're the car is at time x. In algebra we can find the average velocity between A and B1 by subtracting the miles traveled by the time taken to travel those miles. However, how can I find the exact velocity (Instantaneous velocity) of the car at point A? If I make the time interval between A and B small enough as to make close to an instant, then I can Algebraically compute an approximation of instantaneous velocity.

 

In integral calculus, the questions searched are a little different.

 

What is the area of an exotic shape? And from what I have studied, exotic shapes just mean shapes other than the ones we have formulas for. We have formulas for squares, circles, triangles, cubes, spheres, pyramids and if my memory doesn't fail me, cones. I bet we also have formulas for many other shapes, but how about the area of the shape in the chart below.

 

How do we measure this shape's area?

 

Well, we could inscribe it in a grid of squares of a given size and count the squares that the shape fills.That would give us an approximation of the area we are looking for.

If we start to make the squares smaller and smaller, more of the shape is inscribed within it.

We can repeat the process of making the squares smaller and smaller, to get better approximations. If we make those squares infinitely small, the value of the area of the shape will be so close to its real area that the difference would be negligible.

 

Therefore, the previous act of getting the tangent of the curve and now the act of getting the area of an exotic shape, relied on the same procedure to get answered. We used infinitesimally small numbers. In fact, I have run across various references to calculus as infinitesimal calculus. If we want to make the jump from algebra, which can only give us approximations of the answers to these questions, to calculus , where we can get the exact answers we are looking for, we must go through Limits.

 

And so must I, next time.

 

What do to think?

 

 

 

Beginning Calculus in fits and starts

After 10 weeks of taking a class, it has been hard for me to figure out a way to start Calculus. The past few days I've been collecting resources to help me study. I have videos, ITunes courses, websites and more. What I now need is structure.

 

I guess I got too used to being given material instead of going to look for it. Maybe I am just holding myself back in fear of what's to come. Two posts ago I, finished with a line that sums up my feelings. I illustrate it bellow.

According to some accounts, ancient map makers used to put inscriptions at the edges of their maps that read: "beyond here there be (insert your mythological beast here)". I guess it was their way of saying, "don't know what's beyond so it must be sea monsters, dragons and some other weird stuff." It could have also meant they feared what was there.

 

Do I fear calculus?

 

A little. But I fear not knowing what's beyond even more.

 

It's about time Imjust dive in. Training's over. A part of me knows I am ready for this.

 

So here it goes...

 

Statement of accomplishment

I got my statement of accomplishment from the UCIRVINE pre-calculus class I took at Coursera. It looks clean and simple. It has my name and the signature of both professors in the class. The only grievance I have is the note at the bottom.

 

It reads:

"PLEASE NOTE: THE ONLINE OFFERING OF THIS CLASS DOES NOT REFLECT THE ENTIRE CURRICULUM OFFERED TO STUDENTS ENROLLED AT THE UNIVERSITY OF CALIFORNIA, IRVINE. THIS STATEMENT DOES NOT AFFIRM THAT THIS STUDENT WAS ENROLLED AS A STUDENT AT THE
UNIVERSITY OF CALIFORNIA, IRVINE IN ANY WAY. IT DOES NOT CONFER A UNIVERSITY OF CALIFORNIA, IRVINE GRADE; IT DOES NOT CONFER UNIVERSITY OF CALIFORNIA, IRVINE CREDIT; IT DOES NOT CONFER A UNIVERSITY OF CALIFORNIA, IRVINE DEGREE; AND IT HAS NOT VERIFIED THE IDENTITY OF THE STUDENT."

This is my first Coursera course so I do not know if all other universities use the same language. If they do, shame on them. I understand all the caveats they must state, specially the part about this course being in no way eligible for college credit and the fact that in the free version of the course they could not verify your identity. I guess what bothers me is that they focused on all the negatives, none of the positives.

 

I will make a suggestion in how to phrase the bottom of those certificates.

Please Note: This online offering while covering essential topics of the subject studied is different from the curriculum offered to students enrolled at UC Irvine. The statement is given to the student named above under the assumption that said student abided by the honor code explained in the syllabus and handed in their own work. Unfortunately we cannot verify this students's identity. This course is not eligible for college credit, grade or degree at UC Irvine. This statement does not affirm that this student was enrolled as a student of UC Irvine.

But maybe I'm just overly sensitive. Good news is I passed the course...and at least I can verify my identity. At least I hope I can. Let's check:

 

Fernando Santiago = the guy who spend 10 weeks doing an UC Irvine precalculus class at Coursera.

Me=Me

 

Identity verified.

 

This post should give me closure.

 

Your thoughts?

Autopsy of a test result: How panic, exhaustion and arithmetic don't mix

Since my school days I've had this rule on test results: If you can understand why your answers are incorrect, you are OK. The reasoning behind it is that if you are able to figure out what went wrong and why, that means the learning had taken place but there was a mistake along the way to execution. On the other hand if you have no clue as to why your answer is incorrect, that should send you back to review the concepts being tested.

 

As I related in my previous post I scored a 24.5 out of 34 in my Pre-Calculus final. The one that was 2.5 hours long and I had to take twice back-to-back because I thought I had set the time running on the final attempt by mistake. I took this week to understand what wheat wrong in those 10.5 items I got wrong. The answers surprised me. But they really shouldn't have.

 

In one of my Encouragement pictures posted here there is a quote that reads: 10 out of 9 times it's arithmetic that gets you. No one seems to get the joke... But this test is testament to that. 9 out of the 11 items missed were due to arithmetic, but not the way that you think. Here is the list of reasons my answers were incorrect.

 

1. Wrote the wrong sign when copying problem.
2. Used full angle when I needed the co-terminal angle. Which I knew was only in Q4 or Q1.
3. Wrote the wrong sign when copying problem.
4. Wrote a 28 that looked like a 78, I couldn't read my own writing!
5. This time I used a sign incorrectly (summed -3+10 and wrote -7).
6. Used a parenthesis instead of a bracket in an interval, even when I knew the number was included in the interval.
7. Error copying line in a problem, I wrote a 5 instead of a 1 in the line below.
8. I got the answer write but the program would only accept the variables py as p*y. A clearer head would have realized this.
9. Boldly stated that the square of 9 was 49...which is the square of 7.
10. Forgot to find the squarer root of the hypotenuse in a Pythagoras theorem solve. I am actually good at solving these problems, it was a huge oversight.
11. I was too exhausted to simplify the solution of a half-angle cos identity.

With the possible exception of 2 and 6, the rest of my mistakes were due to my exhaustion and my panic to solve the test. Most of them were due to sloppy, hurried writing. The others were due to foggy thinking. I could find none that showed a flawed understanding of the material. In fact when I got back to doing these again for review, the moment I found my mistake was invariably followed by a "well, duh!", or "that was a stupid mistake". None of them were followed by a "why is this incorrect".

 

Before you think I am saying that I deserved a higher score, rest assured I feel I got the score I deserved. The mistakes I made were mistakes nonetheless, and they prove I need to pay more attention to what I am doing. And prove that I shouldn't take 5 hour long tests at 3am on a Saturday!

 

What do you think?