On learning Calculus

I started this journey with a the following concern: would I be able to learn Calculus, or is it something beyond my reach?

I think I have answered that question this year. I am quite able to learn Calculus and I have actually learned some. My statement of intent was "to learn and master Calculus", which now sounds like a very bold statement. I believe I would have needed many more years to master a discipline as vast as Calculus. However, my new understanding of it fills me with satisfaction. I have indeed attained closure on a topic that was as personal as it was academic.

What do I take with me? Quite a lot actually, from the importance of starting with a good base, to the realization that I learn best with the structure of a course than on my own. I also take limits both as a mathematical concept as well as life concept. One helps you explore infinity, the other, I realized, is mainly in your mind. I take differentiation and integration, since at times is important to know the guiding essence of a thing and at others the power of togetherness.

I also learned a lot about change.

Change of rates as well as change of heart. Change as the definition of what we are and were we are going.

I am happy I did this project. It has been hard, it took a lot of time and effort. I made some hard sacrifices as time. But I made it.

I want to thank my wife María del Carmen for her love and support, and also my daughter Amanda for putting up with me calculating away some Saturday mornings. I could not have done it without you. You are, ahem, integral to me. I love you.

So here it is, the end of this blog, I guess. The last entry. Thank you Calculus, it has been a blast.

 

The fundamental theorem of Calculus

Here I am. At the end of my project. My next to last post. And my topic is the fundamental theorem of Calculus. Like the name says, this theorem is a pretty big deal. Let's start by what the Theorem says according to Wolfram Alpha:

I can proudly say I actually understood some of that, but at first could not yet grasp the implications that statement had on all I had studied so far.

Here is Dr. Fowler from Mooculus.com explaining this theorem and it's implications brilliantly:

Now here is what (I think) the fundamental theorem of calculus means in my own words. If you want to integrate acontinuous function on a closed interval, instead of doing the limits of the Riemman Sum applicable, just find the anti-derivative of the function you need to integrate and substract the result of the evaluation of that anti-derivative at the befining and end points of your closed interval.

Or in better words, forget about integrating, just anti-diferentiate!

In my last two posts I had been searching for the area under the curve of x² from the interval 0 to 2. And I had to do a bunch of sigma calculations and set up Riemman Sums and then even take a limit in order to get to 2.667 square units which is 8/3.

The fundamental theorem of calculus is, as I see it, a reward for all my efforts. It is a way of saying:"Fernando, you have toiled and fret, and sweated over these sums and spend countless pages calculating and recalculating all these stuff. You have earned a shortcut." Why thank you very much calculus!

Do you want to check out my new super power? Ok.

The FTOC is telling me that to integrate from the interval a=0 to b=2 of the function x² all I need is an anti-derivative of that formula which I will proceed to evaluate at both points of my interval and then subtract. What is an antiderivative of x²?

Technically there is a + C after that formula but I am assuming it is 0, check Dr. Fowler's video again for that to make sense.

Ok, so now I have my antiderivative x³/₃ and I am ready to evaluate it at 2 (b=2) and at 0 (a=0) and then take the difference .

Now let's plug it back into the formula for the fundamental theorem of calculus.

And finally we get:

There you have it, the elusive 2.667, the area under the curve of x² From the interval starting at 0 ending at 2.

Wow, that took way less effort than before. However, If I had not passed through all those previous steps, all that trouble and effort, I would not have appreciated the beauty, simplicity and deeper meaning of what I have accomplished.

This is my next to last post in this blog and it is a fitting one. I cannot begin to express the emotions I feel at the moment. I have attained great insights in this journey. I now have a deeper understanding of math and the world around me.

Let me know your thoughts in the comments section.

 

On integrating and finally integrating

I wish I could explain how satisfying it is to finally learn the concepts of integration in calculus. For years I have seen how people would represent calculus with a picture of a curved function with the area under it shaded, and for the life of me, I could not figure out how they would calculate that. Now I know. The relative simplicity of the process has a ring of truth, and symmetry, and beauty that threw me back to discovering geometrical theorems in high school.

Well, here it goes: Integrals as I understood them.

Imagine you have a formula that is continuous at least for a given interval [a,b]. I chose again x² in the closed interval form 1 to 2. I already tried to find the area under that curve in my post on Sigma.

By using 10 rectangles of width 1/5 and height x², I was able to approximate the area under the curve to be 2.28 square units an underestimation of the true area. However, I posted that I could get better approximations if my rectangles had been infinitely small.

To get those better approximations I need to improve the sum I used before, pictured to the right. And in order to do that, I would need to convert it into a Riemman Sum.

The toughest part of doing integration is to set up the correct Riemman Sum for the purposes intended. I struggled so hard with this part that I want to give you this video to follow just in case I mess up. Here is the general formula for a Riemman Sum:

Since I am doing a right Riemman Sum, I will use this version of the formula. Where a and b are the interval of my function, and n represents the times I will be cutting that interview. A right Riemman sum will give me an overestimate of the area under the curve, which will complement the underestimate of 2.28 I got

 

The first thing you need to do to set up a Riemman sum is deciding what your interval is (in this case is from 0 to 2) and then decide how wide you want your divisions within that interval to be. I want my intervals to be 1/5 units wide, but that is not important right now, just remember there are ten 1/5 divisions from the interval 0 to 2. Then for the height of my rectangles I chose to evaluate the function x² on the right hand side of those intervals. With those steps selected then I follow the x² rules for Riemman Sums.

A Riemman Sum is the addition of the formula I want to evaluate (x²) at the specific cuts I made. The first point on my interval is a=0 and the last point is b=2. I want to divide that interval in n cuts of a certain size. The formula for the cut size is:

Now here is the formula for the height of my rectangles.

So putting all the steps together here is the formula for my Riemman Sum

And that is actually the hard part, for me at least. The rest is arithmetic.

So the answer I got was: 8/3 + 4/n + 4/3n²

But what does that mean? Remember when I said I wanted the width of my rectangles to be 1/5 units and that it meant I would get 10 segments from interval 0 to 2? Well, if you substitute n=10. The area under the curve it gives me is 3.08 square units.

Now since this is a right Riemman sum I know it is an overestimate. My last attempt in the Sigma post was equivalent to a left Riemman sum which gave me an underestimate of 2.28.

If I take the average of these two numbers I should be able to get a better estimate of the area under the curve: (3.08 + 2.28)/2 = 2.68.

And 2.68 is very close to the true area under the curve which is approximately 2.667. Now, how can we get there?

Well supposed that instead of splitting my interval of this Riemman sum into 10 pieces I split it into 100, n=100, what happens then? We get 2.708 instead of 3.08. And what if n=10,000, that would make our rectangles very, very small, we then get 2.670! That is very, very close to 2.667

And what if n=infinity?

Then the n in this formula 8/3 + 4/n + 4/3n² would be so small (and therefore the width of the rectangles would be also so small) that the only effect relevant in 8/3, and guess what 8/3 comes down to: 2.667 approximately.

 

And what did we just do? We just took a limit.

Whoah! What?!!

Yes, we took the limit of our formula to get the true area under the curve. And that my friends is called integration.

To integrate is to do the following:

To take the limit of the Riemman Sum you are working on as n approaches infinity.

In fact the definite integral is a normally written as a variation of the formula above.

The elongated S just means, take the limit of the sum of f(x) times the change of x from the interval from a to be, as that change of x gets infinetly small.

And there you have it. Integration via the sum of infinite rectangles.

This one was a tough one and there are some considerations to this integrations stuff, but you can review them here.

Let me know what you think in the comments.

 

Area under the curve: Sums and Going Sigma

At last I reached the integral part of Calculus. I was eager to get there since it dealt with a topic which I find fascinating, how can I find the area of a curved object.

You see, area for me,and for the rest of the world, is width x height. Which is pretty simple if you are dealing with rectangles. However, what happens when you are not dealing with straight lines and need the area of a curved region. Well, apparently you just part from what you know and build a bridge!

What I mean by that is that if I wanted to find the area under a curve like the one in the picture opposite. I can draw rectangles under the curved region up to the function and approximate it's area by summing the areas of all the rectangles I drew. That is a neat trick that has served humanity for centuries. And the cool part is that if I can make the rectangles thin enough, I can get a better approximation of the area I am looking for.

If the concept of getting better approximations by making an interval smaller and smaller a sounds familiar it's because we saw it back in limits and derivatives.

So how do I go about summing all these rectangles? I mean of they are 5 or 10 it's simple enough to do it by hand, but what is they are 100 or a 1000...or n rectangles!

That is were sigma comes in.

Have you met sigma? That weird looking capital E that might hunt your math nightmares. It turns out it is tame enough once you get to know it.

 

Sigma is just notation for adding something over and over again. I loved it when Professor Fowler at Mooculus compared it to using a "loop" in programming. It basically encodes and bounds a sum for a specified interval. One of the things I remembered having problems with was the notation used in sigma. For some reason back in college I found it difficult and unapproachable. Now it looks quite simple.

Let's imagine I want to add 1+2+3+4+5+6+7+8+9+10. If I have to write that sum over and over again it gets kinda tiresome. So I will use sigma as shorthand for that sum. But first, let's think about what I am summing: the number (whole numbers; integers; in order) form 1 to 10. So if I had a machine that would spit out a one then a plus sign then the next number after one and so on, I would replicate that string of numbers above. Well that is what sigma does, let's see:

Whenever you see that weird E, just imagine it is saying "please add the result of whatever formula is after me, starting with the number that is under me in the variable and repeating the process as many times as the number above me while making n to be a whole number greater after every repetition." Well, at least it said please.

Okay, first we have that n is the formula we are summing. Then under Sigma is an n=1, meaning the first result is 1. The it is asking us to repeat the process 10 times, that is the number over sigma. However, after each repetition we must make n a number greater than before.

So we start with n=1, then add n=2, then add n=3 and so on until we add n=10, 1+2+3+4+5+6+7+8+9+10! For a better (much better) explanation you can go here.

So if I take the example above, and lets say I divide the area under the curve into 10 sections of 1/5 square units, whose height is the formula f(x)=x² evaluated at those cut points.

Therefore, the first rectangle would have area 0² times1/5=0, the next would have area (1/5)² times 1/5=.008, the one after would be (2/5)² times 1/5=.032 and so on. We could represent that sum this way, at least I think we could:

I Sigmalize we are saying square the 5th of whatever n is from 0 to 9 (the height of our rectangles), then multiply it by 1/5 ( their width) and Sum all 10 results.

This will give you an answer that the area under the curve is approximately 2.28 square units.

That is not bad as approximations go. If you see the two pictures bellow, I shaded the 2.28 square units and pasted them to cover my rectangles.

But as you can see, I could not cover all the area under the curve. There are some triangles that go from the base of the rectangles up to the curve that I cold not cover. To cover all the area under the curve I would need to make smaller and smaller rectangles. In fact if I could make those rectangles infinetly small, I could approximate almos exactly the area under the curve.

I'll give you a preview. The area under the curve is actually closer to 2.666, but we will need to use something I have been looking forward to learning for a long time: Integrals.

Let me know if I got this right in the comments.

 

 

Antiderivatives and when all starts to fit together

As I near the end marker of this journey of learning calculus I meet anti-derivatives. And I am glad I did. Somehow this concept helped me put into perspective all the concepts that have come before it and makes me feel I am back on track again. Just in case you were wondering where I lost my way, it was somewhere into L'hopital's Rule.

First things first: What is an Anti-derivative?

Well an anti-derivative answers the question of: what is this formula a derivative of? Or from what original formula could we have gotten this derivative. For example, if we have x² (the squaring function), it's derivative is 2x. Therefore, x² is an anti-derivative of 2x. (For formal definition go here)

Notice that I wrote that x² is an anti-derivative of 2x. This is important because there can be many anti-derivatives for a given function. If we consider this formula for an anti-derivative: ^xn+1/_n+1+C, (which looks a lot prettier in pictures, see bellow) there is a constant C that is introduced. The way I understand it is that there is only so much information a anti-derivative can give you. In order to recover a specific formula from it's derivative you need to know where that functions "started".

In the anti derivative formula, if x=0 the all we get is C. For example, imagine that x is time, then x=0 is time 0 or your starting point. At that starting point then f(0)=C.

To complete the argument above, let now imagine this scenario. I know that -2x+5 is a derivative of a function I am interested in knowing. Using the ati-derivative formula I get that the function I am interested in is:

But what is C? I have no clue with the information I was given. Let's say I know C is a whole number between 1 and 4. A graph can show me what to expect the graph to be.

But until I know what that constant actually is, I will not know the original formula.

In the picture opposite I have 4 possible graphs, each passing through the y (vertical axis) at 1,2,3 and 4. All 4 graphs are exact copies of the formula I am looking for, but they "start" at different points C.

For a great explanation about how C is relates to anti-differntiation in terms of position and velocity. Check out this video from my Mooculus course. I liked this video not only because Dr. Fowler seemed to have had had too much coffee, but because his explanation incorporates the steps to solve an anti-differentiation equation that has "physical" applications.

Anti-differentiation is an important bridge in my road to understand calculus. At least that is the promise that was made by my professor when he introduced the topic. Whether that is the case or not, I find it fascinating that having information about a function, I can derive other functions that are related, and give me additional information about the original one.

Let me know what you think in the comments.

 

Words of Encouragement

 

My understanding of the chain rule and 5 videos in case I blew it

A few weeks ago I was introduced to the chain rule in my Coursera/Mooculus course. And I found it to be one of those concepts that is straight forward to understand but hard to our into practice.

The chain rule deals with composition of functions. In other words, it deals with the derivative of functions within functions.

The best example I can think of to explain this is,the following: Imagine you are a salesperson whose job is to call customers, set up an appointment for a consult, give them a sales presentation and close a sale.

That sale depends on how many presentations are given, which are dependent of how many appointments were made, which are in turn dependent on how many calls were made. If you had a formula that described this process and you wanted to know how do a change in calls affect sales you might need to use the chain rule.

Let's imagine that such a formula exists. We will use the following variables for it: Sale (S), Presentation (P), Appointments (A), and Calls will be our (x). The formula is S(x)= P(A(x)), in order to know how changes in x affect S(x) we will need to differentiate the function with using the chain rule:

S'(x) = P'(A(x))(A'(x)) which is the same as saying we are taking the derivative of the outside function evaluated at the inside option and we multiply that with the derivative if the inside option.

Let's imagine than in the example above these formulas can be substituted for S(x)= √(3/4x). How will a change in x affect S(x)?

S'(x)= [1/2(3/4x)^-1/2](3/4) and simplifying we get:

S'(x)=	3
	8√(3/4x)

 

Let's imagine the sales person makes 100 calls a day according to the original formula he will get approximately 8.66 sales. What if the sales person makes 50 extra calls by what amount would sales change?

S'(150)=	3
	8√(3/4(150))

 

According to the derivative sales would change .03535 times multiplied by the 50 extra calls which would yield approximately 1.76 extra sales. Now that's the way I understood it. But, since I might be wrong, here are five videos explaining the chain rule.

Professor Jim Fowler produced this amazing video explaining the concept. It is 10 minutes long but for someone strugling to understand what the chain rule is and how it works, its worth the time.

 

Here is the chain rule introduction by Khan Academy.

Chain rule introduction:

 

I liked this example from That Tutor Guy

 

Another straightforward example from justmathtutoring.com

 

This one is from the IntegralCalc channel in YouTube

 

Reference for derivatives and limits

The hardest things  for me to find while studying are good reference sheets. Some people might object to their use and see them as crutches one uses instead of trying to understand the subjects studied. I see them as facilitators. I love to look at reference sheets for the big picture. There I can look for pattern and similarities between the concept studies. And instead of just relying on them, they make me want to explore more and go further.

 I include a Slideshare presentation with a reference sheet on derivatives and limits in the hopes that other feel as I do and can find it helpful. Let me know what you think in the comments. If you cannot see the slide, follow the link at the bottom of the page.

Calculus Derivatives Limits from surya_putra_2009

Words of Encouragment

 

On derivatives and rote learning

The derivative of a function gives us important information about the function being examined. That is a very cool idea. It is basically metadata about how the original function will behave. The derivative of a function, using a very basic explanation which is what I can manage with my knowledge, describes whether a function is increasing, decreasing, and/or changing direction, in other words, it gives you the rate of change.

The derivative also gives us the slope of the tangent line for any point along a function that is differentiable. In other words, if we know the derivative and a point along the function, we can calculate the tangent line for that point.

It bet there a lot of other things the derivative tells us, but these two are the ones that fascinated me the most.

The more I think about it the more it sinks in that calculus is about change. Or in the words of professor Jim Fowler from Mooculus, about how "wiggling the inputs affect the outputs".

Let's do an example: Imagine we have a toy rocket which we shoot up into the sky and watch it fall to the ground. A formula that describes the trajectory of the rocket is: y=-x^2+4x where x=time in seconds and y is feet. In one second it's 3 feet high, in 2 seconds it's 4 feet high. It's average climb from zero to 1second is 3 feet per second and from 1 to 2 seconds it's 1 feet per second. But what how fast was the rocket climbing at exactly 1 second?

The derivative can help us with that. Using the derivative rules:

The derivative of -x^2+4x is -2x+4. And once we know the derivative, and understand that the derivative is the slope of the tangent line at a given point, and remember that the derivative at a point gives us the instantaneous rate of change of a function, we can do some algebra to know how fast the rocket was climbing at exactly 1 second:

-2(1)+4= instantaneous speed

-2+4= 2 feet per second

So at 1 second the rocket is climbing 2 feet per second.

What happens at 2 seconds?

-2(2)+4=0

At 2 seconds the rocket is moving at 0 feet per second, therefore it's standing still.

And at 3 seconds?

-2(3)+4=-2

Interesting, at 3 seconds the rocket is not climbing but declining at 2 feet per second.

We know the rocket hits the ground at 4 seconds. How fast was it going?

-2(4)+4=-8

The rocket hit the ground declining at 8 feet per second.

When we look at the graph we can see these numbers make sense. The parabola starts steeply (the rocket is climbing fast but slowing down), then levels off to a point where there is no more climb (the rocket stands still), and then the parabola declines steeply, (the rocket falls accelerating until it reaches the ground). That was really cool. Now if we look at the graph formula of the derivative by itself, it would have told some key elements of the original function.

-2x+4 is a line. And this line is positive from time 0 to 2 seconds, at 2 seconds it crosses the y axis and is negative until 4 seconds. Looking at that graph and using the first derivative test, we can conclude that from time 0 to 2 seconds the original graph -x^2+4x is increasing in the interval, that at 2 seconds there is a local extreme value where the graph changes direction, and we can also see that from 2 to 4 second the graph is decreasing in the interval. What this means is the if I had the function of the derivative but not the original function I could make some intelligent guesses as to how the original function would behave. Now that to me is amazing.

Like I mentioned before, I have been taking a calculus course for about two months now. Therefore the basics of the derivative is something that has long passed. If fact, it took me a couple of hours to find an explanation of it that I could use in order to post it. This problem got me thinking about the depth of my learning. I know I already covered some of these insights in earlier posts about learning for testing and grades, versus learning to understand.

You see, I was feeling quite confident going week to week in my calculus course, at least at the beginning. That was because I was doing the minimum study required to pass the quizzes. And then, when the concepts started to stack up, I found myself lost. I fear this is what is wrong with the way we educate in Puerto Rico and the United States. If we are teaching to tests, then the students will study to pass those tests and might not explore what the knowledge in itself means. That might mean that the best product we could be creating is excellent test takers instead of excellent thinkers.

 

The last 6 months

I have always found it amazing how time vanishes. Six months ago I was talking about continuity, and here I am today looking back in disbelief that so much time had passed. All I did back then was take a summer break. I thought to myself after five full months of math I would take pause for a month or two. And so I did.
To anyone looking at this blog I have abandoned my goal after less than half a year. But like an unknown function, just checking the endpoints will not tell you what is happening along the way. Therefore, I am a happy to record that I have been doing some Calculus behind the scenes.
On July 24, I receive a comment on this blog saying this:

You might be interested in https://mooculus.osu.edu/

I followed the link and was blown away by what I saw. It was just the course I needed and I could do it through Coursera, the same platform I took my pre-calculus course.
The course is from Ohio State University and it features,I kid you not, the coolest professors I have ever met: Dr. Bart Snapp and Dr. Jim Fowler. Their lecture videos are short, clear and, more times than not, hilarious. I find it increadibly comforting to watch a video on a frustrating difficult topic that is explained with such weird, electric enthusiasm. I particularly enjoy Dr. Fowler's lectures, which are the majority, he is so intense and smart and lives math!

That is what I have been doing since the end of August. I got through 8 weeks of following the course and doing all my quizzes. I even took my fist mid-term. In all this time I have learned about, derivatives, differentiation, the chain rule, the power and product rules and L'hopital's rule. That's when I started to run into trouble. I didn't have a lot of time to practice more and the concepts started to gang up on me. I'm sad to say I could not keep up with the course since then. But a have continued watching the videos.
I have returned to this blog through a series of fortunate events. In the intervening time, I started taking a certification course in online teaching. I am on my sixth and next to last week. The assigment for this week: to create a blog or continue one that is already created. So here I am, I had planned to return in the middle of December to wrap it up but this feels better. It feels like destiny.
By the way, I just realized that the person who had left me the comment of July 24, was Dr. Jim Fowler.
Woah. Mind. Blown. Thanks, Professor.
Let me know what you think in the comments.

Continuity- both for limits and life

If you can draw a graph on a piece of paper without lifting your pencil once, that graph is continuous. In other words, there are no holes, jumps or asymptotes that block its process. My project of learning calculus is, by this definition, not continuous.

Continuity seems to be important enough in Calculus that every book I've seen has a section dedicated to it. However, it's heartening that, as I have learned, continuity is not crucial for determining limits. As long as the graph is approaching the same number from both sides there will be a limit even if there is a hole at that number. Therefore, I estipulate that as long as I am approaching my goal of learning calculus this year, I am allowed a couple of holes here and there.

When a graph is continuous, the limit of a function as x approaches a is f(a). In other words, if I know function x+1 is always continuous, then the limit of x+1 as x approaches 2 is f(2) which equals 3. The information that the graph is continuous let's me know that I can substitute the number I want to take a limit of into an equation without running the risk getting an undefined expression like dividing by zero.

Most of the calculus I have been doing so far deals with discontinuities. And I imagine that in the end it is discontinuities that will be a major part of my studies.

For a good brief intro to basic calculus and discontinuities you can try this video. I found its series in YouTube and really like it since it reminds me of the format of my coursera Pre-Calculus videos.

I plan to minimize future discontinuities in my studies, if possible, and get back to a more productive rhythm. This calculus stuff is getting ever more fascinating and I want to get and see it all.

Let me know your thoughts.

 

Limit existence and infinity

One if the awesome things about studying on my own, is the flexibility of choosing my own schedule and resources. The drawback, as I have mentioned before, is that different resources have different points, methods and even explanations. If your resources also vary in timeframe, those issues become more apparent. Let's take for example the limits of 1/x as x approaches 0 from the negative and from the positive side.

If we look at a graph of 1/x we can see that as x approaches 0 from the negative side, the graph starts to approach the y axis but not touch it. Since x cannot be 0 (1/0 is undefined), then the y axis, where x=0, is an asymptote, a line that will never be touched by the graph. That means that our graph will get closer and closer to the y axis for all infinity without touching it. Therefore the limit of 1/x as x approaches 0 from the negative side, in the strictest definition of a limit, does not exist and that is how my text book defines it.

My online class however, also defines this limit as negative infinity and as positive infinity when you approach 0 from the positive side. The Profesor explained that it was a more informative way to describe the limit of 1/x as x approaches 0.

My problem came along when I did my textbook homework after my online lesson. While I was checking my answers I noticed I had gotten two incorrect ones. They were the limits described above. In the textbook the right answer was that the limits did not exist, I had answered that they were positive and negative Infiniti. Being a student on my own, I had to retrace my steps, watched my class again and do a lot of research to find out if I had gotten the right answer. When I reviewed my online lesson I heard the explanation the professor made about these limits not existing and how using infinity gave us more info about what was happening.

If I had not been able to re-watch my class, I would probably still be looking for the answer. In a traditional classroom my doubt would have been put to rest in a second by the teacher. On my own I had to figure it out myself. I see advantages and disadvantages to both situations.

How about you?

 

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?