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 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.

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?

 

 

 

Of humbleness and understanding your domain

On New Year's Day, I felt really confident on my quest to learn Calculus. I would say over confident. Even after getting excellent advice on where to start (Pre Calculus), I pushed on with the Calculus books I had. Then I hit the wall. Yes, The Wall. The wall of realization that I was getting nowhere. The wall that has a sign that says:"if you are here and do not know what to do next, you missed a step, go back and return when ready."
Hitting that wall was (and still is) painful because of my pride.
 
Had I been humble, and realized I had not done math like this in a long time, I would have started from a math place I am still comfortable with. But no, I believed myself to be just a little fuzzy about quadratic equations and trigonometry. Oh, how wrong I was. How wrong was I? Would you like to know? Ok, let's give you an example of a phrase I found in the MIT Calculus for beginners page that baffled me.
 

The equation ax² + bx + c = 0 can be rewritten (when a is not 0, after dividing by a) as

That my friends is just an algebraic manipulation. A simple exercise in the grand scheme of things. Still, I could not follow the steps taken to get there. That was my wake up call. I need to go back to pre-calculus, back to functions, back to the last place I feel comfortable with.

That's how I landed in general function theory. I mean I work with functions all the time. How many calls have got in a weeks time? How much have we paid in cable over the last year? How much has our kilowatt hour fluctuated in 3 years? I love graphing things out, so I should start there, right?

Wrong.

Because when I tried starting with general function theory I ran into this gem in wikipedia when reviewing domains (which the course tells me is very important for pre-calculus and calculus problems)

For a function

defined by

, or equivalently ,

the codomain of f is , but f does not map to any negative number. Thus the image of f is the set ; i.e., the interval [0, ∞).

I must have stared at those statements for 5 minutes straight, before realizing that I had no idea what they meant. The English I got...but the symbols baffled me. After much rumination, and 6 hours of sleep, I went back to them and did what I should have done from the start. Went even further back to get the notations right.

Translation (as best I could come up with): , means for a function with a domain (the universe of numbers you will input into a function) of the Real Number (numbers that you can put on a graph) and a codomain (all the numbers the function could assumed as defined even when the function won't output all values in that codomain) that is also all Real numbers.

That translation took me an hour, and half a dozen websites to produce...but hey, that's progress.

Let me know if I am on the right track!

 

A little background for all this

Learning Calculus has been my goal for a very long time. When I arrived to the University of Puerto Rico in 1996, I held a medal for Mathematics, among other subjects, and an aced AP advanced math test that propelled me for classes ahead of other first-years. I was very excited to take my Pre-Calculus course. It was nerve-racking because I was selected to take a condensed section of Pre-Calculus, that was supposed to be two semesters, in one term.
 
Then, about a third into that semester, my life changed dramatically. My father, who was my math mentor when I thought no one could get me out of the woods of polynomials, died in a car accident. That left me and my younger brother parent-less. We had lost our mom 3 years earlier.
 
I read somewhere that Calculus is the study of change...well no calculations could have helped me cope with what I had gone through that day. I battled on for the rest of the semester but it was hard. I just couldn't concentrate on my classes. I got a C in that Pre-Calculus class. I was in such denial of what had happened in my family that I credited the grade to my lack of ability to understand calculus.
After that my life took me away from mathematics. Whenever I face problems that involved math beyond algebra, I shunned them. Still, a part of me knew I was just shortchanging myself.
 
So little by little, I have come back to the world of math. Actually my favorite tool for exploring it has been excel. Once you get me started with equations, its hard to quit them. Now after 17 years I made myself a promise. I would learn and master Calculus because I owe it to myself. I owe myself to know if I truly have a limitation in learning it, which will put me at par with most of the wonderful people I know, or if it was something that I can do after being ready to.
 
That is the question this journey will answer.
 
Please fell free to comment. I don't want to take this journey alone.
 
 

Back to Basics

Well, I got my answer from Quora as to where should I start to learn Calculus and, as I feared, it's clear, balanced and reasonable.

"I would retake Pre-Calculus first. Not because you got a, "C," in it in college but because Pre-Calculus is largely review of all the material you will need to know for Calculus.I would [not] try learning by yourself; you may lose motivation that way. Instead, sign up at a local community college and take classes there. The textbook is probably going to be your biggest expense." From Henry Maldonado @ Quora.

As much as I would like to think that my Math is where it was 17 years ago, I know there's a lot I have forgotten. So pre-calc it is...
 

Ok, so where should I start?

Let's try the collective wisdom of Quora:
 

 
I always feel self conscious when I use Quora, I usually get a swift lesson on English grammar and the correct way to ask a question. Still, it's the best place to go for crowd sourcing answers...now I wait.
 

Letter of intent?

Calculus was first discovered, if I may use that word instead of invented, either by a 23 year old Issac Newton or by a 28 year old Gottfried Leibniz, in 1666 or 1674 respectively. Now, approximately 34 decades since that discovery, a 34 year-old, husband, father and professional, is trying to learn it...for no practical reason, other than to reach a long-delayed personal goal