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.