Today is the 40th anniversary of the first manned lunar landing (1969) .. a historic event for techies .. cuz it was techies who put us there. And techies used » calculus to put a man on the moon (some 240,000 miles away).
At the heart of calculus lies the notion of » limits (a concept I'm quite familiar with, unfortunately).
Calculus was invented to solve (among other things) the problem associated with finding the instantaneous rate-of-change .. as visualized by the slope of a line tangential to any given point on a curve (of a graph).
To calculate a slope, you might recall (dust off them algebraic brain cells) we pick two representative points, find their difference and divide the » "rise by the run" .. the change-in-Y / change-in-X (.. commonly referred to as "delta-Y over delta-X").
As the change in the X coordinates (recall from basic Algebra) gets smaller and smaller, we get closer and closer to determining the slope (rate-of-change) at a particular point.
The problem however .. is that a point has no size, so the "change" or 'difference' (in the X coordinates) becomes zero. And dividing anything by zero is a major mathematical no-no. (Defined as "undefined" .. a mathematical black hole that will crash your computer.)
See t=13:00 here, and especiaaly t=13:30.
The concept of » limits was introduced to address this problem. Imagine standing in your living room, and walking half the distance to the furthest wall. Then walk half the distance again. And again & again.
Each time, you keep getting closer & closer. But .. you'll never actually reach the wall (cuz you keep going only half the distance). A hundred years from now, you'll be very, very close (to the wall), but still not quite there.
Getting closer & closer to the wall is analogous to decreasing the size of the difference between the two X coordinates along a curve plotted on a standard graph (which contains an X & Y axis). But the limit (drum-roll, please .. here it comes) is » the wall! .. even tho, in reality, you never actually get there.
That's why the notion of a limit represents a mathemetical "concept" (not reality). If you think about it, you can't really have an instantaneous rate-of-change (.. cuz nothing can change in an instant, cuz an instant contains no time). And the word 'rate' implies "per-unit-something." That 'something can be (and often is) » time.
That's also why the result is called/termed a 'derivative' .. cuz you can't get there with conventional mathematical manipulations. It's kinda like what that old farmer told me down South when I asked for directions » "Son, you can't get there from here." =)
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In the same way (like our wall), the instantaneous rate-of-change of a function at a particular point on a curve will never actually really get there. But we can see where it's headed. (Toward its 'limit'.)
Some Calculus textbooks use the word 'intend' to teach the concept of limits .. as in, "Where did the function (at a particular point) intend to go?" The furthest wall is where you intended to go. But I don't care much for that description.
The "intended" value of a function at a particular point on a curve corresponds to its limit and represents the instantaneous rate-of-change at that point. That value is calculated by "taking the derivative" .. a function of calculus which is actually surprisingly easy to do, mathematically speaking, compared to the associated algebraic gymnastics.
Derivatives & Integrals
Once the prof finishes teaching the concept of limits, and moves on to derivatives, he rarely returns. And that's where you spend the rest of your time (in first-semester Calculus). But limits really represent the foundation of Calculus. It's how (and why) Calculus becomes Calculus.
The opposite of a derivative is called an integral (« the big S-shaped thingy) .. kinda like what division is to multiplication. An integral calculates the area under a curve. I used to love getting amped on expresso and cranking out integrals for hours. It's like yoga for your brain.
Might be worth mentioning, as a side note, that finanical "derivatives" (backed by US mortgages) are what brought the US economy to the brink. But that's another discussion. My point however, is that the reason something is called a derivative is cuz it's not really.
One of the definitions for the root-word 'derive' is » "Develop or evolve from a potential state."
Key word » potential. Someone who has the 'potential' to grow 6-feet tall is not (in reality) 6-feet tall.
Every definition suggests in some way a "going beyond" the original source. And it seems those departures can be either circumspect (mindful of the risks) or reckless (particularly when money is involved).
My Experience with Calculus
I took the Integral-class (second semester) during the summer .. compressed schedule, 8 weeks, 4 days/week (Mon-Thur) .. like drinking water thru a firehose. Relentless. Intellectually invigorating. A mad sprint.
I had only 2 semesters of Calculus, cuz that's all my degree-path required. So I don't know anything beyond derivatives and integrals. (Calculus is typically a 3-semester course.)
My advice to those gearing up to take Calculus » review your Trig functions (sine, cosine, tangent, etc.). Know them cold. Trying to learn Calculus with weak Trig skills will be little fun (like a root canal). And learn how to use a graphing calculator.
I got 'A's in both my Calculus courses (Math 3A & 3B). In the first semester, I got the highest grade in the class, passing a little Asian girl at the very end (final exam) who was slightly ahead of me the whole semester.
In the Integral class, we had 2 students from Iran who had taken Calculus in high school there. They killed everybody and would even argue with the prof (who happened to be the dept head .. Dr. Hada).
The first prof taught concepts and devised exams that tested your understanding of those concepts. The second taught to the exam. The first method was better, which is why I have a good grounding in the concepts of Calculus. Tho many would/could not pass with that method, hence the reason for the second (.. teaching to the exam).
The topic of Calculus came up with a friend I was having coffee with at the coffee shop recently. He has both a Bachelors & Masters in Philosophy from UCLA (which is supposed to be a top school for philosophers) and is always reading some gnarly book.
I asked if he had Calculus in his degree-path. He responded by saying he wasn't sure.
"How can you not be sure?" I blurted out. "That's like saying, 'You're not sure if you've ever had a root canal.'" =)
He laughed and said. "No, Guess I haven't."
Here's a movie about walking on the moon back in '69. Might be worth seeing tonight for its nostalgic value. Nixon was president then. The Billboard Top-100 were » these songs. Notice the Beatles were together and Elvis was alive. (Hey, maybe he still is.)
Meanwhile, über-geeks had emerged, squinting like moles, from their secret hideaway where they were building a rocket for Apollo 11.
Bit off more than I 'intended' to chew with today's entry. (I do that sometimes.)
For more along these lines, here's a Google search for » moon lunar landing 1969 apollo 11
Here's another for the terms » introduction calculus limits derivatives