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So, in order to delve more into the Calculus
(yup, that’s right, it was originally called “The Calculus” meaning “The Calculator”!),
we need to be able to zoom in infinitely close to a certain point to see what’s really going on there,
and add up infinitely many little areas to get the whole area,
or, well, just tons of infinite stuff, really.
And so before we can really get into the Calculus,
we really need to understand the whole concept of infinity and everything about that.
Now there was this ancient paradox called Zeno’s Race Horse Paradox, dating around 450 B.C.
Basically, it went like this: You’ve got a racehorse in a 100m race.
Now the racehorse first covers half the distance to the finish line (which is 50 m) in say, 4 seconds.
And then it would cover the next half (25 m), in about 2 seconds.
And then it would cover the next half, 12.5 metres, then the 6.25 metres,
then 3.125 metres, and so and so on, each time covering half of the remaining distance,
and this way, never, ever actually getting to the finish line. ??
And this is a paradox, of course,
because everyone had seen the races and knew the horse always DID cross the finish line!
And yet if you looked at it with this halving idea,
the horse never seems to get there.
And, they never did solve the paradox,
and hence why this is the end of your Calculus course. Congratulations.
Okay, they DID actually eventually get around the paradox,
but it took them hundreds of years to do it.
And what they did essentially, is come up with the idea of the limit.
That is, if you add up all of the distances, to infinity (as in, ALL of them) then eventually,
it adds up to a finite number, 100 m.
So the 50 + 25 + 12.5 + etc,etc,
all they up to infinity with add up to 100.
And similarly, if you did this with the time it took
and added those up to infinity (4 + 2+ 1+ 0.5 +0.25 + ...) etc.
they would add up to a finite number as well, 8 seconds.
So in this chapter, we’re going to start with limits and then get right into the Calculus.
Set your pacemakers, ‘cause here we go.
In this lesson, we’re going to explore the basic idea of a what a limit is.
To start off, have a look at this curve and drag the point around a bit,
and as you do, notice the tangent line at that point.
Notice how the tangent line changes steepness depending on where the point is?
Oh, I should give you the definition of a tangent line.
A tangent line at a certain point is a line that best approximates the steepness of the curve at that point.
So here’s a simple question:
What’s the exact steepness, or slope, at a certain point?
Now up to now, the only way we can do that is to estimate the tangent line,
So we can approximate with the tangent line will be,
and then do a little slope triangle thing rise over run,
and calculate our approximate slope.
But if we’re talking about the orbit of a satellite in outer space,
or the strength of a bridge under the stress of a semi-truck,
approximations will get people killed!
And we don't want a killed anybody here, in this course.
So we need something better. Enter, the limit.
So for Ex. For the curve y = x2, what is the value of the slope when x is 1?
Okay, here’s how we’re going to do this.
We’re actually going to start with an approximation of the tangent line by picking another point close to x = 1
and then we’ll connect them with a line, which will give us a secant line.
(Remember, a secant line is a line that connects two points on a curve.)
Now we have the secant line, which is an exactly tangent line but it is short of close.
Then we’ll calculate the slope and that’ll be our estimate for the actual slope of the tangent line at x = 1.
Of course, you can see that it’s a pretty poor estimate of the actual tangent line.
What we need to do is move the point closer, and then we’ll have a better estimate.
Actually, why don’t you play around with that other point
and move it closer to x = 1 to see how the secant line can approximate the tangent line
if you move it close enough.
When you’re ready to proceed, click to continue.
Ok, so here’s how we’re going to do this mathematically.
First, we’re going to go back to that other point we picked,
say it has an x-value of “a”,
and we’ll get its corresponding y-value by putting it into the equation for the curve, y = x^2,
and we get a y-value a^2. And so we have the two points (1,1) and (a, a^2).
Now, can you see what we’re going to do next?
We’re going to set up the slope equation using those two points,
and then, can you guess it?
the suspense is building, we’re going to make “a” get very, very close to 1,
and we do voila, we’ll have our slope of the tangent line!
If you didn’t catch that, here’s what I mean:
And now we do we just make “a” get closer and closer to an x-value of 1.
For example, what if a value was 1.5?
The our slope would be
1.5^2 – 1 / (1.5-1) = 2.5
What if a were 1.3? Then we’d get a slope of 2.3
What if a were 1.2? Then the slope would be 2.2
the slope would be 2.1
What if we kept going, and a was like, 1.005?
Then the slope would be 2.005.
Etc, etc, and you can see that the slope appears to be getting close to a LIMIT of 2.
But rats, we still can’t just plug in “1” for “a” because we’d get an error,
with a zero in the denominator.
But wait! Look at this, we can factor the top!
Now, we still can’t plug in 1 for “a”, but that’s ok,
because we’re simply getting INFINITELY CLOSE to “1”, not actually getting there.
And so we could say something like this:
(thought it doesn’t actually get there)
Wow, that was satisfying.
Now, you might be like, ok, that was just lucky that we could factor the numerator there.
Nope, turns out you can always do some kind of math thing,
usually factoring, to make this kind of thing work,
and then you just plug in your value for a, and you’re done!
Now, just before we finish this lesson,
I would like to show you how we say that limit thing properly. See where we have
mtan = msec when a gets infinitely close to 1
well, we’d actually write that like this:
mtan = lim msec (“The limit of the slope of the secant line, as a goes to 1” – for me to say)
= lim (a + 1)