9/6
We began class with a grueling, forty minute quiz on all 198 trig properties of the unit circle... glad we all studied! Just kidding, it was a timed two-minute quiz on only a few trig values, but if any of us didn't do well, we are welcome to take a longer version covering all 198 for a re-quiz sometime this week!
We then went over our class objective, which was to “develop an intuitive understanding of limits”, and got started right away with the warm up!
For the warm up, we looked at finding the limits of both regular and piecewise functions using our calculators and geogebra.
The first function we focused on was:
We noticed a huge gap in the graph at x=1, and agreed that because the denominator of the fraction (x–1) would equal 0 and therefore create an undefined point when x=1, this made sense.
We then played with transformations to change the dimensions of this function, and also discovered that it can be represented by a piecewise function f(x):
Note: this piecewise function fills in the gap at x=1, which our original function lacked.
With this graph, we could see how the two functions worked to form the graph:
Blue: our piecewise function
Black: the regular functions from which we drew our piecewise function
We then learned about one-sided limits, which mean that the left side of the function approaches a different number than the right side as it approaches the limit. For example, f(x) has a one-sided limit because from the left it approaches x=1 at f(x)=1, and from the right, it approaches x=1 at f(x)=3.
This led us to an entirely new concept regarding limits:
A LIMIT ONLY EXISTS IF BOTH SIDES OF THE FUNCTION APPROACH THE SAME NUMBER AS THE LIMIT GROWS CLOSER.
So, for example, the limit of f(x) as x approaches 1 does not exist because both sides approach x=1 at different values (1 on the left, 3 on the right).
We also learned that functions can have positive side and negative side limits. For example, the limit value for f(x) as x approaches 1 from the left side is 1, and from the right side is 3. If the side-specific limit values are different, the limit cannot exist.
The next function we explored was:
We also learned that functions can have positive side and negative side limits. For example, the limit value for f(x) as x approaches 1 from the left side is 1, and from the right side is 3. If the side-specific limit values are different, the limit cannot exist.
The next function we explored was:
To create a slider for f(x) on geogebra, type into the input bar: (k, f(k))
Or, if you were graphing g(x) rather than f(x): (k, g(k))
The next function we looked at was:
And we contrasted h(x) with the piecewise function p(x):
As with functions f(x) and g(x), we noticed that the only difference between functions h(x) and p(x) was that h(x) did not have a value at x=1, whereas p(x) had a value of 2. The hole in the graph of h(x) occurs at y=3, though, so the limit value of h(x) as x approaches 1 is 3. The graphs of h(x) and p(x) are shown below:
graph of h(x) and p(x)
We discovered that limits are the perfect remedy for that tricky division by zero problem! They explain what happens as functions approach undefined points.
We then worked through an example to get more familiar with nonexistent limits. We used the figure:
Finally, we took a quick look in our textbooks on pages 60–63, which contained “Limit Rules”:
We finished up by getting a quick head start on our IW problems, remembering that on Monday there will be a quiz on IW's 1, 2, and 3.
We then worked through an example to get more familiar with nonexistent limits. We used the figure:
Without even graphing the equation, we knew there would be a hole at x=2. This is because when 2 is plugged in for x on the denominator of the fraction, the denominator = 2-2 = 0. After noting that, we graphed it:
From the graph, we can tell that the the left and right sides of the graph approach x=2 at different points, meaning that the limit does not exist!
Finally, we took a quick look in our textbooks on pages 60–63, which contained “Limit Rules”:
LIMIT RULES
Sum Rule
The limit of the sum of two functions is equal to the sum of their limits.
Difference Rule
The limit of the difference of two functions is the difference of their limits.
Product Rule
The limit of a product of two functions is the product of their limits.
Constant Multiple Rule
The limit of a constant times a function is the constant times the limit of the function.
Quotient Rule
The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.
Power Rule
Provided that the base which is being raised to the power is a real number, the limit of a rational power function is that power of the limit of the function, provided the latter is a real number.
We finished up by getting a quick head start on our IW problems, remembering that on Monday there will be a quiz on IW's 1, 2, and 3.
~Rebecca
P.S.
Next class’s scribe will be Anna! :)
P.P.S.
Screenshot cred to Francie for some of the images, and for letting me read her notes!
=x^{2}+2(\frac{|x-1}{x-1})-1 "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
P.P.S.
Screenshot cred to Francie for some of the images, and for letting me read her notes!
UPDATE!!!!!!!!!!
WOW. Long time, no update. So now, looking back, the concepts from this post seem really simple (simple posts, and yet the explanations are completely, utterly, INCOHERENT. I mean, I guess that's probably because I had absolutely, positively no F*@#ing idea what I was doing, but oh well.) So now let's take a look back...
Wow, remember that quiz we took? The 2 minute one that we thought was terrible? We had no idea what was to come... at least OB let us all retake Quiz Numero Uno until we got 100's. Yay! Free quiz points! I hope everyone was able to take advantage of that.
The biggest difference between us "November AP Calc Students" and those "September AP Calc Students" is, basically, that know how to do magical things with derivatives. We can find derivatives! We can find 2nd, 3rd, 4th, and 5th derivatives! We can take derivatives of exponential and trigonometric functions; we can take derivatives of those wonky curves that have both x's and y's in their equations. We can take inverse derivatives! And we have rules for finding derivatives... yes, many magical rules... power rule, quotient rule, product rule, chain rule! We can use derivatives to find tangent lines, normal lines, and even limits. Yes, LIMITS- the exact subject matter of this feeble little post from way-back-when in September.
So how about that first example, huh? You know the one I'm talking about...
We noted that annoying "gap" in the function at x = 1, and decided that the limit did not exist, because the left and right sides of the function did not approach the same "y" value as the "x" value approached 1. This was pretty smart (way to go, us), but now that we know about derivatives, we also know that where x = 1, f(x) is indifferentiable (that is, there is NO DERIVATIVE for f(x) when x = 1.
And what the F*@# is a derivative, you might ask???? Actually, you probably wouldn't ask that, because you're all super smart now and know exactly what a derivative is, but for the purpose of this post, and because I want to be thorough, I'll just provide a definition anyway. Actually, I'm going to provide two definitions of a derivative. That's right, TWO!!!!!! Actually, they're both the same definition, essentially, but each one looks at the issue from a different perspective, so the one which is most helpful to you guys is a personal choice, or may depend on what exactly you're trying to do with a derivative.
Okay. Okay. Okay. Here are the definitions:
A mathematical derivative is:
a) The instantaneous rate of change of a function at any given point, with respect to its variable
b) The slope of a line that is tangent to a function at any given point, with respect to its variable
Phew! Now that we've reviewed what exactly a derivative is, we should probably review another awesome thing we already know: Where functions DON'T HAVE derivatives. This seemed a little complicated to us at first, but actually, it's common sense. Since a derivative is a rate of change, it can basically be equated with the term slope. And where might a function lack "slope"?????? Um, where it doesn't exist! DUH! Therefore, functions lack differentiation at CORNERS, CUSPS, DISCONTINUITIES AND VERTICAL TANGENTS. Bam. I just scribed that off the top o' my head (that's how far we've come). So, we can conclude that functions aren't differentiable at limits.
OMFG SEW COOL.
That definitely applies to that example we were just talking about (i.e., that function is indifferentiable at x = 1, because there is a discontinuity).
That actually applies to, like, EVERY SINGLE PROBLEM WE WORKED WITH THAT FIRST DAY.
Yeah, I know, this is all pretty obvious now. But if we had known it back then, just imagine how easy Calc would have been! How many quizzes we would have aced, for all our glorious derivative knowledge is practically worth its weight in gold. Except derivatives don't weigh anything. Damn.
To sum up, (hehe, math joke, "sum") I'll just remind us of those limit rules, which like pretty much everything else on this post, seemed really complicated back in September but is pretty much common knowledge now.
We've pretty much figured out that the limit rules can be remembered without memorization. WHY? Because they're common sense. But I'll still re-scribe them, just for a quick refresher.
Sum Rule
The limit of the sum of two functions is equal to the sum of their limits.
Difference Rule
The limit of the difference of two functions is the difference of their limits.
Product Rule
The limit of a product of two functions is the product of their limits.
Constant Multiple Rule
The limit of a constant times a function is the constant times the limit of the function.
Quotient Rule
The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero.
Power Rule
Provided that the base which is being raised to the power is a real number, the limit of a rational power function is that power of the limit of the function, provided the latter is a real number.
See, it's funny because at this point, these rules have been driven into us so far that we juggle them around, left and right, without even a second thought.
Okay, so it's time for a link, because I didn't include one last time. But what should I link to? Well OBVIOUSLY it should "link" (hah, sew punny) limits and derivatives together.
So first, here's a little instructional video about LIMITS! (Because we haven't already had enough of them, and we like to spend our free time on Friday afternoons watching extra videos about them) It's just a guy doing multiple examples of problems regarding limits, and hopefully it will help cement the idea if you're still trying to wrap your mind around it.
And here, second, is a video of exactly WHAT a derivative is, in case you guys are still completely lost, which I really hope you're not, because then we're probably all failing:
Hopefully these two videos are pretty easy to understand, and hopefully they'll give you all a good idea of the relationship between limits and derivatives.
Personally, I think this guy is pretty great at explaining things. For some more videos of his (and he has videos on everything from chain rule, to implicit differentiation, to photosynthesis to a review of FDR's New Deal), go to http://www.khanacademy.org. Seriously.
Okay, that's it! This post is officially updated. Happy end-of-first-quarter, mathletes.
~Rebecca
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