O'B explained that while we can usually do our IW at our own pace, we're going to have some assignments this semester that actually have to be done on time. I know, it's crazy. Who does he think we are? He explained that he needs us all to be at a certain baseline, and to do that we have to gain a common understanding of some mathematical concepts through getting our IW done on time. It's pretty unfair, but also pretty logical.
We then logged onto the blog to check our answers with the correct answers for last night's IW. This was quick, as we were just looking to see whether we were in the ballpark/understood the concepts properly. O'B encouraged us to highlight the questions which we just plain couldn't understand, and promised we would discuss them later as a class.
After that, we went over all parts of the homework in detail. The answers and explanations are already available online, so I figure it's not very useful to take up blog space with the specific work for all these problems. Instead, I'll just give you guys a few of the highlights.
Ok, so first thing: motion problems have three layers to them: they can be...
NUMERICAL
GRAPHICAL
ANALYTIC
This first problem gives us just a table of numbers, so it's a numerical problem. We'll deal with the other two types in a bit. The table looks like this:
Okay, so when you're doing motion problems, the first really important thing is to figure out what you're dealing with. Is the question asking you about velocity? Acceleration? Distance? Displacement? Figure it out and get your units straight. That's the second really important thing: using the right units.
In this problem, we're dealing with velocity of an object at different times, and the velocity is measured in meters per minute. Ok, now that we've got that down, we can get to the meat of the problem. First, what's velocity? Velocity is basically the speed at which a particle is traveling. Speed is synonymous with rate of change, and rate of change is synonymous with slope, so basically, if the position of an object over time is represented by the function f(x), the velocity of that particle over time can be represented by the function f '(x).
So, we're calling the velocity function f '(x), and we're told that the velocity function is differentiable. This is the third really important thing of this scribe post: if a function is differentiable on an interval, then it's also continuous on that interval. Please note that this doesn't work the other way around; continuity doesn't guarantee differentiality.
We actually used the magical properties of Geogebra to plot these points on a graph, giving us a visual representation of velocity (on the y axis) over time (on the x axis). It looked like this:
At this point, O'B told us something really exciting: you can use Geogebra to create a function perfectly tailored to include a certain set of points! OMFG NO WAAAAAY SEW COOL.
You just have to type into the input bar: "fit poly" and enter your "list of points", the degree to which you want to raise your polynomial, and BAM. It will fit a polynomial function to your list of points, drawing it on the graph and spitting out the corresponding equation in the key on the left. So just to recap, this is what you would type in:
Now, O'B told us a super fun fact: when trying to pick the degree to which you want to raise the polynomial, think about the number of points you have. If you have 6 points that you're trying to have fit, like we do in this problem, then a polynomial that raises x to the 5th degree will perfectly fit that set of points!
After we input that into the wonderful Geogebra, it worked its robot magic and did some super cool formula stuff to give us this function, and corresponding graph:
Again, this graph shows velocity over time, and is tailored to fit to the points originally given to us in the table. Now by turning this table in to a visual graph, we have officially made this a graphical motion problem. Yeah, that's pretty amazing, but it doesn't stop there, no way. By giving us an actual function to work with, Geogebra has given us a freaking analytical motion problem. That shit cray.
In the words of O'B,
"THIS IS THE F*CKING TRIUMVIRATE OF AP CALC, GUYS."
Well, that's a loose translation.
Basically, if you can discuss a motion problem numerically, graphically, AND analytically on the AP exam in May, you've got it in the bag. Well, at least you have this unit in the bag. O'B then asked us to note that "You wouldn't wanna be that schmo from Missouri who loses points because they can't put their math into words. Make sure your explanations are clear".
We then talked about how it's really quite important to remember the Intermediate Value Theorem, and that it only holds if a function is continuous. It can, however, still hold true if a function is continuous but not differentiable.
Oh wow, this part is EXCITING. We're getting EXTRA CREDIT. If you go out and google it, is there some strange function in the wild which is differentiable, but has a discontinuous derivative??????????????????
You don't have to go do it right now, in fact O'B wants us to hold onto it and look it up over the weekend in our ridiculous amounts of free time.
O'B then told us that he hopes by the end of class, we get really confused about motion and speed and really question everything in our lives... whatta jackass. He also hopes we're excited for IW 5.. does he know we're second semester seniors?
Ok, moving on.
O'B wants us to remember that when you're looking for the derivative of a function at a certain value and you have a graph of that function, all you have to do is draw the line that is tangent to the function at that point and take the slope of that line. Don't get caught up in the details of Calc; know them, but also remember the big picture. Use your brains and be logical, I guess.
Then we reviewed the Mean Value Theorem. I'm not going to go over it here, because I'm sure it's detailed on another blog post somewhere, but if you need a quick refresher, take some of your precious free time and watch this video from Khan Academy; it explains it pretty well.
Next, we talked about acceleration. Basically, acceleration is the rate at which the velocity of an object is increasing or decreasing, or in other words, acceleration is the derivative of velocity, and is therefore the second derivative of the motion function.
While talking about Example 3, #3 on the IW, some AP Physics students had some serious anxiety over whether or not the interval we were talking about was inclusive or non-inclusive. O'B told them to calm themselves, we're not going to worry about endpoints too much in this class, and we can probably just assume that it's a non-inclusive interval.
O'B also told us that one common error that students make when considering distance traveled by an object in motion problems is forgetting to consider the points where the velocity of an object is zero: if we do this, we miss the point where the object changes direction, which is important for adding the correct number of units traveled to our distance value.
QUIZ TUESDAY, it's gonna be real fun.
If you need more guidance about the homework, just check out the answer sheet online, it can be found under the "Even Answers" part of the wiki as a PDF. It has really clear explanations and should help you be prepared for Tuesday's bloodbath*ahem*quiz.
THEN WE PLAYED LEAPFROG! WOOT WOOT!
O'B reminded us of a good trick for winning this game: "If you're trying to decide between two cards, remember, always hold up the best card." Thanks, Bill. Thanks.
One question got us really confused; it read: "When the position is decreasing at a decreasing rate, the acceleration is _______." Most of us answered positive, because we figured that the rate at which the position was decreasing was decreasing. This was apparently wrong, because if rate of velocity is "decreasing", acceleration is always negative. O'B agreed that this wording was wonky, though, and we threw the question out. Eddie got mad because he was like the only person who got it right because he was like the only person that didn't think about the fact that it was a double negative. He was pretty sad. Sorry Eddie :(
Then Scotty won and stood on her chair.
It was a fun class. ANNA will be the next scribe!
~Rebecca
*******************40 MINUTE PERIOD*******************
When we came into class, O'B begged everyone to read the scribe post. He's obviously fighting a losing battle, because everyone knows that scribe posts are boring as hell. Then, we went to the iCal and answered the Question O' The Day. Treat and a half right there!
The question asked...
It took some coaxing, but eventually we all submitted answers.
It turns out we were super divided! O'B's nifty pie chart showed a ratio of 9 "True" answers to 8 "False" answers. He called the Damianator in to see the results, and I think she was pretty disgusted with the stupidity of half our class. Which half? We don't know! O'B wants us to do our homework
and figure it out for ourselves, like big kids. He said we'll get the same question again tomorrow and hopefully we won't f*ck up like we did today.
Then, we talked about some things that should probably be cleared up on this post: the first thing we talked about was the actual nature of velocity. #calrobbins helped us out, telling us that...
Speed is the absolute distance of velocity.
Or, in other words, velocity is speed with direction.
I also mentioned that if you want to know a function's derivative and you have a graph of that function, you can just draw a line tangent to the function and find the slope of that line. I'll clarify, that is if you want to find the derivative of a function at a specific point. For example, the derivative of f(c) can be found using this method, if c is a single x value somewhere in the domain of f. f '(x) is actually going to be another function altogether, and to do that we'll need to do more than just draw a single tangent line at one point on a graph. O'B also reminded us YET AGAIN of another fancy way to find he derivative of a function at a specific point super quick: nDeriv! nDeriv! nDeriv!
Then O'B slammed this blog for being too explicit, claiming that "This is a family blog".
All I have to say about that is that I'm sorry his life is so sad that he never graduated from 8th grade and can't handle a few swear words without flipping sh*t.
Again, he asked us to really try to have IW5 done and ready for class discussion tomorrow, because class will be more valuable if we do. Also, we have a quiz, so it's probably a good idea.
Speaking of tomorrow's quiz, it will feature one free response question! OHEMGEEEEEE.
We then did a quick example to illustrate the meaning of the vocabulary word "acceleration":
O'B told us that acceleration is
. This actually means meters per second, per second. In other words, he we decided, the object is having a change in velocity of –4 meters each second.We then finished up by working on IW5, and "thinking math"!
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