….WE’RE FINISHED LEARNING NEW DERIVATIVES!!! (sorta) WOOHOO!!!!!

http://media.photobucket.com/image/haleluya%20cartoon/ottoneb/animated136.gif

OK, so the first thing we did in class today was observe an obese sumo wrestler plunging to his death on a pair

of skis he probably uses as tooth picks.

This was an intro, in some way or another, into a new topic that we will

be studying for the rest of the quarter: Related Rates. I will get to exactly what those are in a bit, but first a

quick review.

(fill in the blanks)

Calculus is the mathematics of _______, and everything changes with respect to ______.

Algebra is the study of the relationship between _______.

Calculus is the study of the relationship between the _______ __ _______ of the variables (echem… derivative)

Related Rates:

Related rates are, (as far as we are concerned), how multiple derivatives are related. These variable derivatives include...

dx/dt= instantaneous rate of change of horizontal displacement with respect to time (or as we will be looking at it for the rest of this scribe post, horizontal velocity )

dy/dt= instantaneous rate of change of vertical displacement with respect to time (or as we will be looking at it for the rest of this scribe post, vertical velocity)

Take, for instance, a cone... (just bear with me, I did this all in Paint)

Say you were to fill that cone with some agua...

It becomes evident that the rate at which the height (vertical) of the water increases is directly dependent on the width (horizontal) of the cone. Both the rate of vertical growth and horizontal growth are unique derivatives, and they clearly have a relationship. (Hence the term related rates).

(answers to quiz): Change, time, variables, rate of change.

Next, we took a rubber pig, Porky, and ruthlessly pinned him against a wall using a long pvc pipe, mercilessly using him as a demonstration for our own convenience, completely oblivious to his own mutilated feelings. The situation is simply this: Porky is on top of the pipe, pinned against the wall. We decide to lower Porky by slowly sliding the bottom of the pipe away from the wall at a constant rate until Porky reaches the ground only to go into years of shock therapy.

The rest of class was dedicated to filling out Exploration 4.9A, which Bru so graciously posted.

The same scenario of Porky's descent is posed in the exploration. (In the exploration, the pipe becomes a ladder and the wall becomes a skyscraper). And the million dollar question that we answered in the investigation is...

How is the rate at which Porky is descending related to the rate at which the bottom of the ladder is moving way from the skyscraper?

In order to answer this, we need to find the velocity at which the bottom of the ladder moves away from the skyscraper and the rate at which Porky descends the building face.

The horizontal velocity is given to be 2 ft/s, the initial horizontal distance from the skyscraper is 10 ft, and the length of the ladder is 26 ft.

In order to find the vertical velocity, we need to find the height of porky at one second intervals. We can do this because we know that the bottom leg is expanding by two feet every second, and the hypotenuse stays the same.

If you are having trouble imagining this, here is a crude visual to help you out. Each colored triangle represents Porky and the ladder's position at different seconds.

Now comes the subject responsible for the largest amount of point deductions on our tests...Algebra!!!

We must use the Pythagorean theorem to determine the length of each part of the triangle at each second.

For instance...

if in the first second, the horizontal length is 10 ft, and the length of the ladder is 26 ft (see image above), we can use Pythag to determine the height of Porky.

a=10 ft

b=?

c=26 ft

b=24 ft

Now, let's take the first second after the ladder is moved.

Because the ladder is moving at a constant rate of 2 ft/s away from the building, the initial length is 10 ft, and this is the first second, the horizontal length of the ladder from the building will be 12 ft from the skyscraper.

And since we know that the length of the ladder itself is constant, we can substitute the values

a=12 ft

b=?

c=26 ft

into the Pythagorean theorem, which gives us a b value of 23.06 ft.

This train of thought can be applied to the triangle every second until Porky reaches the ground, keeping in mind that the ladder's distance from the skyscraper will increase by two feet every second.

(Image taken from 3rd slide of 2-17)

After calculating all of the a and b distance values we are able to answer the initial question:

The rate at which Porky descends is related to the rate of the ladder's movement

by some variable rate.

After we found all of the b values manually, we were asked to find an equation that related the changing a and b values.

For the sake of easiness, a and b can be substituted for x and y, respectively, leaving us with the equation...

To solve the unknown rate of Porky's velocity, we simply implicitly differentiate with respect to time...

This equation relates Porky's velocity and the ladder's velocity. It is called a Differential Equation because it contains a derivative.

So, that concludes Monday's lesson, the HW for Friday is pages 1-3 of the handout Bru gave us in class, Section 4-9A: Related Rates Warm Up.

PS, I kinda lied about being over with learning derivatives, I just really wanted to put a dancing cat in the post.

The next scribe post will be

__Hyunhwa!__

Holy guacamole. This post is awesome. Really good example with the cone. And the cat is pretty sweet... a little creepy, but sweet. Really good use of the equation editor as well.

ReplyDeleteHAHAHA the cat is hilarious. And good post too. I really liked this post because you talked me through the whole lesson, word for word, even included the obese sumo wrestler. And you added some nice images. I think i found one mistake in your explanation of solving for Porky's velocity. In some of the equations you replace dx/dt with dx/dy. Minor typo but it confused me. Really nice post overall. A- on the J-tron foteen scale

ReplyDeleteJ-tron

I really enjoyed the comment at the beginning of the post about the sumo wresttlers it made me laugh. I was really impressed with how well summarized this post was. You covered all of the material and I respect you honesty at the end admitting that we may not actually be done with derivatives. I also looked through out the post and the error i was able to find was the same as J-tron. Phenomenal overall. Also I liked the paint drawings and I was wondering if you had thought of any other examples related rates. Thanks

ReplyDeleteDammitimmad,

ReplyDeleteThis is a well-organized and engaging post. The pictures from paint were a great addition to provide a visual representation of the related rates idea. I particularly appreciated the ladder picture.

I also noted the same error that was pointed out above. It becomes increasingly more important to be precise with your notation of derivatives in related rates problems, as I'm sure you'll discover.

Although it sounds like you didn't cover it in class yet, I'm curious about how you imagine the differential equation relating the rates will be used. Why is it useful to have an equation like that?

Again, excellent work! Enjoy related rates; they're one of my favorite topics!!!

SKS

Awesome scribe post! I really like the detail in which you explained the lesson. The dancing cat and the comment about the sumo wrestler set up a light and funny mood for the rest of the post. All of the diagrams helped me to understand the lesson a little better. The diagram of the ladder's position helped me visualize the ladder's path a bit better and figure out the relationship between the x and the y. The only error I noticed was the same one as J-Tron and Winn Plot which confused me a little bit as well, but other than that amazing job!

ReplyDeleteNice post. I especially liked the narrations on Porky's feelings. The math was also very good. It was easy to understand and if I hadnt been in class, this would definitely help me with the promlem. Good job!

ReplyDeleteDammitimmad,

ReplyDeleteAre you sure this is the right nom de plum for you? Because after this scribe post your name ought to be dammitimgood! This is an excellent scibe post. Besides the one mistake previously mentioned, I have only one small comment. In your wonderful paint example of the chalice and water I would recommend saying that the rate at which the water is poured into the cup is the independent varibale that determines the height of the water. Seeing as we are studying related rates it makes more sense to use the rate at which we pour the water for this example. Although the rate at which the water rises does change due to the chalice's shape, this is not a controllable or changeable rate. Good job on this great post. Sorry for my nit picking on something that isn't really even a mistake, but you didn't really give me much else to work with. You sure deserve a Johnny Mcguire's after this post!

Thanks,

mc Casper

WOW! Great post!

ReplyDeletelove the dancing kitty!

I just noticed one other thing here. I don't think your post is labeled correctly because when I click on the scribe post button it doesn't come up with the rest of the posts like it ought to do. I'm sure Bru has already brought this to your attention. You can press the edit button on your post and add all of the required tags and labels that way. I don't think there are any labels on this post yet. But at least you remembered to name the next scribe, which is whats actually important here.

ReplyDeleteDammitimmad,

ReplyDeleteI mean, I think, I would like to say, I, I don't know. Why you did so good? Tell me the trick. Why that little kitty is so lovely and cute? Why your post is so clear and well-organized? Finally, why don't you add some color into your post? Thanks for spending the time on this post. It is absolutely awesome work, masterpiece.....