A view through the walls of our classroom. This is an interactive learning ecology for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
I knew I forgot to do something yesterday. And that was check to blog -__-"
Sorry that this is very late, some urgent things came up and I wasn't able to get home until 1, so I'll just give a quick run down of what happened since everyone was in class.
Anyways, we worked on another question, this one being about Scuba Steve. There was a lot of information, so the question had to be read carefully to fully understand it.
So as you can see, to solve it, we found that the hypotenuse of the triangle at the beginning is 30 root two. we then found the remaining area near the end of the distance to be x since we do't know how long that distance was. Because triangle is a triangle with two sides equal, the top of it would be 30, thus the 170 - x (it's 170 because 200 - 30)
So that makes the second part of the equation for part a. The next slope we found one of the sides to be 45 feet. So then we found the hypotenuse to be root x squared + 2025. And now we can build our equation.
*Red is what Paul stated. Blue = edited work. The RDT triangle is an example of how to change from rates, to distances, to times.
The next part is done by benchmen (Yes, he's a plural. That's not weird), and he found the derivative of the function, and set it to zero to find x.
Unfortunately, this is as far as we got as we ran out of time. Next scribe will be Ben
The Truth about Bruce Lee and Chuck Norris
Some real fighting, none of that ninja fighting bullet dodging special effects haha.
And here's the youtube video. Its the nom nom bunny we loved =).
And the next scribe is ...hmm..there's so many to choose from! Well..since it looks like its going to rain...and rain starts with the letter "r"...*Stares at scribe list*. I choose Rence, Lawrence!
Alrighty guys, as chosen by yinan I'm the scribe for todays class, which is great, because its a short scribe, and I have to go to work in about half an hour, so it works out perfect.
Anyways, todays class was a relatively simple one, as for reasons unknown to me, Mr. K wasn't there. Instead we had a sub, who's instructions to us were to work on the questions we had been given yesterday (I think? I wasn't all to clear about which questions he was talking about.) So the majority of the class did that. Myself, Lawrence, and Francis used that opportunity to do a bit more work on our DEV and go over some questions from the text that we had been struggling with.
Overall the class was mostly geared towards review and practice questions in preparation for our ever looming AP Calculus Exam (May 6th D:)
So I think thats everything, and its good too, because now I must get ready to go to work :] See everyone tomorrow!
SO theres one swear in it, I didnt notice it until just now. Theres also some mild drug references, although they're in good nature. Basically, make sure theres no young children around and it should be fine :]
Edit: Oh yeah, Kristina's scribbbbeeeeee Sorry for the lateness. I forgot ^^;
In today's class Mr.K went through the our wiki, to show us some tips to compare the difference between two person's edits of the question. All you have to do is to quick the page history, then choose the two that you wants and qiuck compare. The green parts are the parts that are added, and the red is the parts that are been deleated. Also the deline for DEV project is extened, due to the high purssre of the upcoming AP Exam.
As always we did a mini exam to help us review for the exam: Q1/ Find the derivative of both function, which V1=cos(t) & V2=-2e^(-2t) It's in terms of V because the X1 and X2 are the postion fungction the derivative of postion is velocity which is V. Then you let V1=V2, which is cos(t)=-2e^(-2t). You can graph both is the calacror or find the interstion point of the two functions. OR you can let the whole thing equals to 0, cos(t)+2e^(-2t)=0 then find the root of the function. The answer should be D)three
Q2/ In this parblem you applie the derivative test rules. Graph the function, then find where the function have it's roots over the interval [-1,]3]. Then at the three roots find the point that is going from negative to postive, because it's a local minimum. Your answer should be E)2.507
Q3/ From the graph, take a slice and look at it as a disk with a washer in it. Then , cos(x)=x which becomes 0=cost(x)-x. use the area formular the pluge everthing in to find the function . If you done everthing correctly you should end up with C)1.520
Q4/ will this question camed up the third time, I think everyone is good with it. The answer is D)8647 gallons.
Thought I'd do my scribe early to make room for studying =(.
This was a gimme question. Very straight forward. We just have to apply the average value formula:
This was a washer question. Ah good times right? "I don't see the hole!" In washer questions we just subtract big radius squared from little radius squared and integratate. Don't forget your pi.
After simplifying we realize that the function is just a line. So we know I is true since the limit from both sides will be equal since this is a line. II is also true because f(a) is defined. III however is not true because x cannot equal a. So there's a hole in our line.
There was a little typo with the questions here. But it's all better now. The table gives us numerical values so that was a hint that we numerically find the derivative at 1.2. We do that by finding the slope from the left and right of 1.2 by using the f(x+h)-f(x)/h formula.
So we first start by cutting the graph up into 4 intervals. The ending time was 24 and they want it into 4 intervals so 24/4=6. So our intervals are 6 hours apart. To find a midpoint Riemann sum we use the middle value of the interval as the length of the rectangle and of course width is the change in time.
K that's it folks. I think this is my last scribe post =(. The exams getting closer so don't forget to do your three questions a day. Hope you guys are enjoying you're long weekend. Next scribe will be Yi Nan.
Hi everyone, I'm Benchmen and I'll be your scribe for today. I'm sorry if this scribe is boring because I have to finish this quickly so I can finish off a project for another class.
We had another exam practice session today. The question that we did was a very famous one from past AP exams. It was THE AMUSEMENT PARK QUESTION.
Here is how to get the answer for part a.
This part of the question was quite simple, but there was a part to this question that may mess up some people. Let me show you how to answer it than I'll talk about that tricky part. To get the answer integrate the entering function for 9 to 17 with respect to t. You do that because since the entering function is a rate (derivative) function and if you integrate a derivative you'll get the total change of the parent function which in this case is the total number of people that entered the amusement park. If you do that you should get 6004 people.Never forget the units. You will know what unit should be part of the answer if you understand the concepts and/or technique used to find the answer.
One tricky thing that people had trouble with was understanding what the question was asking for. Some people included the Leaving Function. If you do that you are solving for how many people were in the amusement park, not how many people entered it. Those are 2 totally different things since there are entering and leaving functions.
Part a) involved the process from part a) plus a little simple multiplication. Since there are 2 costs for tickets at 2 different time intervals, you will need to do 2 integrations. So integrate the entering function for the first time interval (from 9 to 17) to get the number of people and multiply by the cost to get the amount of money made for that time interval. Do that again for the second interval and add the results together and you should get the answer in the image above. Again, don't forget the units.
This part of the question involved an accumulation function. As you can see this accumulation function represents the total number of people in the amusement park over a time interval from 9:00AM to x o'clock because the function involves the integration of the difference of the Entering and the Exiting functions. The question is asking for the derivative of the accumulation function. So if you differentiate the function you will get the integrand of the accumulation function but respect to the variable limit instead because an accumulation function is a composite of functions. If you differentiate a composite of functions, you must apply the chain rule. The differentiation of the accumulation function above results in the differences of the Entering and Leaving functions which is the change in the number of people in the park. If you did the math correctly your answer should be the answer in the image above. For the Free Response questions on the AP exam, a word answer is required. The word answer should be very specific but not long because this is math class not english class.
The last part of this amusement park question is an optimization question where you are looking for what time is it where there is a maximum number of people during the open hours of the park. So what you have to do is differentiate the accumulation function from part c) and find where the resulting function is 0. A function has a maximum or a minimum where ever the derivative has a root or is undefined. When the critical number is found, do a line analysis of the derivative to find where the parent function is increasing or decreasing. If the function is increasing on the left and decreasing on the right of the critical number, the critical number found is a maximum. The time when there is a maximum number of people is about 15.7948 hours after midnight(as in 12:00 am of the current day).
The averages of past AP exams are found on the next slide and as you can see as a class (average) we are just above average which is good because that means we could be expecting a 3-4 on the exam.
Thats my scribe and sorry if it was horrible. Very busy since the APs are coming in about 2-3 weeks.
Remember, try to do 3 AP Free Response questions a night. Get constructively modifying the wiki questions.
Related Rates: Area, Perimeter, and Diagonal of a Rectangle
Related Rates: Shadows Including Similar Triangles
Related Rates: Distance and Velocity and Spongebob!
Exam Review Intro
We do our exam review in three ways:
Doing mini-exams pre-test style
Doing questions in class related to rusty topics
Doing old exam free-response questions and study how each question evolved throughout the years and do 3 questions a night
Today's class we decided to do #2, reviewing related rates.
Related Rates: Area, Perimeter, and Diagonal of a Rectangle
The length (L), the width (W), and the rates of which the lengths (dL/dt) and widths (dW/dt) are changing.
By convention, we're going to designate an increasing rate as a positive rate and a decreasing rate as a negative rate.
a) We know the formula for the area of a rectangle as A = L * W, where A is area, L is length, and W is width. Since we're looking for the rates of change (That's the definition of a derivative!) we differentiate the formula, with respect to time.
Since L * W is a product, we use The Product Rule to differentiate.
Since area, length, and width are all with respect to time--meaning that area, length, and width are functions of time--we must use The Chain Rule to differentiate.
Answer: dA/dt = 32 cm/s^2; increasing
b) We know the formula for the perimeter of a rectangle as P = 2L + 2W. Differentiate the formula using The Chain Rule, since P, L, and W are with respect to time--a function within a function! Then plug and chug.
Answer: dP/dt = -2 cm/s; decreasing
c) We know that the diagonal (D), the length (L), and the width (W) are related in The Pythagorean Theorem: the square of two sides (in this case, L and W) equals the square of the hypotenuse (D), so D^2 = L^2 + W^2. Differentiate the formula using The Power Rule and The Chain Rule.
Answer: dD/dt = -33 cm/13 s; decreasing
Related Rates: Shadows Including Similar Triangles
We know the height of the lamppost (L = 16), the height of the man (M = 6), the rate at which the man walks toward the streetlight (db/dt = -5), and the length of the man's shadow from the base of the lamppost (b = 10).
Since the man is walking toward the lamppost, by convention, the rate is negative.
b) Using similar triangles, we can see that the ratio of L to M equals the ratio of s to b+s. We simplify our proportions and differentiate to determine the rates of change (That's the definition of a derivative!). To differentiate, we use The Product Rule and The Chain Rule--Refer to Related Rates: Area, Perimeter, and Diagonal of a Rectangle for reference. We plug in the numbers to obtain ds/dt.
Answer: -3 ft/sec
a) Note this question is underlined in blue. The exam would never ask you to do part B because you need to do part B anyways to answer part A. The rate of the tip of the man's shadow is dP/dt. To obtain dP/dt, we differentiate P = b + s.
Answer: -8 ft/sec
Related Rates: Distance and Velocity and Spongebob!
HOMEWORK!
HOUSEKEEPING
Next scribe is bench.
Wiki constructive modification due Sunday midnight.
Developing Expert Voices projects due soon.
AP calculus exam is in two weeks.
Three AP calculus exam free-response per night.
Homework: Olympic Spongebob!
We will be reviewing applications of derivatives (related rates and optimization), applications of integrals (density and volume), and techniques of antidifferentiating (integration by parts).
I really need to stop with these late night scribe posts.
For the record, Joseph is scribe, and the list REALLY needs updating.
Question (1)
In this question, the answer should be straight forward. If you approach X from the left (low), then the largest integer value is 1. This is demonstrated in the first line. If you approach from the right (high), the largest integer value is 2. These are also known as roof and floor values. Because the right approach (floor) is not equal to the low approach (roof) then the value cannot exist, meaning the answer is (E).
Question (2)
For this question, we need to absolutely remember what the definition of a derivative is. Because this is it! Written right there. 1 is the value of f(x), which is also equal to sin(π/2). You will never actually see this, but you need to recognize it. Once you see it, this question becomes easy. The derivative of sin(π/2) is 0, because at sin(π/2) you are at the top of the curve and the tangent line (derivative) will be perfectly horizontal. Thus, the value must be zero, or (C).
Question (3)
Just keep deriving. You’ll get there eventually. (C)
Question (4)
A painfully simple solution to this question. To find a line, you need to have the slope and a point. You are given the point. Therefore, logically you should find the slope. Derive the equation of your curve, and you know that equation also equals the line with y = 1 and x = –1 (the equation for any line is y – y1 = m(x – x1). Solve for m with algebra. Once done that, do some rearranging with algebra, but the answer should become obvious as being (A).
Question (5)
Speed is the scalar version of velocity, which has vector (direction). When velocity is negative, you are moving backwards. Because speed is scalar and has no direction, it is the absolute value of velocity, so in terms of speed even if you are moving “backwards”, your speed will be positive because you are moving period. So if you re plot the absolute value of the graph, you will see the greatest value is at t = 8 which is answer (E).
Question (6)
If at t = 3 our object is at the origin, and its velocity is positive, it means the object is moving away from the origin in the positive direction for 3 units (the triangle formed by the graph from 3 to 6 is base 3 and height 2, so (1/2)2(3) = 3). So, we want to find where the object moves in the negative direction for 3 units, Since the value of the triangle created by the graph from 6 to 7 is base 1 and height 2, the area and total distance changed is 1 in the negative direction, meaning we havent gone backwards enough to be back at the origin. If we find the area from 6 to 8 then the base is and the height is 4, so by t = 8 you’ve moved 4 units negatively, which is past the origin, but we can logically deduce that at some point between t = 7 and t =8 we had moved 3 units negatively and thus returned to the origin. So, logically, the answer is (E).
Question (7) Free Response
For (a), you know S(0) = 6 so use that to solve for C. Quick math determines that C = 6 because e ^(0k) = 1 so 1C = 6. You also know that consumption doubles every 5 years, so S(5) = 2(6) and 2(6) = 6e^(5k) so e^5k = 2. Take the ln of both sides, so ln2 = 5k, and solve for k.
In (b), you want the integral from 3 to 13 (remember that your function counts years from 1980 so 1983 – 1980 = 3 as your starting value, and you want a 10 year period so 10+3 = 13 as your other limit) of the function. Calculator please. Really, you dont want to integrate by hand, but be my guest.
(d): “The integral of S(t)dt from 3 to 7 gives the total amount of cola consumed in the United States in billions of gallons per year during the time period from January 1, 1983 to January 1, 1987.”
Remember guys to start making your significant edits to the wiki solutions manual, and to at least try 3 exam questions a night. This is crunch time!
I dont really have a new funny/interesting youtube video, so here’s Intergalactic.
Is keeping first things first important as you come down this last stretch toward to AP exam? And if it is, how are you at making sure you tackle those 3 suggested review problems every night?
I’m hoping that in my sharing one of my experiences, you can find some bit to help guarantee your success on your test day! In early 2003, I too faced an important test date and I too appreciated the opportunity and felt I had to find a way to achieve. Achieving this goal was too important to me and my test if I passed would lead to a national teacher certification (only 40% of test takers passed). I was really trying to be superwoman—just as I imagine many of you work at being superteens. I thought I was prioritizing but ooohh---a to-do list with 20 items all the time!!! Urgent “stuff” kept happening and I was always responding to that. Does that sound familiar?
I was overwhelmed!!!! I sat down and broke my test review into manageable chunks (I had 2 months until the test) and put them into my planner. I did that first because the goal to pass the test was so important. And I planned to turn off instant message, not answer the phone, or have the TV on during the review sessions. I had tried before but was always interrupted by the phone or my students on instant message with questions about our studies.
Then I looked at my other “stuff”, and categorized it: vital, important, or nice. Then I took the vital “stuff” and categorized them again: vital, important, or nice. And I let go the nice. So my house wasn’t very clean during the process, and we didn’t have gourmet dinners. But the laundry was done and we had quickie suppers. My students’ work was graded but I didn’t plan any big field trips or projects during that time. I set aside a time every second night to evaluate their work. As I look back now, I prioritized, and then prioritized again. I did a mental daily check of my goals and made every effort not to be dragged down by urgent if it didn’t help me achieve my goal. Of course, I had to be flexible at times. I couldn’t always follow the plan exactly. But since I knew where I was going and I had planned for time to get there, my review was accomplished by “the day”.
I truly believe, that with good preparation and putting first things first, you'll too feel that great rush of a job well done, and a goal achieved when you learn your scores. I share these experiences, knowing that you are planning and reviewing, but wondering is there one little piece here you could use to help you on your way? Or can you point all of us to some tips that are really helping you manage to put first things first?
Almost forgot my youtube contribution. Thanks for reviving it Jamie :). And yeah, my reaction to the upcoming exam would probably be something like the Troll 2 scene XD.
And as usual, if you're too lazy to check the slides, the next scribe is Paul. Wiki questions are due at midnight tonight also, so get moving everyone.
So, as much as I'd like to say that I totally did this last night, I totally didn't. I 100% completely forgot, so here it is now, along with a special edition post test thought analysis kinda deal.
So Differential equations. For the most part, I found this unit quite easy conceptually. I understood most of what I was supposed to do and when and things, but I would always make stupid mistakes which cost me the whole question. I think this is mostly a matter of needing to do more questions so that I get out of the habit of overlooking those little bits. That said I don't think I did so well on the test D:
For the most part all I can really say about the test is I blanked. I was feeling pretty confident in the concepts and applications of them, and then as soon as I sat down to write the test, it was like someone opened a window in my brain, and whoosh, it all flew out. bleh, to say the least.
Anyways I'm looking forward to reviewing further, and covering all the loose ends I have with the course, as at this point in time it feels like I have alot ;P
Like I said in my last BOB post, the differential equations unit is our final unit before the exam. We just did a test today.
I thought the test was quite easy. I completed the test with about 20 minutes to spare so I decided to do all of the free response questions. Yes, I did omit one of them.
How did everyone feel about the test?
The exam is coming in about a month. I feel pretty good but as a review I would like to go over the accumulation functions as well as the related rates problems. The related rates problems are giving me the hardest time in this course.
How does everyone feel about the exam coming up? Is there anything that you would like to go over?
BOB, again!!! Almost forgot to do it. OK I went through the Chapter yesterday, everything seems to be easy and also hard. I'm having trouble with most of the word problem question. In this unit you always have to think through the whole question and find the giving data. It's really easy to miss some parts. OK I have to o back to review now, good luck everyone!!!
I actually quite enjoyed this unit. It was a great finale to the cours.e I liked the end of this unit in particularly, but I didn't appreciate Euler's method because I kind of still don't understand it. I kind of don't remember it. ALl I vaguely remember is that it was a method used in slope fields?
The other thing that I had trouble with, was the basic math. Simple precal. I seem to be over thinking things. Not really anymore, since I've been practicing, so I feel really confident about this test. That part at least.
Newton's law. Again, the only thing was the basic math, but now, the bigger problem to me is how to set up the function. But seriously.. I've never felt better about any other test. Let's hope this feeling stays this way and hope that Euler's doesn't get out and take me down.
I understood most of this unit, and most of it made sense. Looking at the pre-test, the thing I want to work on most is identifying what kind of solution I should use for a problem. But I liked how a lot of the stuff we did before sort of mixed together in this unit, and it was almost like a review of older stuff (particularly, logarithms...). I found I sometimes got held up on the pre-calculus, which means I might want to brush up on that.
Also, Newton's Law of Cooling was a little confusing, but I think I get it. I only run into problems when solving for C or pinning down the original function when given a point on the original function.
I think the test should go well for me, and hopefully well for everyone else. Good luck guys!
So at first the unit started off nice and slow. Quite easy to understand for the most part. I liked it. Then it was weird because somewhere along the way in the middle of the unit, it was like my mind just blanked out and everything was blurry and unclear. Mind you this is just me.
The Eulers Method I will study up a bit on because I don't quite remember it... At all.
Newtons Method of Cooling I believe I had the easiest time with (being that it was taught last) and that I seemed to catch on how to do it. I felt comfortable doing the questions and they just seemed tedious, and sometimes mind wrapping.
When we first started this unit, it was quite easy. We learned a lot of stuff in this unit from our previous units. The Euler's Method when done by hand was tedious, but not hard. Slope fields were pretty cool to draw.
We went into the velocity and acceleration problems next. It's much easier to follow the question by writing down everything that's given in the question, and from the initial information, finding hidden information by doing a bit of work.
Newton's Law of Cooling was not bad in class, but looking back I can't remember much about it. I still have trouble on when to use certain equations and using the calculator to solve certain problems, so I'm still sort of worried. I just wish we had a class today for review or maybe a formula sheet, I suppose this would help by refreshing my memory on the different equations. Good luck on the test everyone.
So this last chapter was a nice closing to the course. What I had trouble with this unit was separating the variables. It wasn't the calculus that was messing me up but the algebra. After doing a lot of questions with separating, I'm used to doing them now without making mistakes. Some other things that have been messing me up on a lot of questions is antidifferentiating. I know it should be a breeze but I just can't remember that darn + c. It looks like it just doesn't belong there XD.
K so studying for the next 2 hours. I really wanna do good for this last chapter :(
Today's class was dedicated to the pretest. It was pretty straightforward and we are capable of getting nearly 100% on questions like these. I mean, I can say for at least myself that I seem to be over thinking things. Or if not, my mistakes seem to come from errors... that shouldn't really be errors. Most of my other mistakes are a result from not continuing my work. I'd have the right idea or I'd have the right concept, but I wouldn't think it through. The steps of these questions are like putting ideas from different units together. I'd get stuck somewhere.....but hey.. this isn't a BOB, or maybe I should just merge it in. Meh...I'll probably do one later.
Anyhow, I guess I'll explain the pretest. If you can explain something, your understanding of it can be deeper. In this question, an equation is given for acceleration. As well as that, initial values are given to help determine the equations for velocity and position.
The objective of the question is to find an equation that follow the conditions of the initial values and to help find the correct equation, it's important to remember, that acceleration is the second derivative of position. Ergo, the method of finding the position under these circumstances is antidifferentiating; once to get the velocity and another time, to get the position. C is the correct answer. Here, a derivative is given and it's asking for the AVERAGE rate of change. As soon as this is indicated, so is the pattern. The general formula of the average rate of change is 1/b-a multiplied by the integral of the function from a to b, with respect to x. From there, it's as easy as "plug and chug" to get the answer which happens to be A. This is the multiple choice question that confused me. Here, it really helps to write down the information that the question is giving you and think of other ways to skin a cat. One of the things the class learned was why Leibniz notation is very much appreciated. The question tells us that the slope of the tangent line [derivative] of y is equal to ey.
From there, we separate the variables and antidifferentiate, of course, not forgetting the "plus C" for the constant. In order to determine what value the constant is, we plug in the initial value to find the constant and it results in finding a general differentiable equation-- the answer for this question; letter c.
This is the last of the multiple choice questions. Another question that is easier to solve in Leibniz notation. This may look like a mini monster of a question, but one of the first things to do is to algebraically massage the equation to separate the x's on one side and the y's on the other.
Then, anti-differentiate both sides and find y. Again, to determine the constant, plug in the initial values given in the question and plug in that constant in order to determine a specific function, rather than settling with a family of functions caused by that "plus C".
Finally, there was the three part open response question.
For the first part, I was ashamed of myself... I spent ten more minutes on this part of the question than I should have. All of the needed information was already given in the question. All I needed was to plug the values in. In part B, I got the right idea but I didn't get the right variables for Leibniz notation and I felt like i was going in circles. But I HAD IT!!! This question is following a very similar pattern to those questions from the multiple choice. Isolate variables, integrate each side to get the parent function... or the family of parent functions and solve for the constant to get the explicit equation. Another way of checking if the equation is correct is to plug in the initial values and see if they are equal to each other. Finally, since we've found the parent function in part B, all we have to do for this question is plug in the time it takes and determine the velocity.
There's the pretest for you. A quick reminder that Mr. K mentioned that he will not be here tomorrow and on our test day.. which is the next day after tomorrow. There isn't going to be a class tomorrow so.... enjoy.. your hour? haha.
The rest of the slides are question that are probably homework/review questions in preparation for the test.
Also.. our assignment.. Claim your wiki-questions soon!
Next scribe is....Kristina.
ZOMG>>>!! I almost forgot about our youtube tradition [that was broken...]
This one is really.. just.. incompetent.. but I love this movie.. it's been so long.. since I've seen this.
This unit was a breeze after what we've done in the past units. The thing that I liked about this unit was that it combined almost everything we learned in the past units including the derivatives, integrals, and all of their applications. The slope fields part of the unit was something that I was looking forward to since the beginning of the course. Some of the students that took the AP Calc Course last year showed me the kinds of things that I will be learning and I saw the slope field. I was confused at how something like that could be drawn on the Cartesian Plain. Well now I know.
This is our last unit guys and it will be time to do some more crazy studying for the remaining of the month before the exam. With enough studying we will be ready to take on the exam.
Good luck on the test on Thursday (?) and on the exam next month.
Looking back, this unit wasn't that bad. Most of the content in this chapter were things that we should already know, such as the first and second order equations. Those parts weren't so bad to deal with. I found myself becoming quite fond of the Newton's cooling law questions as well. They were fun to do once I actually started getting past my pre cal problems with them.
The slope field and Euler's method things were so so for me. I admit having some trouble seeing a certain function within some slope fields, it takes me a while for the more difficult ones. As for the Euler one, doing that manually was a real pain in the butt. There was so much more room for errors that I had to restart the assignment that was given around that time about five or six times. Yes, I messed up that much.
And of course there was the differentiating by separating the variables using the Leibniz notation thing. I really didn't like the Leibniz notation at first, but I can see why we should learn to love it. Admittedly, I did fail hard at recognizing where to do the separation of the variables on the pre-test. Maybe I'm just too used to the other notation better, or I was thinking too hard but I think I should be able to get it on the test! Other than my utter lack of ability to spot when to separate, I'm confident in actually differentiating using this method when I actually do spot it.
Emm..I think that's it. Yes, good luck on the test on Thursday my fellow peers. I shall now go back to cry at the fact that I can't edit my Question 9 on the wiki page.
The intro exercises were simple at first since it's a review of chapter four, including those questions where we had to antidifferentiate the acceleration due to gravity twice to get displacement, until they started asking questions that I haven't learned until later on in the chapter. Luckily, I caught on, but, unfortunately, I still forget the +C when antidifferentiating...and it becomes more of a problem when there are more than two +C's for those questions where we antidifferentiate both sides after separating the variables. And antidifferentiating e^-t...I'll be prepared the next time I'm asked to antidifferentiate e^-t.
Solving differential equations graphically via slope fields, as I remember, was the easiest out of the three ways to solve a differential equation, since all there is to do is draw lines at each (lattice) point on the graph OR use prgmSLOPEFLD to draw the lines for me.
Solving differential equations numerically via Euler's method was a bit too much to absorb at first, but reviewing Newton's linear approximation technique was helpful since those two methods are similar.
Solving differential equations symbolically via separation of variables was a doozy, although I personally prefer this method over the numerical method, just because. Comparing the separation of variables questions in our book to the questions on slides, to me the slides were harder (and more helpful) since the slides had just more.
I'll admit the pretest wasn't my best pretest though, but I did keep track of where I went wrong! The main thing I need to watch out for are those +C's!
Today in class we were finally introduced to our Wiki Solutions Manual, a little project where we thoroughly solve one question from a set of ten given. We post the solutions on the blog in the first week, and in the second week we find a mistake or simply make another person's solution more elegant.
Easy enough, and this will prepare us for the DEV projects that are soon arriving deadline. We will use LaTeX code to make the equations a lot easier to write out, on the page itself on the right side bar, there is a link that gives all the codes right below the "sitmo" box.
The goal of this project is to come up with perfect solutions to the calculus questions posted, through various minds working and correcting a single problem. You may correct an already corrected problem if you believe something has to be corrected. You may check if a question has been edited by simply clicking "page history" on the post itself.
If you're still not sure on how to do this project, there is an explanation and example on the wiki page itself. Remember, first come, first serve as far as the questions go. The tab for the Wiki Solutions Manual is found at the top of the blog by the banner.
We will have a pre-test tomorrow, on our unit of Differential Equations.
We had a quiz today and we were allowed to use calculators.
Find what x is equal to, when y is equal to one, by inputting the y-value into the given equation. Find the derivative of that equation, and bring the y-derivative value to one side, then input the points that you found from the first step. We do this because the derivative on a point on a graph is the slope.
Question 2.
The area under a graph is found my integrating it over a certain interval. We're trying to find what interval that is. If the area under both graphs is equal at a certain interval, we equate them and find the integral. Then we solve for k.
Question 3.
Integrate it. Use chain rule. The interval given is 1 to 2x.
Question 4.
We found the area from the x-axis to the graph for each part. I`m not quite sure how we did this. But I`ll find out.
We also did question 3 of our worksheet we got on Wednesday when there was a substitute teacher. We didn`t quite finish in class, but I`m sure we will go over the question in tomorrows class.
Good-luck on the pre-test everyone and the next scribe is Jamie.
Solution 1, is pretty straightforward. Since the variables are already on their respective sides, just antidifferentiate. Since we’re solving for C, we’ll put that on the left and move the rest to the right. Simplify. As Benchmen pointed out, because C is a constant, even if you multiply it it will remain C, hence the part that says “let (2) C = C”
Solution 2 is a little longer, but also pretty straight forward. Organize your variables to their respective sides, integrate/antidifferentiate both sides. Take the ln of both sides, and solve for A which is e^c. The solving for A is a bit of simple algebra.
Solution 3 I tried twice but kept getting stuck. I figure the way to solve it is to follow the same steps as in 2, IE organize and integrate, and once you have an equation (this is where I got stuck) solve for t using the given values.
Solution 4 I have no clue, sorry guys. Feel free to post your solutions in the comments.
Solution 5 I know HOW to solve, but cant actually do it because I get stuck (as you can see) at the point after integrating. However, this is very much like question 3, where you are given inital values. In this question, you would organize, integrate, and then use the inital values to pinpoint the graph since when you integrate/antidifferentiate you are finding the family of functions and you want the specific function that passes through a given point.
Solution 6 - 10 use Newtons cooling law.
As usual, this has been a late scribe post, although from what I remember on Thursday you guys are probably ahead of me on this stuff anyway. See you all tomorrow.
Oh and the next scribe is Francis.
Wednesday, April 8, 2009
Scribe again!!!!! OK, we start with finishing the honeybee problem. To be honest I didn't really get it, it's kind of hard for me to understand. I will skip that question.
Then we went to the quiz, parties exam questions.
In this question you use 2ND derivative test to find the concavity of the original function. I will use Mr.K's work red part is the 1st derivative, then he used green to find the 2ND derivative . After that he use the number line to complete the 2ND derivative test.
For this question there isn't any correct answer for it,I will explain it any way:
A graph will help to find how the triangle look like on the graph. Use the area equation: 1/2bh then find a(3) =54, because the coordinates of the triangle is (x, 27-x^2) use it to build the area equation. Then use the first derivative test to find the max and min, evaluate the area from then.
This is the mean value question that we have to use the formula to help us to find the answer. 1st we plug the values in to simplified it then evaluate it from 0 to pi/3. Find the anti-derivative of tan(x) by using substation. If you done every thing correctly the answer will be B.
i picked A for this one and I think I'm not the only one who gets wrong on this question. It's not as easy as I thought. We forget the dx part that it counts as part of the substation. That's where the 1/2 comes from, u find dx to replace dt, because x is use to replace 2t+1. Then you work thought that you should end up with D.
Mr.K will be away tomorrow and the next couple days, someone else will be here for him but he doesn't know who.
Next scribe will be...paul......lucky get the long weekend to do!!!!
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*Red is what Paul stated. Blue = edited work.
*Red is what Paul stated. Blue = edited work.
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