The Return of Optimization!

Optimization Review

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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!

Mini Exam Review Time

AP Calculus - Mini Exam Review

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WARNING: CONTAINS SOME FOUL LANGUAGE

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.

BOB: Differential Equations

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.

Scribe Meets BOB!

Pre-Test: Applications of Integrals

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Yes, so I'm going to input my BOB in the same post also! So here I go!!! Also, what the heck is that ugly bar on my front slide? Oh well, whatever. I'm just glad slideshare finally worked.

Anyways, I thought this unit was pretty hard to grasp at first, especially the rotation parts. I couldn't see how they made the washers and shells at first but after doing lots of practice with them, I can see the patterns! They're beautiful by the way. Yes, so the pre-test also gave me some confidence for the test tomorrow since I did pretty well. Although, I kind of blanked out and couldn't finish my long answer. I basically just left the integral but didn't go on with solving for k. Silly me, har har. I won't do it for the test though.

As for the hardest parts for me. It is definitely the density stuff like that Boston by the Sea question we took so long to finish. Although I'm still kind of iffy, I think I'll manage.

Yeah, so in case you didn't look through the slides, the next scribe is Lawrence.

And to follow with the recently started tradition, here's a youtube video. I'm in a massive nostalgia stage right now, heck, I even watched the first episode of the first Power Rangers again when I got home today. Great stuff man. Anyways, here's my contribution...its a bit of what's to come...if you catch my drift >_>

Varied Axis of Rotation

Varied Axis Rotations

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Again, if you're too lazy to look through the slides. The next scribe is Joseph.

Solids of Revolution: Introducing the Washer

Solids of Revolution

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If you're too lazy to bother looking through my slides, next scribe is Yinan.

BOB: Chapter 7

Hello everyone. Is everyone ready for the test?! I think I am... Anyways, I liked this chapter a whole lot more than the last few. Finding anti derivatives using substitution and integration by parts was hard at first but after more practice and help from those spare days, I think I got them down, for the most part. Though I still have problems finding out how to rearrange the function so that my "du" can fit into it when doing substitution. That's the most tricky part indeed. I will also need to remember my "+ C"s for sure, but after having done some more practice on anti differentiation, I'm sure I won't forget about them this time... I hope.

Yep, the midpoint, trapezoid, and error stuff seem to be okay. But we'll see if I really understand them on the test.

Okay, good luck on the test everyone!

More Fun with ArcTrigs! =D

*Yawn* Hello everyone, I have just woken up from my very long nap and my back is still kind of sore from slipping on the road today, but I am here now to scribe for today's official unofficial day!

Starting off, we were supposed to go onto some new stuff today but since none of us did our homework from yesterday, we ended up going through those homework questions instead.

Having already done (a) yesterday, we continued on with (b) and (c).

• Since the arctrig function is the argument of the outer trig function, we can make this equal to x
• We then took the argument of tanx (Opposite/Adjacent) and formed our triangle - Quick Note: To get a better visual of which Quadrant you're in, draw your triangle as if it were in that Quadrant. Most of the triangles drawn on today's slides didn't follow this but Mr. K said it was better to do it this way
• After making the triangle from the argument of tanx, you can now solve for secx
• Follow the same procedure to get (c)

These questions are a bit trickier, considering the fact that you have to remember your sine/cosine/tan(?!) dances.

• (a) is easy enough to do in our heads since they are easy angles that can be taken off the unit circle
• After finding the arctrig values within the brackets, take the sine of them and from there it should be smooth sailing

• (b) is not as simple as (a), since you don't have nice pretty unit circle values this time
• Like the previous question, you take the arguments of the outer function and make them equal to, in this case, since there are two arguments inside, we shall make one equal to x and the other equal to y
• Make your triangles for x and y
• You should now notice that you have enough information to complete your cosine dance! All that's left is to input the values and then do basic algebra. Tada!

• The part in the box was just some small explanation on the different between an arctrig and its inverse.
  
• As you can see, if the whole expression is to the exponent negative 1, then that is the reciprocal
• Do not confuse the reciprocal with the arctrig
  

• Same deal as (b) with this question.
• No need to make a triangle with arctan since taking the tan of an arctan will just leave you with the tan of that value anyways!
• After making your triangle for y and finding the value for tany, you can now use the tangent law thing (I've forgotten the name for these things...addition law?) and solve. The part on the left side was just Mr. K building his tan law thing.

• This question basically gives away to you what quadrant x is in. Since it says that x > 0, that means that it will be in quadrant 1 for both questions
• From then on, its mostly the same procedure as the last few questions. Take the inside argument and make it equal to y (x is being used this time so use y instead...or whatever variable you want)
• Make your triangles
• Solve
• ???
• Profit!!

There was also this cool little trick that Mr. K taught us to solve for squaring two-digit numbers. I'm going to use 12² as my example:

• Take the nearest multiple of 10 of the number, n, you are trying to square, in this case its 10
  
• Find the value needed to get back to 12, in this case its 2.
• We then add 2 to 12 to get 14.
• Multiply your (n+value) by (n-value). So from our example we should have (10)(14) = 140.
• Now go back to the value we needed and square that. It should be easy since its only a one-digit number. From our example, we get 4.
• Add the number you just got to the product we made and voila! We got our answer!
  

That was all for today's class. Homework is to graph the three arctrig functions from memory. Don't cheat! As for the last scribe for this cycle, that shall be Yinan! Now have a nice day as I go back to rest on my back and maybe kill some more brain cells with my super long naps.

BOB: Integrals (Ch. 6)

So, after a bit more last minute studying, I'm finally here to BOB!

For the most part, I found this unit to be easier than the last few we've gotten. It was also a helpful review and helped clear some of the fogginess I had with the first Integrals unit.

What I found to be the most challenging parts were the Accumulation parts at first but now I think I kind of got that down now. I also remember looking at some questions in the exercises where it talked about Non-Odd or whatever it was and being completely confused so I'm still kind of iffy on those parts. I also remember being confused about that rational rule thingie (I also saw it in the exercises..it was asking to explain how this is greater than that in words or something like that >__>). Yeshh..

The easiest parts were definitely finding the derivative and such from the integral and also finding the area between two curves, which I found kind of fun :P.

Well, then. There are still some things that are still a bit foggy like that part with underlying functions..but if I think hard enough I can get it to wrap around me head. Also, after looking at the pre-test and such, I can say that I'm in good shape...for now. The last question was a bit hard but the rest was pretty good. Anyways, I should get going now! Good luck on the test everyone :D

J2K Timeline [SORRY FLJ for the letter stealing. At least we acknowledge our lack of creativity for the name and at least we used a number.]

Wow. Compared to Team FLJ, just wow. Sorry we had to ruin it by not having our own graphic and colour to this post, but at least the work is done. Or this part, anyhow.

Here's our timeline, also subject to change.

JANUARY 8: Making of timeline COMPLETE

Record copied examples of questions for project reference. COMPLETE

• Copying a rough draft of 6 questions from textbook/resources to use as references for the types of problems to use in DEV project; chosen based on difficulty we had with these questions so that we’d spend time learning on how to solve these problems.

JANUARY 20: Brainstorm on possible ideas for project to have more appeal for learning.

FEBRUARY 5: Continue brainstorming possible theme ideas and have made the decision on what to use.

FEBRUARY 6: Create 6 questions from reference problems.

• Start the process of creating six of our own word problems that relate to a theme; using the type of questions we chose, with first, learning how to solve these problems.

MARCH 15: Have the majority of the 6 created questions solved.

• Answer as many questions as possible and make sure that we're getting the right idea by using resources, such as asking other people, reading textbook, internet and asking Mr. K for greater understanding.

MARCH 20: Preparation of extra matierial.

• This is for the "look" of the project, the part that will make it visually easier for people to learn the material. Basically getting aspects of the theme together using videos, slides, pictures etc, that abide by copyright rules.

APRIL 12: Finish answering and explaining questions.

• By this time, hopefully we'll have all of the questions answered and explained thoroughly enough to be taught to readers of our project.

Media is cited and sourced ready to put into DEV.

APRIL 17: Merging of all aspects of project for preparation of final structure.

MAY 1: Finalization, adding finishing touches getting ready for immediate publication on the blog.

MAY 2-3: Submit to Developing Expert Voices blog as a big finale.

There's our timeline and we hope to follow it, and finish this ASAP.

BOB: Applications of Derivatives

Hmm, well after doing the Pre-Test, I can say that I'm kinda in good shape for the test. The only problems I really had with it was the long answer question, specifically the last two parts of it. Although, from what Benchmen told me, I had the last part kinda right. >__>

As for this unit overall, the major problem that I had, and I'm assuming the same goes for the rest of the class, were the Optimization Problems. Just when I thought the Rates of Change questions were bad, these types of questions had to come into the big picture. Really, although I'm pretty confident I can do the square/rectangle ones and NOT the inscribed stuff since those are really hard for me -__-;

Yes...as for the rest of the unit. I can say that I really enjoy finding critical points for some reason, plus I was able to learn how to use chain rule on the natural logarithm because of critical points questions, so I feel quite fondly of them :(

Mhm, so good luck on the test everyone and if it ends up not being tomorrow don't shout at me again like last time haha XD

Post Holidays Day 2

Since there were a few people who were taking their Provincial English Exam today, all we did in class was Page 293 (Supplementary Problems) ODD.

Might as well add the fact that the test is tomorrow (According to Benchmen, don't blame me if I'm wrong again XD). Yes, so don't forget to BOB for those of you who haven't yet! :D

Next scribe will be Shelly.

Big and Tall...Oops, I mean Small :P

Today's class started off with Mr. K giving us a small chat over the tags on our blog. Since we had so many due to most of us tagging our work wrong, he went over what the right tags to use for our posts were. Here are the main three tags we are to use every time we post something onto the blog:

• Name
• Type of Post (Scribe Post, BOB, On My Mind, or Links for Learning - Explained in some detail later)
• Title of Current Unit (They are shown in the book, or, you can see the slides for them)

Now to get into more detail about the Types of Post, especially the last two which are indeed new to all of us.

• Scribe Post: Self-explanatory, this is what we use if we're posting the scribe for the day
• BOB: Use this to tag Pre-Test thoughts and such, we should know how to use this by now
• On My Mind: When posting something of interest that isn't a scribe post or BOB, use this tag. A good example of this tag would be one of Benchmen's standalone posts that wasn't a BOB or scribe post (See post after the last test date where he was asking for opinions on the test)
• Links for Learning: Mr. K will mostly be using this tag to post links that will help us with current or past units (Will have the unit tag along with it). I believe students may use this as well.
  

We then moved on to talking about the video assignment, "What is a Derivative?", that was previously talked about on the day Mr. K came back. The details are as followed:

• Create a commercial describing what a derivative is
• Must describe using little to no algebra
• Cannot use derivative rules
• Must be 30 seconds MAX, there is a 10% deduction for every second over the max time limit
• Must be posted as a video response to Mr. K's video which will be posted on youtube (Will be posted on blog with the tag "Calculus Commercials"
• This will not only be open to us, but to the whole youtube community. Who knows? We might even go viral!
• No specific due date but must be done by the holidays!

After that, we then moved onto a few questions that was review for yesterday's class on limits.

Like yesterday's questions, first start off by expanding the numerator and denominator. Once done with that, you multiply the numerator and denominator by 1/x² / 1/x² since that is the highest degree in this question. This, as a reminder, reduces the highest degree to 0, leaving us only with 6/2, along with the remaining parts of the numerator and denominator divided by a degree of x. Those go away as the limit reaches infinity and we are now left with the horizontal limit equaling 3.

An easier way, as shown boxed in red would be to just take the coefficients of the leading terms and dividing them since all we care about are the coefficients. It saves us time and thus makes a seemingly long question a lot shorter.

Now onto a quick review of the 1^st and 2^nd Derivative Tests.

• 1^st Derivative Test: Finds critical numbers, as well as looking for where the parent graph is increasing or decreasing
• 2^nd Derivative Test: Determines the concavity of parent graph along with finding inflection points

In reality, these two tests both do the same job. That same job being the fact that they both can be used to max or the mins of the parent graph. Although, they do require different procedures to go about it.

From there, we then started the new stuff. Optimization Problems. These are basically problems where we are looking for where a function is big and small (max and mins). An example would be owning a business and trying to make as much money as possible while keeping production costs at a minimum.

There are six steps to solving Optimization Problems. They are as follows (Click to enlarge image for easy viewing):




We then went on to using these steps to solve a problem.

An open box with a rectangular base is to be constructed from a rectangular piece of cardboard 16 inches wide and 21 inches long by cutting a square from each corner and then bending up the resulting sides. Find the size of the corner square the will produce a box having the largest possible volume.

First use Step 1 to find what we are trying to maximize or minimize, in this case it is the volume.

We then follow Step 2 and write an equation for what we found in Step 1, which is the volume. So...



I've included a diagram so you can get a better grasp of the situation, plus they look pretty :). We then move onto Step 3, which is take the optimization equation (which was made in Step 2) and turn it into a two variable equation so we can easily get the derivative. So just substitute in the values from the diagram into the optimization equation and you'll get this:

We need the domain of c first so we can then use the quadratic formula on the derivative function and then take the value within the domain. That will be the size of the corner of the square which will provide a maximized volume, thus answering our question. If we continue on and do the quadratic formula, we would then find out that the only value that fits into our domain is 3. Thus, the size of the square that will provide us the maximized volume is a 3x3 one. Tada, we're done!


Well that was fun, the next scribe shall be, following Francis' lead on taking the next person on the list, Joseph. I refuse to call you Master Joseph :D. Oh yeah, homework is Chapter 5.4 Questions 1-6. Alrighty then *puts in PROPER tags*...*leaves*

BOBby Flay: Differentiation Rules

Well....this unit was kinda half and half for understanding for me. The first half of the unit was simple enough for me to understand since it was mostly just learning the rules of differentiation and then applying them some functions. Although, admittedly, the chain rule took me a while to properly use and I remember looking at the quotient rule and thinking "Whoa.." but that got better after Mr. K taught us the song (Which I probably didn't sing enough since I forgot the LO-LO part during the pre-test XD).

Anywho, the only thing I can say that is really difficult for me are the rates of change problems. The easiest being the triangles, although still hard to think through. Its especially discouraging to think of the fact that we are only given 15 minutes to do these types of questions during the exam, while for me, it takes usually about an hour to thoroughly think things through and actually get the right answer. I was also confused about the rates of change question on the pre-test since I didn't know what to do with the graph. It was just like "Hey, look a graph...hmm..its about 5pi at 4 seconds. Now what?". Maybe I'm making these out to be harder than they are, but I guess I'll just have to do a crabapple load of these types of questions till I can finally get it within 15 minutes >__>.

Okay, I should stop now before I write an essay again. Now back to reviewing..and good luck everyone! d=(^__^)z

Friday's Scribe

I am the scribe for Friday and sorry for saving this till the last day..but I'm here now!

Anywho, we started off the class with a rates of change question in which we were split into three groups. The question was incredibly long to read and it was about a dingo?..named Wallaby. Instead of explaining the question to you, here it is!

Willy the Wallaby is a hard worker, but not devoted enough to work at the circus for almost 70 hours building a tank just so he could be shot out of a cannon like a metal ball. He is convinced there is a better job out there (and he wants to at least keep a shred of dignity). So Willy joins up with his friend Roo and his old coach Spike to seek out a tough guy name Wyman Weasel who has been sending out letter for a covert employment operation offering great pay for the ones with enough "guys and smarts" to find the secret hangout. The instructions began at the intersection of Crime Dr. and Twotough Blvd. The streetlight on that corner is 36 ft off the ground. The only ones who can get the job have to figure out the rate they have to be moving toward the south end of Crime Dr. in order to have their shadow changing length at a rate of 11/35 ft/sec. Once the applicant found the rate, the next instruction was to move at that pace and an escort will approach. The escort will take any animal that moves at the right pace to the clubhouse door. The final password to get in is the rate of change of the tip of the shadow. Willy and his friends want your help to finally get the good paying job they deserve. Find the rate Willy and his friends must move if they are all 4 ft 6 inches tall, and then find the rate of change of the tip of the shadow.

OKAYYY!! First, let's draw a handy dandy diagram. Its important to get the visual first when doing these types of questions to see what you're working with.


Now, we are given the rate the shadow, s, is moving, which is 11/35. We also know the height of Wallaby and his pals, which is 4'6, convert that 6 there into feet and we got 4.5, as shown in the diagram. We now have all the given elements to this question and we are asked to find the rate Willy and his friends are to move with that given height, in other words, we are looking for dd/dt. The second part asks for the rate of change of the tip of their shadows, so the last piece to this puzzle would be da/dt.

From there, we can then take a closer look at the diagram. Notice the relationship between the bases of the two triangles compared to a. That's right!

We can also derive the derivatives from that relationship in respect to time. Now with that, we can then move onto the the next inspection of the diagram. There are similar triangles, so let's make our ratio and solve for the rate of change of d.

If you're wondering how we went from line 2 to line 3, we basically just multiplied both sides of the equation by 36 so that it is easier to solve for dd/dt. This also helps save time, since on the exam, we are only given about 15 minutes to solve these questions (We took more than 30 mins to solve it in class XD).

Alright! Now that we have the dd/dt, we can now solve for da/dt. All we have to do is put dd/dt and ds/dt back into the equation dd/dt + ds/dt = da/dt. Voila, we're done! You've now successfully helped Willy and friends :D!


Now, the next part of class consisted of us trying to find the root, using Newton's Method (x₀ was -0.5) for the function:

f(x) = x + sinx - 2x + 1

Let's simplify this first...

f(x) = sinx - x + 1

Find the derivative next which is...

f'(x) = cosx - 1

Now we can use Newton's method to find x₁. If you don't remember, the formula for Newton's method is as followed:

Using it is as simple as it looks like. Take the negative of x₀, in this case it is -0.5, and divide that by the derivative of x₀. Then add x₀ to that to get x₁. As you can probably see, it doesn't equal zero, so we have to keep going. Instead of using x₀, now you use x₁. If that still doesn't equal zero, then you keep going with x₂ and so on till you get zero.

Mr. K also taught us how to use Newton's method using our calculators.

-First you have to graph f(x) into Y1.
-Then, you graph f'(x) into Y2.
-Now 2^nd QUIT and press VARS, Function, Y1.
-Put an open bracket from there and input the value for x₀.
-Close the bracket, then press STO-> (above the On key) and then Alpha A. This stores that value into A.

Go back to step 2, instead of going to Y1, go to Y2. The rest is the same till the last step, you'll be pressing Alpha B instead to store it into B. We've now got all our elements needed to solve using a calculator.

From there, first press the negative button and then Alpha A. Now divide that by Alpha B. Then add -0.5. Yippee Kayeeh! You've now found x₁ using your calculator! Now repeat the steps using the consecutive x values found till you get it to equal zero!

*Pants* Alrighty now, the test is on Wednesday! Be sure to get your BOBs out by then. Also, if you haven't done it by now, Chapter 4.7, questions 11-22 ODD...I believe. Okie dokes, I am done. Oh yeah the next scribe is ...........Yinan. Have a nice day. =)

Long Lost BOB: Where Are We?

Oh BOB, how I missed you so but not as much as how I, and I'm sure my fellow peers would agree, missed Mr. K. Welcome back you snapping clickity click math teacher we all learned to love.

Anywho, back on topic. Where am I?...is the big question. To put it bluntly, not very far. The first unit was obviously pretty good since it was just review of last year's stuff and the only problems I could say that I had with that unit was trying to jog my memory (I am easily forgetful of things). Overall, nothing more to say here...moving on.

Chapter 2.......TT___TT. I can still safely say that I don't quite understand this unit all that much. I understand that a derivative is the slope of a tangent line and all that jazz, but when it comes to graphing the derivatives of functions...oh boy. Graphing was always my worst enemy and it didn't help that I didn't get how to graph the derivatives in the first place. The only thing I'm confident about graphing for derivatives is the fact that if there's a critical point on the original function, the graph of the derivative will be at 0 there since the slope of the critical point is 0.

Now the last chapter. I can admit that it was a bit easier, but a bit doesn't cut it when I still don't understand derivatives. It especially didn't help when the test's last few questions required knowledge of derivatives and how to GRAPH it and stuff. I really raged hard at that and I'm getting kinda depressed just thinking about it again. Anyways...*sigh*...I understood most of the concepts for this chapter and last year's calculus blog definitely helped with that so I guess there's nothing more to say on this side.

Yep, that's all I have to say really. The current chapter is actually going quite better than the previous two, although I'm kinda iffy on those rates of change comparison things. I think getting more practice would help for sure since I can say that I understand the differentiation rules (at least I think so).

Kay, this is getting a bit long and I know Mr. K doesn't want an essay so I shall stop there. Most of my feelings were expressed and letting anymore be known wouldn't be so pleasant. So toodles for now. :D

Pre-Test: Definite Integrals

Today we had our pre-test for the unit Definite Integrals. Since Dr. Eviatar is currently smart board handicapped and cannot post any slides, I shall steal the one from last year's blog :D

AP Calculus Slides October 30, 2007

View SlideShare presentation or Upload your own. (tags: apcalc07 definiteintegral)

Anywho, the first two questions are basically the same type of question. All you have to do is input the functions given into your calculator and then use fnInt to get the answers. Easy peasy.

The next question is basically just thinking each of those answers through and if you could understand what each are trying to say, then you'd find that the only true answer is D.

Question 4 is a bit trickier. First you have to make the function so that you could write it into the definite integral form. After constructing the function from the info given [(5t+4)-.5t²], you then make your definite integral and add 10. The reason you add 10 is because at t = 0, the tank should contain 10 gallons of water. From there, you just solve and voila, you should get the answer on the slide.

Number 5 is just a simple analysis of the graph given.

Part A for the last question shouldn't be too hard if you had been keeping up with your homework. The reason why the interval shown is from 0 to 4 is because it takes 4 years to get to 1996 from 1992.

Part B just deals with finding the values for f(t) for the first 4 intervals. You just start with f(0) and go up to f(3) then add them all together. d(-_-d)

Part C was just asking to basically write a story about how the oil consumption was changing during the interval given in part B.

Alright that's it. The next scribe is not Justus, but Joyce. Good luck on the test tomorrow everyone and a reminder to the scribe that you don't scribe on test days! =D

Working with Derivatives

Hello again everyone, tis I, Kristina, back for another scribe post! Today we had the wonderful Mrs. Karras as our sub for today, along for the next 3 days.

Anywho, we mostly worked with the formula of a derivative (slope of a tangent line). If you've forgotten what that formula was, here it is again:

While Mrs. Karras finished writing that up on the smartboard, someone asked to re-explore what a limit was. She reviewed it to us briefly, explaining that a limit basically means that something is approaching another without touching it. An example she gave us was that she could get as close to Paul and invade his personal space as much as she wanted but she couldn't technically sit right on top of him since there was basically no room to do so and such, also, he may have exploded =(

She also bestowed us the gift of notes as to what the derivative of a function is in more detail. Due to her not being able to post the slides with these notes on it till Thursday, I shall kindly post them here for anyone that missed it or was not quick enough to take them down.

"The derivative of a fcn (a gift she bestowed upon us, it is short for function) at a chosen input value describes the best linear approximation of the fcn near the x-value under consideration."

With the help of this graph (I forgot the arrows at the ends of the parabola xD), we can see that in this case "a" is the inputted value mentioned in the above note and the ..faded...red line is the linear approximation of the fcn near that inputted value, "a".

"For a fcn with a single real variable, the derivative at that point equals the slope of the tangent line of the graph."

Meaning that we can input that value into the derivative formula since the definition of a derivative is that it is the slope of a tangent line.


Wahoo, now for some examples she gave us!

In the first example, she gave us the function f(x) = 2x² - 5x + 6. We were told to find f'(4). Solving this is as easy as bringing paper plates and cups for Pi Day (Yes, I'm thinking about it already). Back on track, all you have to do is input the value 4 into the derivative formula shown at the top of this post. If you're still lost, here is what it should look like as you work through the problem:

Tada! The derivative is 11 since as "h" approaches 0, it is shown that the value is close to 11 thus the answer. Simple algebra, right?! And since some people may make mistakes in their algebra during a question like this, here's a song to help you (I was bored so I decided to search "math" on imeem TEEHEE)

Kay, so it doesn't exactly help with the algebra...sorta, but I thought it was kinda neat so HA!

Back on track! The second example we got asked us to find the derivative of f(x) = x² - 3x for a number "a". How we solve this is the exact same as the last example, except we are working with a variable, "a". Once solved, we should find out that the answer is:

Now, if we were given a question where it asks us to find the derivative f'(1) for the same function given in the last example, you can do it the long and tedious way (but not hard!) by using the same procedure as example 1, but you don't have to! This is because we've already found the derivative for f(x) = x² - 3x earlier, which is 2a - 3. We can simply think of that as a function and then input 1 into that and voila! Simple and easy, also elegant.

For our last example, we were given the function f(x) = x² and were asked to estimate the derivative at the points {-2, -1, 0, 1, 2}. First we must graph this. After you have graphed it, draw the tangent line to those points and then just take an estimate at what the slope of that tangent line is. To check your answer, just input the points into the derivative equation and that's that!



Now that that's out of the way, the next scribe is.. Not Paul. Thanks for the confusing name, Paul :(. Oh and starting from page 100 up to 105 is review on some of the stuff I've talked about. Also, homework is on page 105. There are 8 questions but just do enough of them till you understand it. Anyways, I'm done for tonight. Have a nice day :)

September 15, 2008

Hello everyone, its Kristina with today's scribe post. Today we found out at the beginning of the class that we were behind in our studies since we were supposed to be halfway through chapter 1 but were not! After finding this out and a little bit of chit chat, Ms. E (I don't know how to spell her name, sorry =( ) told us to work on Exercise 1.4 in the book which was on Exponential Functions. Yes, more review stuff that we should already know.

To help jog your memory about Exponential Functions, here's a link from our blog last year that leads to the page with all the posts tagged with "Exponential Functions".
http://pc40sw08.blogspot.com/search/label/Exponential%20Functions

Also, while I'm at it, here's another link leading towards the page with all posts tagged with "Exponents and Logarithms" since some of the stuff in Exercise 1.4 deals with some of the things in that area.
http://pc40sw08.blogspot.com/search/label/Exponents%20and%20Logarithms

Well then, that is all for today. Next scribe is Joyce. No hard feelings since I put your name in red okay? There's no deep meaning to it =).