**sigh** It's over.. One week has flown by. But it still seems so slow...Winter is quite...well, I hope you just end. I want it to be...another season. Another season where people can be outside. But meanwhile.. the majority of the break, some people are bored to death, doing house chores, or just finishing school work, if we don't go out at every opportunity we get.... but my point is.. while some people are doing that, other people in the world are doing this.

Oddly, this video made me happy. It's a great holiday vid, despite the random clothes swapping. It scarred me at first, but...then I just kept looking.. I love this song. Especially after you, Kristina... **glare**. I'll probably still be singing this on the first day back. Anyhow, here's the vid:

FAB 5 - JINGLE MY BELLS FA DAYS by Mark "Markooshka" Kanemura and friends.

That made my days...

## Saturday, December 27, 2008

## Thursday, December 18, 2008

### Determining the Properties of the Parent Function Using the First and Second Derivatives

OVERVIEW:*************************************************************************************

- Calculus Commercials: How Many Ways Can You Define the Derivative?
- Mathematical and Computer Science Definitions of "AND" and "OR" Notes (slide 10)
- Determine the Properties of the Parent Function Using the First and Second Derivatives (slide 2 to 9)

1. Calculus Commercials: How Many Ways Can You Define the Derivative?

A continuation of last class' lesson, but first, we watched calculus commercials, which can be accessed here:

http://apcalc2008.blogspot.com/2008/12/gauntlet-is-thrown-down.html

The class agreed that the time restriction of 30 seconds to define the derivative was challenging, but there are our finished products! Note that people have commented on our projects.

*************************************************************************************

2. Mathematical and Computer Science Definitions of "AND" and "OR"

Let

A = go to the store

B = wear the hat

One day, you aren't sure if you should go to the store or wear the hat. So you think and think and think, but in terms of mathematical and computer science definitions, you notice that their definition of AND and OR are different from the AND and OR of English. The OR used in math is the inclusive OR. The OR used in English is the exclusive OR.

Here is a list of all the possibilities:

- You go to the store and wear the hat. (A = true; B = true)
- You go to the store but not wear the hat. (A = true; B = false)
- You don't go to the store but wear the hat. (A = false; B = true)
- You don't go to the store and you don't wear the hat. (A = false; B = false)

A ^ B = satisfy the conditions of A AND B

*************************************************************************************

3. Determine the Properties of the Parent Function Using the First and Second Derivatives

Using the derivative rules, we found the roots of the function.

Remember, k is a constant, not a variable!

We let k equal a negative number, like -1 000 000, in f' and discovered that to the left of 1/k, f is increasing, so f' there is positive.

We let k equal a positive number, like 1 000 000, in f' and discovered that to the right of 1/k, f is decreasing, so the f' is negative.

Since f is increasing to the left of 1/k and decreasing to the right of 1/k, we can imagine that at 1/k, there is a local max.

Remember, k is a constant, not a variable!

In the previous slide, we evaluated f' when 1/k>0.

In this slide, we evaluated f' when 1/k<0. style="font-style: italic;">

Since f is decreasing to the left of 1/k and increasing to the right of 1/k, we can imagine that at 1/k, there is a local min.

We have now obtained as much info as we can from the f'. On to f''!

Using the derivative rules, we determined f''.

We determined the roots of f'', so we can determine the sign of f'' on either side of the roots; this info can be used to determine f's concavity to either side of the roots.

We let k equal a negative number, like -1 000 000, in f'' and discovered that to the left of 2/k, f'' is positive, so f is concave up.

We let k equal a positive number, like 1 000 000, in f'' and discovered that to the right of 1/k, f'' is negative, so f is concave down.

Remember, k is a constant, not a variable!

In the previous slide, we evaluated f'' when 2/k>0.

In this slide, we evaluated f'' when 2/k<0. style="font-style: italic;">

Synthesizing all that we know, there is that list of properties f has.

- We know where f is increasing or decreasing at certain intervals.
- We know f's concavity.
- We know where f changes sign.
- We know f's local extema.

END NOTES:

- Homework: the rest of 5.5 Antiderivatives
- Looking forward to the rest of the presentations tomorrow!
- Ending our scribe for 2008 is Francis.
- Reminder tomorrow's class will range from 10 to 25 minutes, with the Holiday Inn Gym Riot, starting at 10:30 AM, hosted by the DMCI Student Council. We will be called down to the gym between 10:15 AM to 10:30 AM. See you there! =)

Labels:
Applications of Derivatives,
Calculus AB,
Scribe Post,
zeph

### Today's Slides: December 18

Labels:
Applications of Derivatives,
Mr. Kuropatwa,
Slides

### Ryon Lovette - Equation

So I was bored while studying, and I found Math and R&B. =)

Equation - Ryon Lovette

Equation - Ryon Lovette

Couldn't help to notice you standing in the hall way. Tears rolling down your face, girl. I heard him tell you that he was sorry for breaking up with you But you're way to beautiful to cry. Soon as he walked away, I came up from behind. I told you a joke to make you laugh. I'm nothing like him girl, you do the math. (Chorus) Me plus you, I'll take that number. Multiply your smile, minus the drama. Give me a fraction of your heart. I'll solve your problems. Now put that together. We make up a perfect equation, equation, equation. Me and you make up a perfect equation, equation, equation. Me and you make up a perfect equation. | And you can count on me. Forget about your past. I'll always put you first, never put you last. Girl, you're my absolute value. Have no fears, I'm here to save you. Promise I won't change, I won't play those games. Even when it rains, my feelings they stay the same. Put me to the test I haven't failed yet. Trust me I'll pass girl, just do the math cause. (Chorus) (Me and you make up a perfect equation) You deserve the best, nothing less from me. Hold your hand, open doors, treat you like a lady. All your dreams will come true, baby when you're with me. Don't fight yourself from him, I'm the half you need. (Chorus) |

## Wednesday, December 17, 2008

### The Absence of Evidence

We started off the class with a pretty funny clip which featured Calculus and the Numa, Numa song. Basically the person in the video showed two pieces of paper, one a function and one an anti-derivative.

Following that, we got into the stuffing of the class. We reviewed derivatives and what they are in order to grasp the concept of an anti-derivative. We were told to remember something when we do anti-derivatives, which is "If you can do something, you can undo it." For anti-derivatives, this means that any derivative can be anti-differentiated.

A key point is to know that an anti-derivative is a family of functions, however, if you have a given point, you can find one particular function within the family of functions.

An

**Indefinite Integral**is not like a definite integral. It picks our a family and if a point is known, it picks a particular one. This sounds a lot like an Anti-Derivative.

Moving along, we are told that for every derivative rule, there is an anti-derivative rule, as this is automatic when a derivative rule is made.

So, knowing that, can we find the parent function with first and second derivatives?

Let's use an example.

If you know the derivative of sine, you know the anti-derivative of cosine.

Therefore if you know the derivative of sine, you know the anti-derivative of sine.

In words: The derivative of sinx is cos x, so the anti-derivative of cosx is sin x + C where C is an arbitrary constant. This basically means that C can be any value, because the function can exist in a number of places in the "family of functions."

The derivative of cos x is negative sin x, so the anti-derivative of sinx is negative cosx + C.

After this, Mr. K gave us a question so that we can apply this.

Consider the function (not the way it's supposed to be written but sitmo won't let me write it properly.) * "k" is a parameter and can be any value.*Using the first and second derivatives, describe the features of the family of functions generated for different values of "k".

Bench did the honours of doing this question.

Using the quotient rule and the chain rule...and factoring out...and reducing...and then we let it equal 0, so that we can find x.

o = 1 - kx

kx = 1

*Remember that critical numbers are at zero and undefined points, however, for this function there are no undefined points.

Therefore, this function will have no discontinuities (cusps, corners, magic tricks, hat tricks, pen tricks, kick flips etc.)

So by the first derivative test, when x = 1/k, the function is at a max.

Okay, it's 12 AM. I need to study for Physics... Ugh.

Next scribe will be Zeph!

Rence ~ Out

kx = 1

*Remember that critical numbers are at zero and undefined points, however, for this function there are no undefined points.

Therefore, this function will have no discontinuities (cusps, corners, magic tricks, hat tricks, pen tricks, kick flips etc.)

So by the first derivative test, when x = 1/k, the function is at a max.

Okay, it's 12 AM. I need to study for Physics... Ugh.

Next scribe will be Zeph!

Rence ~ Out

Labels:
Applications of Derivatives,
Rence,
Scribe Post

### Today's Slides: December 17

Labels:
Applications of Derivatives,
Mr. Kuropatwa,
Slides

### Math Fun

Here are some videos I found on Youtube. Its fun trust me :P

I don't persinally use this but it is cool...

http://www.youtube.com/watch?v=9qQAYEYLCoU

http://www.youtube.com/watch?v=I9t-gYnPNaw

http://www.youtube.com/watch?v=IIwlBjNLpjI

His profile:

http://www.youtube.com/user/glad2teach

That's all =D

I don't persinally use this but it is cool...

http://www.youtube.com/watch?v=9qQAYEYLCoU

http://www.youtube.com/watch?v=I9t-gYnPNaw

http://www.youtube.com/watch?v=IIwlBjNLpjI

His profile:

http://www.youtube.com/user/glad2teach

That's all =D

Labels:
benofschool,
Calculus AB,
On My Mind

## Tuesday, December 16, 2008

### (Di)Lemma's, Proofs, and Other Things.

By now it should be no mystery to everyone that I, (Hi I'm Justus, glad to meet you) am the scribe for today. Now I regret to inform you that this scribe post might now be wholly as spectacular or colorful as the one I last left you with, and this is due to the fact that well, I just don't have that much time on my hands today ;p As a result you get the scribe post you may or may not be about to read. I apologize in advance for any trouble this may cause.

So without further ado, onward with the show!

Where to begin...where to begin...

This is indeed a hard question, since we really didn't start with anything in particular, but instead jumped all over the place across a few topics we all felt would be good to talk about. Well I'll just start with what I have first in my notes here.

All right, so to kick things off we started with some more proofs (Or rather, logical explanations) for the differentiation rules (Which Jamie so graciously typed up for us in a lovely format.)

The first (according to my notes) was;

Now Mr.K explained to us that it is quite possible to go through the algebraic explanation of this differentiation rule, but also said that it is much simpler, so just remember the graphs of sine(x) and cosine(x)

Sine Graph

Cosine Graph

Looking at the sine graph as the graph of some function we want to differentiate, we may pick out the important points/bits of information found within the graph itself. By doing a quick glance you can find the local minimums and maximums, as well as the inflection points for the graph. These will be used to construct the derivative graph of sine(x).

Our experience with the graphs of functions and the graphs of their derivatives tells us that the red points are the local minima/maxima and that the blue points are inflection points. We know that because the red dots are local extremum, they will become roots for the derivative graph of f. We also know that the blue point(s) is a maximum because the slope at the inflection point is positive.

At this stage it should be easy to see, that connecting these dots gives you a derivative graph identical to cos(x). Therefore;

We also dealt with the derivative of cos(x) being -Sin(x) following basically the exact same steps. Those can be seen in the following image.

Moving along nicely then :]

The next topic of discussion, is kind of a twisty windy, jump off into several lemmas, resulting in an organization dilemma, and me having to sort out some thoughts kinda topic. For the most part, it involves this;

and the question, how would you find the derivative of that logarithm? This is indeed a tough question, as we didn't (at that point in time) have any differentiation rules for logarithms of any base (which in reality, is what we were trying to find at the time.) However, by using the change of base law, and some knowledge about differentiation rules with did know at the time, Mr.K showed us the way.

LEMMA TIME

(something about lemmas imagery here)

Change of Base Law, how it works, and what it looks like on the inside.

So, say you have a logarithm base "a", that equals "n", that you would like to change to base "c."

You must first start by re-writing the logarithm as a power (In this case, a to the n equals b.)

The next step is to take the log of both sides giving you the following;

After you have completed that, you would use the power law to bring down the "n" exponent on the left side leaving you with something you can work with.

After that, all you have to do, is isolate N. Looking back at the beginning we see that in the very definition of what N was, lies the logarithm we wanted to change the base of. What we're left with, is N (aka. our logarithm) equals a new logarithm, now with a different (hopefully more useful) base.

I hope that made sense ^_^;, just in case, here's all the steps together, so you can see it as one fluid thing, instead of a bunch of seperate steps.

1.

2.

3.

4.

5.

Back To Our Regularly Scheduled Programming.

So we have this logarithmic function

which we would like to find the derivative of. The first step would be to apply our newly a wholly understood change of base law to end up with this;

From here, we will pull out 1/ln2 so that we have;

the reason for this lies in the fact that although ln2 is a logarithm, it is still just a number, and therefore, a constant. By pulling it out of the whole thing, we get the nice and easy to differentiate, lnx by way of the constant multiple rule. For those of you who may be having trouble remembering exactly what that is, it basically means that;

"The derivative of a constant times a function is the constant times the derivative of the function."

As a result of this differentiation rule, all that remains is to find the derivative of lnx which we know to be 1/x

*Note* We did a Lemma for the differentiation rule for lnx but when I looked at my notes I couldnt really makes heads or tails of it. I'll need to check it over with Mr.K and get that up here once I understand it. Sorry guys D:

The result, is the derivative, which in this case, happens to be

Now if we think back to the beginning, the primary goal was to see if we could find a rule for all logarithmic functions, and to this end I believe we achieved our goal. By simply substituting a variable (lets say a) into the spot of 2, you get the rule for differentiating logarithms.

With that stuff out of the way we moved onto the proof for d/dx of tan x, which happens to go something like this.

After converting tan into something more friendly, you simply apply the quotient rule.

Tadaa! If you wanted to find the derivative of say Cotangent, you would do the exact same thing. Turn cotan into Cos x over sin x, and apply the quotient rule from there.

Another neat little tidbit of information for you. According to Mr.K, the derivative of all "co" trig functions is negative. Just thought you'd like to know ;p

Now, after alllll that stuff behind us, we got back to antiderivatives. Remember that an antiderivative lives on the idea that anything that can be done, can be undone. However, with derivatives, because of how we figure them out, there is a small kink. Any constant in a parent function, is lost when you take the derivative of that function. Essentially, you may know a functions derivative, but you may NOT know its vertical positioning by the same method. In effect, when you differentiate a funtion, you are finding a family of functions, not just one. Now there is a way to figure out what the constant was, if you have a point from the graph you'd like to single out. These are called intial value problems. Lets see an example.

Because we know an antiderivative gets rid of the constant, we need to add that back in. We do so with the value "c", seen above.

Know this is called the inital value theorem because we have a value at the beginning (initial) of the problem. In this case its the value of f at 1

From this point we substitute the initial value into the antiderivative and solve for c.

And thats really all there is to it :] In case that didnt make much sense, here's the slide that explains it better then me I think D:

Okay guys, I think that about wraps it all up for tonight. Not gonna go into a big spiel this time, its late, and I'm tired, and I have basketball tomorrow, and the day after, and the day after lol.

SO with that said, lawrence shall be scribe for the next day (which is technically today.) Alright? alright. Night all :]

So without further ado, onward with the show!

Where to begin...where to begin...

This is indeed a hard question, since we really didn't start with anything in particular, but instead jumped all over the place across a few topics we all felt would be good to talk about. Well I'll just start with what I have first in my notes here.

All right, so to kick things off we started with some more proofs (Or rather, logical explanations) for the differentiation rules (Which Jamie so graciously typed up for us in a lovely format.)

The first (according to my notes) was;

Now Mr.K explained to us that it is quite possible to go through the algebraic explanation of this differentiation rule, but also said that it is much simpler, so just remember the graphs of sine(x) and cosine(x)

Sine Graph

Cosine Graph

Looking at the sine graph as the graph of some function we want to differentiate, we may pick out the important points/bits of information found within the graph itself. By doing a quick glance you can find the local minimums and maximums, as well as the inflection points for the graph. These will be used to construct the derivative graph of sine(x).

Our experience with the graphs of functions and the graphs of their derivatives tells us that the red points are the local minima/maxima and that the blue points are inflection points. We know that because the red dots are local extremum, they will become roots for the derivative graph of f. We also know that the blue point(s) is a maximum because the slope at the inflection point is positive.

At this stage it should be easy to see, that connecting these dots gives you a derivative graph identical to cos(x). Therefore;

We also dealt with the derivative of cos(x) being -Sin(x) following basically the exact same steps. Those can be seen in the following image.

Moving along nicely then :]

The next topic of discussion, is kind of a twisty windy, jump off into several lemmas, resulting in an organization dilemma, and me having to sort out some thoughts kinda topic. For the most part, it involves this;

and the question, how would you find the derivative of that logarithm? This is indeed a tough question, as we didn't (at that point in time) have any differentiation rules for logarithms of any base (which in reality, is what we were trying to find at the time.) However, by using the change of base law, and some knowledge about differentiation rules with did know at the time, Mr.K showed us the way.

LEMMA TIME

(something about lemmas imagery here)

Change of Base Law, how it works, and what it looks like on the inside.

So, say you have a logarithm base "a", that equals "n", that you would like to change to base "c."

You must first start by re-writing the logarithm as a power (In this case, a to the n equals b.)

The next step is to take the log of both sides giving you the following;

After you have completed that, you would use the power law to bring down the "n" exponent on the left side leaving you with something you can work with.

After that, all you have to do, is isolate N. Looking back at the beginning we see that in the very definition of what N was, lies the logarithm we wanted to change the base of. What we're left with, is N (aka. our logarithm) equals a new logarithm, now with a different (hopefully more useful) base.

I hope that made sense ^_^;, just in case, here's all the steps together, so you can see it as one fluid thing, instead of a bunch of seperate steps.

1.

2.

3.

4.

5.

Back To Our Regularly Scheduled Programming.

So we have this logarithmic function

which we would like to find the derivative of. The first step would be to apply our newly a wholly understood change of base law to end up with this;

From here, we will pull out 1/ln2 so that we have;

the reason for this lies in the fact that although ln2 is a logarithm, it is still just a number, and therefore, a constant. By pulling it out of the whole thing, we get the nice and easy to differentiate, lnx by way of the constant multiple rule. For those of you who may be having trouble remembering exactly what that is, it basically means that;

"The derivative of a constant times a function is the constant times the derivative of the function."

As a result of this differentiation rule, all that remains is to find the derivative of lnx which we know to be 1/x

*Note* We did a Lemma for the differentiation rule for lnx but when I looked at my notes I couldnt really makes heads or tails of it. I'll need to check it over with Mr.K and get that up here once I understand it. Sorry guys D:

The result, is the derivative, which in this case, happens to be

Now if we think back to the beginning, the primary goal was to see if we could find a rule for all logarithmic functions, and to this end I believe we achieved our goal. By simply substituting a variable (lets say a) into the spot of 2, you get the rule for differentiating logarithms.

With that stuff out of the way we moved onto the proof for d/dx of tan x, which happens to go something like this.

After converting tan into something more friendly, you simply apply the quotient rule.

Tadaa! If you wanted to find the derivative of say Cotangent, you would do the exact same thing. Turn cotan into Cos x over sin x, and apply the quotient rule from there.

Another neat little tidbit of information for you. According to Mr.K, the derivative of all "co" trig functions is negative. Just thought you'd like to know ;p

Now, after alllll that stuff behind us, we got back to antiderivatives. Remember that an antiderivative lives on the idea that anything that can be done, can be undone. However, with derivatives, because of how we figure them out, there is a small kink. Any constant in a parent function, is lost when you take the derivative of that function. Essentially, you may know a functions derivative, but you may NOT know its vertical positioning by the same method. In effect, when you differentiate a funtion, you are finding a family of functions, not just one. Now there is a way to figure out what the constant was, if you have a point from the graph you'd like to single out. These are called intial value problems. Lets see an example.

Because we know an antiderivative gets rid of the constant, we need to add that back in. We do so with the value "c", seen above.

Know this is called the inital value theorem because we have a value at the beginning (initial) of the problem. In this case its the value of f at 1

From this point we substitute the initial value into the antiderivative and solve for c.

And thats really all there is to it :] In case that didnt make much sense, here's the slide that explains it better then me I think D:

Okay guys, I think that about wraps it all up for tonight. Not gonna go into a big spiel this time, its late, and I'm tired, and I have basketball tomorrow, and the day after, and the day after lol.

SO with that said, lawrence shall be scribe for the next day (which is technically today.) Alright? alright. Night all :]

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