Mean Value Theorem

f ' (c)= [f(b)-f(a)]/ (b-a)

So the Mean Value Theorem states that if you are given a function that is continuous and differentiable (which just means no jumps, holes, or corners) the slope of the secant line and the tangent line are the same at point "c". This means that they are two parallel lines.

Alright, so let's try this out. We will be using a very familiar graph to make things a litttle easier to see and understand. (At least i worked for me)

We have the graph f(x)= x² on [0,2]

So a=0 and b=2. Let's get this numbers in the formula [f (b)-f (a)]/(b-a)

1) [f(2)-f(0)]/(2-0)= 4-0/2=2 so m=2

2) Now we will replace x by the two we just got into the f(x)>>>>f(2)=4

Now we have a point and a slope so let's use the point-slope formula

3) y-4= 2(x-2) which is y=2x. This is our secant equation. (The green line on the graph)

We know the f '(x) = 2x so f '(c) =2c

We also know that 2c= slope and we know the slope is 2. Therefore c=1.

Now we find out what f(c) equals when c=1>>>>>> its 1 so we now know the tangent line has to go throught the point (1,1) with the slope 2x. We do the point-slope formula.

4) y-1= 2(x-1)

y-1=2x-2

y= 2x-1 (It's the red line)

So why doesnt this work on a function that's not smooth? Let me show you with an example AKA another graph.

As you can see the tangent line doesn't exist at x=0

What is f '(x) saying about f(x)

This is the graph f '(x) which simply means that what you see on this graph are the slopes of the graph f(x). So what is this graph telling us about our invisible f(x).

1) Where is the functions f(x) increasing? Where is it decreasing? And how exactly do we know?

Ok, so as I mentioned before, this is the graph of the slope of f(x) so where is the slope increasing? F(x) is increasing from (-2,o) U (0, 2). When f '(x) >0 the function f(x) is increasing. -2,0,2 are exclusive because at those points the slope is zero. F(x) is decreasing from (-∞, 2) U (2, ∞) because when f '(x) <0> the function f(x) is decreasing. I a visual learner so for those like me here you go. Hope it helps.

2) Where is there an extrema?

So, to answer this question you must already know that to find an extrema you have to find either the end points or the critical points. Because this graph has no end points we have to find critical points. A critical point is found when f '(x) is either Zero or Undefined. As you can clearly see on the given graph f '(0) is zero so your extrema is at (0,0).

3) Where does f(x) concave up? Where does it concave down?

F(x) is concave up when the slope of f '(x) is positive which is approximately from (-∞, -1.25) U (0, 1.25) and concaves down approximately from (-1.25, 0) U (1.25, ∞)

4) So what does f(x) look like on a graph?

It has to be a x^5 because f '(x) is x^4 and the anti derivative has to reach x to the power of five.

Stiff Muscle

1. Which mindset do you think you are a part of when it comes to "intelligence"? According to the reading, what tells you that you are of this mindset?

When it comes to intelligence I think there are some things you can learn and some things that take longer for one to understand. Well, when I started reading, I noticed that I leaned more on the fixed than the growing mind set when it came to math. I have always felt that my math skills aren't that great. For this reason, I sometimes have my doubts and ask myself why did I even take this class. I'm usually not a quitter and enjoy the academic challenges that are put upon me, but with math I don't see it as a challenge. To me calculus is beyond what I can do and the only reason I stayed was because my Math Teacher truly believes I am capable of doing this and succeeding.

2. How has this mindset helped or hurt you in math?

Well, it surely hasn't helped me at all. It has actually held me back from what I could probably do. When it comes to math I seem to be easily discourage and I know this isn't the way it should work but I can't seem to help it. I just say "Math is NOT my subject" and stick with that. It's hard to change the habit of saying that. I've said for quite a while I'm beginning to believe the my subconscious believes it.

3. What is your reaction to finding out that the brain is just a big muscle that can be trained?

My dad has told me that before so it wasn't a big surprise. Maybe my brain is just one of those stiff muscles that needs more time to train when it comes to math.

4. How do you see this new piece of information affecting your future?

I guess the more logical answer to this question would be "Yeah I will definitely change and this information will help me in the future" but my problem hasn't been not knowing this information. I guess it would be believing it and putting it to use. I really do want to be better in math but sometimes I get mad at myself because I can't seem to understand the material as fast as other, as fast as I want to understand.

I guess I should go easier on me and believe in myself more than I do...like Ms.Hwang believes I can.

My Math Skills and The Limits

Well, as some (Ms.Hwang and the rest of the world) may know, my math skills aren't great. In fact they kinda suck a little and like in all the other sections we have gone over, i have a couple of questions.

I'm still a littel confused about the asymptopes. Like the whole "finding the horizontal and verticle asymptote." Ummm... let me find a problem like the one I am speaking of ...

Find the verticle asymptote of the graph f(x)=tan x/sin x.

Also..

The last two on last night's homework. It also asks me about asymptotes. This time its horizontal.

f(x)= 4x^3 -2x+1/x-2 (#43) all #41-44

Ok so that is an example of what i dont get. Natural logs and e's still sort of intimidate me but after going hysterical and mad because i have to solve a problem with either i get a hold of myself and do it. I just have to remember e is just a number and not a variable. With natural log i just have to remember its just the inverse of e.

Limits aren't as hard as i expected them to be. It was hard in the beginning because i had to accept the fact of end- behavior model. The whole "the closer it gets to infinity the rest of the terms don't matter they'll equal the same in infinity" it was hard but i just reminded myself Calculus is magic and magic shouldn't be question just do it hahaha. But with a little more practice and my questions answered i think ill be fine.

Its not that i slow down in math because i have reached my limit...its the speed bumps that do it. I just have to fix the street I'm in.

NENANI =D

OMG OMG OMG OMG OMG!!!!!

I'm screaming to the heavens and stressing like never before. When people ask me what I want to study I tell them "Umm...something in the medical field." Do you know how many professions are in the medical field...a lot.

Some majors that catch my attention are psychology, biology, and ...I know this may seem as a major way off topic but my mother always told me music heals wounds...Voice and Opera

Ever since I started taking AP Psychology i have been fascinated by the way our genes and environment affect the way we grow mentally. I love the fact that some behaviors I would observe in others had names, theories and causes. Their are many careers that one can pursuit with psychology such as a clinical psychologist, experimental psychologist, and so on.

Since I was small, I would always wonder how it was that doctors knew what was wrong with someone with only signs and symptoms to guide them and treat their patients. I'm a big fan of hectic life (i have one). It might sound like I don't because I complain a lot but after I get through the day I feel stronger than the day before. The life of an ER Doctor seems like the job I would love. Not only is it good pay ($162000...nice huh?) but I would be a licenced miracle worker. I know sometimes it might be a very sad job because obviously I won't be able to save every one's life (ya win some ya loose some right?) but at least I'll win some. (Death is inevitable)

Now, music has been part of my life since elementary 3rd grade when the choir teacher was about to loose her job because not many children were joining the choir. It was during PE while I was playing handball with my friends. She decided to go around our free time to recruit kids for her class. She came up to me and asked if I liked singing (at that time I only sang in the shower) and I said "Yeah...I guess." She told me to sing "Twinkle Twinkle Little Star" and because I wanted to impress my friends I tried my hardest. She said, "You have a beautiful voice." Now...I don't know if she was just BS-ing me but I bought it. I was in choir throughout elementary, learned to play the trumpet during middle school and choir again in high school. I love it.

The question that proceeds "What do you want to study?" is "Where?" I have a few places in mind.

UC BERKELEY!!!!!

Ok...what do I know about Berkeley is that HAMILTON GOES THERE!!!!(This fact is only directed to those who know this Genius.)

It has about 35,409 students as of Fall 2008 including 25,151 undergraduates.

It seems to have more girls than dudes (53% female and 47% male (Fall 2008).)

It has 130 academic departments and is a great choice of school if planning to study medicine.

The GPA requirement is average 4.0-4.2 (I'm not there yet)

UCLA!!!!!!

Before I found out about Berkeley, UCLA was the first one on my list. What I found out about UCLA is that it is a great school for people who are interested in the medical field.UCLA enrollment was about 26,000 for undergraduate students, and 12,000 for graduate students in Fall 2008. To pay for their tuition they offer loans, grants, scholarships and work study. And if you are in state living on campus fees and living expenses are around $19,824 annually.

UC Riverside!!!!

Although this university isn't as well known as the other UC's it still considered a great school medical school. Most of its focus is about medicine.UCR is widely recognized as one of the most ethnically diverse research universities in the nation, UCR's current enrollment is 17,187 students, with a goal of 21,000 students by 2010. Most of their population is Asian followed by Hispanic and then Caucasian. They have 80 Bachelor degree programs and 38 Ph.D degree programs. Sounds good to me.

My Hints and Tips

Transformations

The way i seem to remember things about transformation is that they always seem to be counter intuitive. When you think you have to move x number of units to the left it happens to be the exact opposite of what you have to do. When you think you should make you graph shrink it happens to have to be stretched out. It's quite confusing to me (but so is the rest of math). Let me provide and example.

Example: f(x-7)

This function you can automatically classify it as a line and the minus seven means you just have to move it seven units to the right even though you would think the exact opposite because it's a negative number.

If you want to try it out go to this site
http://www.mathsisfun.com/data/graph.html
(props to Rocio Rodriguez for providing it)

Let's say they give you a function like this f(1/2x)
You would think that it would be smaller because of the the small number. But guess what, THAT'S WRONG!!!"
The function graphed would actually stretch out more that a function that said f(3x). The function f(3x) would actually look more compressed than a function of that said f(1/2x).
(It's crazy but that's math, it's like magic...2+2= Fish)


Trigonometry

Alrighty then, moving on to the trig stuff. Well like Ms. Hwang said, "You have to know your unit circle!!!" When asked to recall the unit circle, my hand twitches and automatically wants to draw it (seriously, no joke).
I haven't reached the stage where I can visualize it, i kind of have to draw it. Memorizing the unit circle took about ummm...25 minutes or so of practice. The way i memorized it was kind of complicated to be quite honest (I'm like that, i over think things). Well for the radians there is a pattern (if you haven't noticed) the denominators repeat themselves like mirror images. Starting at zero and moving counter-clockwise they go 6,4,3,2,3,4,6 only the outcast number in this case changes in this case it is the 2. The numerators are a different story. The numerators for the top half of the circle seem to have a jingle. Try it its pi, pi,pi,2, 3,5. For the bottom half i had to learn it myself with no jingle =(. The coordinates are the easiest things to memorize, the also reflect like mirror images. You just have to remember the first three on the first quadrant and the rest is just repeated (you just have to remember that you still have to change the negative and positive coordinates but that's easy.

When it comes to graphing sin, cos and the rest its a about studying them for me (which reminds me i still need a little work)

What worries me?

Well let's see...i have a list hahaha. I can't seem to graph the inverses like cos-1 x without the stupid patty paper. It sounds silly but only God knows how frustrating it is ( i have to memorize those flash cards). But yeah i think thats about it

P.S: i still am confused about logs

Logs and Inverses

Well let's see what I learn in the week of logs and inverses.

1)When it comes to the inverses of functions it is quite clear to me that the inverse is when you switch the input and the output. I'm not sure I'm explaining it well so I'll use an example.

X____Y

-1 | 3

0| 0

1 | 3
So then instead of -1, 0 and 1 being your x's they will become your outputs...your y's and the same with the y's, they will become your x's. By plotting the points this way you will graph the inverse. This applys to equations as well such as

y= 2x+7 --------->x=2y+7

2) I learned that the inverse of a fuction will reflect about the y=x line

3) I learned that if you do the horizontal test on a function you are able to determine whether it inverse is a function or not. (This is known as one-to-one)

4) I learned that logs are the inverse of exponents and that they can help you figure out what the exponent of a number is.

Ok, i kind of already knew i wasn't going to get this 100% because it's math and well ....it's me and math.

I didn't really understand when it was nessesary to plug in the natural log of something and when i was suppose to use the e.