2005 Free Response Question #5

I hate free responses...

Anyways! Sandy Beach has some problems that needs to be solved! :

Sand being removed:

Sand being added:

A. How much sand will the tide remove from the beach during this 6-hour period?
We use the equation R(t) for this one! Integral of 2+5sin(4pit/25) from 0 to 6. It should output an approximation of 31.816 cubic yards of sand removed.


B. Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.
Since the sand is being eaten [R(t)] at the same time it's being put on the beach [S(t)], we need to combine them to find out how much of the sand remains! Our new equation will be:

Y(t) = [S(t) - R(t)] + 2500

C. Find the rate at which the total amount of sand on the beach is changing at time t=4.
I don't think you include the +2500 in Y(t) for this one because it's asking for the rate of change of the sand... :/ Bleh, confused. Either way, if you plug in 4 into Y(t) without the +2500 then you'll get -1.908750647 which should be rounded out to about -1.909 cubic yards...??


D. For , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
Given that the boundary is 0 to 6, those are our endpoints i assume? I'm too lazy and confused to find the critical points... so using my magic calculator of magic, I found that our critical points are t= 0, 5.118, and 6! ... But I still don't know whether 5.118 is my minimum...

Now then~

*Brings out ipod and begins singing to Help! by the Beatles...*

What's On Your Mind?

PURPLE~

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?

Truth be told, I really don't believe in these kinds of theories or explanations. But for the sake of this assignment, I would assume that I belong in the fixed mindset because of all the obstacles I am avoiding. Not mainly because I think they are 'hard' or 'time consuming', but because I know that my mind can't take any more. I don't give up THAT easily, and criticism isn't always acknowledged, but I do put in effort, just not into the right subjects or matter. I enjoy challenges, but to my benefits or entertainment. Seeing others accomplish is actually encouraging rather feeling it as a threat. Overall, I don't really know whether or not if I'm more of the fixed mindset than growth mindset.


How has this mindset helped or hurt you in math?

I have no clue. Like I said, I don't really believe in these kinds of theories because they don't always apply to an individual. But if I had learned about this beforehand, I would have been discouraged from the beginning rather than attempting to better myself. I may be struggling, but it won't affect me so m


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

I've always known that the brain can be trained but not that I really paid mind to it. But I'm evil to my brain, I feed it dark knowledge and never bother to remember much information besides junk, music, art, and life changing quotes. Thus resulting into a bad trained brain with a wild imagination... with some sort of wisdom about this world. XD


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

I don't and it won't. I can guarantee that. Either because I will pay no mind to it, forget it, or just believe que sera, sera. But if I do somehow believe in this, assuming my nature, I'll be discouraged. But we never know what tomorrow will bring us. Heck, I'm just glad if I get to open my eyes tomorrow...

I'm Over My Limit

Well, this is belated >__>;; Moving on...

Limits.
What don't I understand about them?? Well...
Absolutely Everything!!! :D
...
That would have been my answer a week ago, but not so much anymore.
But I still need help, with the important stuff.

I understand continuous & discontinuous on a graph for the most part, but how about algebraically??
Are there any shortcuts to figuring it out??
What about spotting the vertical asymptotes without going through the process of graphing the whole equation out??

Another big problem is when they throw Sin, Cos, and Tan at us.
Graphing the equations take time and there are some misunderstandings like the limit of sinx/x doesn't exist unless it approaches 0 or something... Don't take my word for it, I'm confused... :/

Word problems are killing meh... >__>

Did Somebody Say College?

What am I going to do? What am I going to be? Where am I going to go? What's the meaning of lief??
STOP PRESSURING ME!!

Lol, but really, thinking about the future is a little scary. It's fun to dream and wonder but what will really become of us? Hopefully everything goes as we plan but it doesn't hurt to have a plan B...or C... throw in a D and E for me >__>;;

MAJORS

• Veterinary Technology - The title says it all. This lets me handle medical technology used on animals as well as providing medical care; Surgery, diagnosis checks, x-rays, blood samples, prescribing medicine, the works. All about keeping an animal healthy. I might get bitten or harmed, but I already have two cat bites so that's no problem. xD I think I have people skills, but not the right one...
  
• Zoology - Studying animals of all kind including insects, fish, and worms! Physiology all over again but for animals, although there isn't any difference besides size.
• Animal Behavior and Ethology - Ethology is a sub-topic of zoology, basically scientifically studying animal behavior. Evolution, neuroscience, and psychology is covered.
• General Arts - Yup. Art. Just throwing in a passion of mine. There are so many types of art, then again isn't everything considered an art? Photography is something I would like to get involved in, especially photographing animals and their habitats. Sketching too!! Animal anatomy sounds fun. Fustrating, but fun none the less.

COLLEGES

• Bel-Rea Institute of Animal Technology - Well known VT university located in Denver, Colorado. Hands-on, real world experiences provided. Offers an Associate of Applied Science Degree in Veterinary Technology and has many programs to chose from. Is recognized for training and student financial assistance by the Department of Rehabilitation.
• UC Davis - Located north of California. Well known veterinarian school. They have a variety of centers on campus ground. The Wildlife Health Center seems interesting.
• Colorado State University - In Colorado. They have a Small Animal Reception and a Large Animal Reception and is known as the Green University. Pure Animal Lovers. RAMS!!
  
• Apollo College - Located in Phoenix, Arizona. Has a Vet Tech Assistant program available. Wildlife Centers are located nearby and internships are offered.
  
• Argosy University - Located in Phoenix, Arizona and in Twin Cities, Minnesota. VT program available in Twin Cities. They have a Blog that updates regularly.

Tips and Hints

I feel like I'm posting false information... Such a high self esteem huh? Anyways, I can't explain really well so don't take my word for it. Really... >___>;;

1. Transformations. Makes graphing an equation much easier if you know how. Easy on some parts, somewhat confusing on the rest. The key to graphing any function or equation is knowing your parent functions.

Equations like f(x)=(x-2)-3 is easy; shift the graph over 2 units to the right on the x-axis, and then sift it 3 units down on the y-axis. But, what if we are to transform functions based off the unit circle like sin(x), cos(x), and tan(x)?

Let's use sine for our formula example:

f(x) = asinb(x-c)+d

• a would tell us how 'tall' or 'short' the graph will be. In other words, it tells us our amplitude.
  
• b makes the function either thinner or wider, thus causing our period to change. To find out our period, we use 2π/b. If a>1, the thinner it is, if a<1>
• c moves the graph along the x-axis. But it gets tricky when changing the input because then you'd have to do the opposite. But this formula actually clarifies which direction to go. If you plug in a negative number for c, you'd get a positive, and if you plug in a positive number you would get a negative.
• d shifts the graph along the y-axis, no confusion there. :]
  

2. Trigonometry. I think of the unit circle, triangles, sine, cosine, tangent and whatnot. What I remember most is SOH CAH TOA. (.___. ;;) Finding the radians and degrees are becoming clearer in a way but I have no advice for that. I remember what Ms.Gapac told us to help us find the positive values; All Students Take Calculus. That goes in the quadrant order, which means in the first quadrant, All (sine, cosine, and tangent) are positive. Sine is positive in the second quadrant, Tangent is positive in the third, and Cosine is positive in the fourth. The reciprocal of a trig function is not the same as the inverse! Eh, not much really to remember right now...

3. What confuses me? What worries me?? MANY MANY THINGS. But trigonometry wise, I can't be confused if I don't know anything :D
lol, I still get confused with remembering how the inverses look like and such. Domain and ranges are a pain to remember too. But we haven't really gotten deep into trig yet so i must wait and see what i must worry about later on. But none the less, I hope that there would be some good advice on trig, it would make it alot easier. :3


Happy Saturday...no wait, Sunday!!!

Logarithms and Inverses

What do I know about Logs?? And Inverses?? If I really think about it...nothing! XD
Naw, jk. But if I actually think about about it, I only know so little... so let's begin!!

Starting with the first part:

LOGS

Logarithms. Log for short! hehe, log... Here are some main point about Logs!

• Logs are exponents! Yup, basically nothing but exponents. Except, just mixed around to look more smarticles! Let's take a look at our example below! In example 1, we see '3 to third power is 9'. In its' log form however, we would say, 'the logarithm of 9 to the base of 3 is 2'.

• There are two kinds of Logs: Common Logs and Natural Logs! :D

• By defualt, log will always have a base of 10, it's just hidden! This is known as a Common Log, which is why when you type in log(100) in our calculators, you get 2 as a answer. Unless the base is specified like the example above, the base of log will always be 10.

• Natural Logs, on the other hand, are different. We see it as (ln) and (e). ln is the inverse of e and vice versa, tha'ts why they cancel out each other. IT'S THE LAW!! x]
  
  
  
  
  
  



• e is an irrational number, like pi!! e = 2.7828183 and so on!! Maybe I should remember e since Cynthia took pi...lol


• Fun Fact: the e symbol is called epsilon in Greek! ;D

Enough about Logs! Let's move on to Inverses!

INVERSES

Now onto inverses! No pun for this one, darn...

• When given a function, graphically, you can figure out how the inverse will look like by switching the inputs and outputs in the table of values. Let's look at this function! Notice the table and the graph: Now look how I switch the inputs and outputs! Now the graph looks like so: So grpahically we can easily find the inverse, but now to find the inverse algebraically!
  
• To find the inverse of a function algebraically, we need to 'solve' for it! Let's use the previous function as an example! Let's take a look: This is our answer, but to make sure, we apply this equation to see if it really is:If the final answer for both is x, then we got ourselves an inverse! Now we verify that the square root of x is the inverse of x^2: So this is our inverse! BUT!! After solving for the inverse, we then must check to see if the answer will be either #5, a function, or #6, a standard equation, from the second bullet.

• If the inverse of a function remains a function, then it is a One-To-One. To know if a function is one-to-one without graphing/solving for the inverse, you simply use the horizontal line test! (It's also not a function because there is two outputs for every input instead of one.) Now we know that the inverse of f(x)=x^2 is an equation, meaning not one-to-one.
• One other note about inverses, to find the domain and the range of the inverse, you just switch it with the origional one. Ex.; f(x) = x^2 D: (all real numbers) R: [0, all real numbers]

f(x) = root of x D: [0, all real numbers) R: (all real numbers)

Now the second part of the Blog!

What did I not get?

• Well, I don't get most of the classwork, but hopefully we'll go over it in class.
• I still have some doubts in logs, solving wise. It's those fractions! -___-
• I need to resist using the calculator when graphing logs too...
• Natural logs still seem confusing.
• Some terms confuse me, I may know what it is and all, but I don't remember what it's called.
• And probably what everyone else has trouble with.... lol....Logs...

Part three!

Now I'm off to comment on your blogs!!

Although I assum they have been answered by now... TT__TT

SUPER SUPER LATE Dx I gotta stop doing that...

Even and Odd Functions

Now presenting my super late blog!! :3

Hehe, funny story... I dreamt i did it... >__>;;

Do not judge me!!

~~

Okay. So we're talking about functions yes? As in:

f(x)= x^2 ...

f(x)=x+3 ...

And so on!

The topic here is telling the difference between odd and even ones... which is pretty confusing for me. But I will try none-the-less to comprehend.

Let's start with even functions!

From what we went over in class I understand that graphically, a function is EVEN when it is symmetrical to the y-axis. Or in other words, when you fold the graph along the y-axis, quadrants 1 & 4 over quadrants 2 & 3, the line of the function will be the same on the opposite side. Like a mirror!

Mathematically the definition of an even function is this:

f(-x) = f(x)

This means that in a function, the output of the f(-x) will be the same as the output of f(x).

[Example; f(2)=6 /and/ f(-2)=6]
But it has to work with all numbers or it won't count!

Some graph examples:

Now for odd functions!!

ODD functions, graphically, are kind of like even functions. BUT they are not symmetrical along the y-axis but at the origin!! (0,0) Lol, for some reason odd functions remind me of Ms. Hwang's clock x]

The mathematical definition of an odd functions is this:

f(-x)=-f(x)

This in particular means that in a function, the output of f(-x) will result the same output of f(x) unit wise but the direction in which it's going is in the opposite direction.
[Example; f(2)=6 /and/ f(-2)=-6]

Just like the even function, this has to work with all numbers used as the input.

Some graph examples of odd functions:

One other thing that I forgot to mention, a function is 'neither' if it does meet either requirements for even and odd functions.
... Yeah, guess that's about it... ^^;

Well, that's all folks!!

Happy Monday everyone! :D

[Yes, I remembered it's not Tuesday...]