Mean Value Theorem

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

1. This means that if you take a function, f(x)
then the slope of the secant line between two points, [f(b)-f(a)]/(b-a)
is equal to the slope of the tangent line at the point c, f'(c)

So for example:
f(x)=x²+1 (to make things easier to see)
f'(x)=2x
[f(b)-f(a)]/(b-a) where a=-2 & b=2 ==> [4-4]/(2+2)=0
f'(c) = [f(b)-f(a)]/(b-a) ==> 2c=0 ==> c=0


Basically, it means the the secant line and the tangent line are parallel to each other.

2. This only works for continuous and differentiable functions because if the function is neither, then either the point c or the slope at c may not exist, as in the function
f(x)=|x| where a=-2 and b=2 (sorry, no graph this time). In this case, the point c would have to be at 0, but since the function is not differentiable at 0, f'(c) does not exist and the mean value theorem does not apply.

f(x) from f'(x)

Woohoo... MORE graphs... (Yes, people, that was sarcasm). Okay, ANYWAYS...

1. f(x) increasing on interval [-2, 2] because f'(x)>0
f(x) decreasing on interval (-∞, -2]U[2, ∞) because f'(x)<0
2. Extrema are at x=-2--> minimum, because f'(x) changes from negative to positive and at x=2--> maximum, because f'(x) changes from positive to negative.
3. For the sake of simplicity I'll round to the nearest whole number...
Concave up on interval (-∞, -1]U[0, 1] because f"(x), aka the slope of f'(x), is positive
Concave down on interval [-1, 0]U[1, ∞) becasue f"(x) is negative
4. I want to say that f(x) might be somthing like -x^5
since it looks somewhat like -x^3 with a bunch of crazy stuff happening in the middle...

Mindsets

Oh wow... that was interesting...

1. First off, this seems like a load of bull to me. I don't know about the rest of you, but my "mindset" changes depending on the subject/topic,it depends on my interest in it, not on whether or I believe the brain is a muscle that can expand or not. Hell, my brain is the last thing I consider when I'm faced with something new. The first thing I think is, "Will this be interesting/useful?" If not, I tend to go towards the fixed mindset, avoid challenges, obstacles and effort. If I do think the topic is interesting or useful, I go towards the growth mindset, I take the challenges as they come and put in most of my effort --- I say most because it's hard for a lazy person to ever put in ALL their effort. In other words, I don't think there is one specific mindset that a person, or at the very least, I, have. It depends on the activity, on whether it interests you or not.
2. As said, I don't believe there is one specific mindset that controls how you learn or think all the time. Isn't this part of the reason we do well with some topics and not so well with others? Personally, I go back forth at times when learning lots of things because some are more helpful or interesting than others. The only reason I even pay attention in class is that I'm sure it will help me when it comes time to take the damned AP test. Once I deem something useless it leaves my mind for a while, until I have to review the topic again.
3. To the mind is muscle thing: My reaction was, "I already knew this." You don't take two years of biology and a year's worth of psychology without learning this. And a couple biological or psychological documentaries tend to include this fact, too...
4. It shouldn't affect me much. Considering I had heard the mind being a muscle thing before, and now, again. Oh, how I hate repetition... As for the mindset stuff, that really shouldn't affect me either considering I don't think it's entirely legitimate.

Algebra vs Calculus

1. What is the DIFFERENCE between finding the limit of a function at x = c and actually plugging in the number x = c? When are the two cases the SAME?

Difference- when you are finding the limit you are finding the y value that wouldbe at x=c, if there was no discontinuity in the function. But, when you actually plug in the number x=c, you will get the output.
For example if there is a removable discontinuity at x=2 in a function, the limit might turn out to be 1, while if you plug in 2 into the function you might get an output of 3.
Same- the two are the same when there is no discontinuity at x=c and the limit is the same as the output of the function.



2. What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?

Similarities- the derivative of a line and the slope of the line are basically the same thing: the change in y over the change in x.
Differences- I'm fairly sure that the differences in the two are in where the derivative and the slope are used.
For example, finding the slope of a parabola, would give you the value of the slope of the parabola's secant line, while finding ther derivative of that same parabola would give you the slope of the parabola's tangent line.

Limits...

Okay well... limits...

Let's see... I think I understood most of the concept of limits. After going through the homework assignments of the past two weeks, I couldn't find any problems that I had left blank... Well, there was one, but after looking at it, I realized I could've just plugged in the number to find the limit. (It was #8 on pg 66, I think the large exponent threw me off haha)

I don't think there was much else...

Wait, one thing: Is there a difference between a limit being undefined and a limit not existing? I thought you could use the terms interchangabley but now I'm not so sure... Anyways, if there is a difference, what is it?


On a seperate note: How can you tell what a graph looks when the equation is in the form of a polynomial divided by a polynomial?
Ex: How can I know what
4x³-2x²+5
3x-4
looks like without having to use a calculator?

Colleges

Majors
Preveterinary studies- not offered in many schools so you usually have to take other classes like biology and whatnot in order to be accepted into a veterinary school. Obviously, help prepare for the path to becoming a veterinarian. Need people skills...
Zoology- study animals. All kinds, not just vertebrates so probably not a good choice for those that a bit squeamish around bugs and worms and such. Study the animal and its anatomy and physiology like the cells and the organs. (Good references I bet...) A bit like ecology in that you study the animals' environment and how they adapt to it.
Wildlife biology- Study wildlife. The animals, the plants, the environment and how they all interact. Study the ecosystems from urban, suburban, rural areas and in the wild. Focus on vertebrates and their environments.

Colleges
UC Davis- In northern California, 15 miles from Sacramento. It has an equestrian and raptor center (raptor as in birds of prey, not the dinosaurs)
American University- Private school in Washington D.C. 80% of the students admitted are from out of state and offers AH DAMN IT! It's religiously affiliated....
Kalamazoo College- In Michigan. Only 32% of students from out of state were admitted and all first year students live in on-campus housing. But don't you just love the name. Kalamazoo. Haha...
Northern Arizona University- rural setting, 74% of applicants are admitted, 30% are from out of state. First-time first-year students are allowed to have a car. Located 140 miles from Phoenix, Arizona and there's a 400 acre forest close by as well as an observatory.

Stupid Amercan University... it was doing so well too... Oh well. The others are just as well.

Tips and Hints

I'm reallly starting to get tired of this...


but ANYWAYS...

1. How do I remember transformations? How DO I remember transformations? Well in order to understand transformations you have to know what the parent functions look like. If you don't your pretty much screwed. I mean, you can always plug in points but that takes a while and is WAY too much work. Now I'm a lazy person. I don't like doing work. I look for shortcuts.

For the sake of simplicity I'm gonna use the parent function
f(x)=sin x
(We all remember what that looks like right, because I'm not using pictures)

I see it this way: It all depends on where the number or change is
In f(x)= sin x + 1 the change is outside of the x. This means that the graph moves up or down, in this case it moves up.
In f(x)=sin (x + 1) the change is linked to the x. This means that the graph moves left or right. Here, the graph moves to the left, because when the change is linked to the x (by parentheses) it becomes like Opposite World. You go the other way. If the number is positive, you shift to the left. If it is negative, you shift to the right.
In f(x)=2sin x the change is... for lack of a better explanation... before the x. Therefore the graph's height increases. Here, the graph's highest point will now be 2 instead of 1.
In f(x)=sin 2x the change is directly in the x. This means that the graph either lengthens or shortens. In this case it shortens (once again, we're in Opposite World; if it's a fraction it gets longer, if it's a whole number it gets shorter)
In f(x)=sin -x the change is the sign of the x. The graph then flips vertically. So with this equation, the point (π/2, 1) would now be (π/2, -1) and so on.

There's more I think but I forget what they are at the moment...

2. Trigonometry... Trigonometry is triangles right? For this I can't really give advice. I don't use tricks, I just memorized the equations and rules... most of them anyway.
Besides, the equations are kind of like the tricks in themselves. Like the ones to find the legs of a 30-60-90 triangle.[If you don't remember: hyp is the hypotenuse, SL is the short leg and LL is the long leg]
It was hyp= 2SL. So the hypotenuse is double the size of the short leg of the triangle and LL=√3SL so the long leg is equal to the short leg times the square root of 3.
And then there's that SOH CAH TOA thing. I believe most of us are taught this when we're first introduced to the idea of finding sine, cosine and tangent of a triangle.
There's other stuff I'm sure but I don't remember it all so (and this is one of the only times I'll ever say this) go back to the notes from math analysis. Study your damn head off if you have to. Or hope other people have hints for this stuff (I know I will... Hell I'll do better than hope I'll PRAY that others have hints for remembering this stuff)

3. Still confused about... well I don't remember much from trig so I can't really be confused. Guess I'll have to relearn it all... Not looking forward to that. Then again, it'll be review so it should be easier the second time around, right? Other than that, no worries and no confusion.