Review Blog

So this week, like sarah stated, we've gone over AB stuff (AKA stuff that cam back to haunt us). Although I thought it was the hardest thing last year, it is relatively easy this time around. Here are some tips that you may need to remember when practicing AB for the AP exam. :P

RELATED RATES:
your steps are:
1. Identify what you have (dr/dt etc.)
2. Determine which formula you are dealing with (a lot of times its volume or area...)
3. You usually take the derivative somewhere around this step (like of the equation...)
4. Plug in
5. Solve

EXAMPLE Air is being pumped into a spherical balloon at a rate of 5 cm3/min. Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.



Solution

The first thing that we’ll need to do here is to identify what information that we’ve been given and what we want to find. Before we do that let’s notice that both the volume of the balloon and the radius of the balloon will vary with time and so are really functions of time.

We know that air is being pumped into the balloon at a rate of 5 cm3/min. This is the rate at which the volume is increasing. Recall that rates of change are nothing more than derivatives and so we know that, dv/dt=5.

We want to determine the rate at which the radius is changing. Again, rates are derivatives and so it looks like we want to determine,
dr/dt= ?
r= diameter/2=10 cm

Note that we needed to convert the diameter to a radius.



Now that we’ve identified what we have been given and what we want to find we need to relate these two quantities to each other. In this case we can relate the volume and the radius with the formula for the volume of a sphere.

V= 4/3 pi r^3

Now we don’t really want a relationship between the volume and the radius. What we really want is a relationship between their derivatives. We can do this by differentiating both sides with respect to t. In other words, we will need to do implicit differentiation on the above formula. Doing this gives,

dV/dt=4 pi r^2 dr/dt

Now I would just plug in giving me:

dr/dt = 1/80pi cm/min.

12/12 post

okayyy, so this past week we have been going over calculus AB stuff, reviewing for the ap test at the end of the year since it is half ab stuff and half bc stuff.
we are done learning calc bc stuff for the year! :D yayyyyyy. hehe

soooo, i shall go over AP CALC AB information for you fellas.

limit rules as x approaches infinty.
1. if top degreee > bottom degree = +/- infinity
2. if top degree < bottom degree = 0
3. if top degree = bottom degree = divide leading coefficients.

when to use l'hopitals rule.
only if the limit is in indeterminate form, then this can be applied! you take the derivative of the top & bottom of the fraction, (separately) then take the limit again & see if it gives you a number/indeterminate form. repeat as many times as necessary.

rate of change problems.
write down all information given to you. figure out what is being asked for. memorize any formulas you may need, (volume of cube, cylinder, area of things, etc.), <-- use any of those formulas needed, plug in all information given, & solve. pretty simple, just make sure you utilize your time wisely and figure out what is being asked of you to find in the problem!


HELP:
please, someone go over any sequences & series stuff from calc bc. it drives me NUTS.
*truth is, I really just need to memorize all the formulas*

Blog instead of Comments

So, since we really didn't do blogs over the holidays there isn't anything to comment on...so I will just review in a blog.

Taylor: they will give you c
f(c) + f^first deriv(c)(x) + f^second deriv(c)/2! (x-c)^2 + f^third dirv(c)/3! (x-c)^3

Maclauirn: is centered at 0
f(0) + f^first deriv(0)(x) + f^second deriv(0)/2! (x^2) + f^third dirv(0)/3! (x^3)

EXAMPLE:
Maclaurin up to the third degree

f=e^x
f(0)=e^0 = 1
f^1(0)= e^x(1) = 1
f^11(0) = e^x(1)(1) = 1

1+x+1/2! x^2
1+x+1/2(x^2)


Now don't forget about sequences and series! my fav :)
HERE ARE JUST A FEW THINGS TO REMEMBER ABOUT THE RULES:

p-series
1/n^p
p >1 CONVERGES
p <1 or =1 DIVERGES

geometric
(5/4)^n
n <1 CONVERGES
n >1 DIVERGES

limit comparison and direct comparison
*you must compare it to something easier
*use a different test
*then use this test to confirm first one

root and ratio
<1 CONVERGES
>1 or infinity DIVERGES
=1 INCONCLUSIVE

alternating series
*must take out the (-1)^n thing
*take limit MUST =0 or CANNOT be used


and something else we covered---

radius and interval of convergence:
*use the ratio test
*take the limit
*set up -1< x >1
*may have to solve

Post.

So we have a test. So here's a little review.

1. A sequence converges if it's limit is a number. It diverges if there's an infinity anywhere in it. For instance:
Given the sequence represented by the equation (n+1)/(n^2), say whether the sequence converges or diverges...at this point you would take the limit as n approaches infinity. In this case if would approach 0 because your limit rules say that if the degree of the top is less than the degree of the bottom, the limit approaches 0. Got it? So the entire sequence converges to 0 (a number)


2. Now for the difference between a sequence and a series. Indeed, I believe we learned this back in Advanced Math, but BRob stressed to us that Tir had issues with it, so might as well knock it in there a couple of times.

A sequence is just a list of numbers...aka...1, 3, 5, 7,...
**Note for this one it would be all odd numbers

A series is basically the same thing as a sequence, except that you have like addition signs in it...for example...3+4+5+6+7..
**Se those addition signs?? yeah, they're the ones you look out for..

3. Okay, so where I got a little tripped up was when we were saying: "If___, then___" But now, I think I've got it right..

Comment if you agree with this, "If the sequence of the series converges, then the series converges"

"If the sequence of the series diverges, then the series diverges"

My question is what are the conditions for ratio and root tests? I forgot those on the last quiz. :D

Post.

So we have a test. So here's a little review.

1. A sequence converges if it's limit is a number. It diverges if there's an infinity anywhere in it. For instance:
Given the sequence represented by the equation (n+1)/(n^2), say whether the sequence converges or diverges...at this point you would take the limit as n approaches infinity. In this case if would approach 0 because your limit rules say that if the degree of the top is less than the degree of the bottom, the limit approaches 0. Got it? So the entire sequence converges to 0 (a number)


2. Now for the difference between a sequence and a series. Indeed, I believe we learned this back in Advanced Math, but BRob stressed to us that Tir had issues with it, so might as well knock it in there a couple of times.

A sequence is just a list of numbers...aka...1, 3, 5, 7,...
**Note for this one it would be all odd numbers

A series is basically the same thing as a sequence, except that you have like addition signs in it...for example...3+4+5+6+7..
**Se those addition signs?? yeah, they're the ones you look out for..

3. Okay, so where I got a little tripped up was when we were saying: "If___, then___" But now, I think I've got it right..

Comment if you agree with this, "If the sequence of the series converges, then the series converges"

"If the sequence of the series diverges, then the series diverges"

My question is what are the conditions for ratio and root tests? I forgot those on the last quiz. :D

birfday blog :)

okkkk, so we had thanksgiving week off. and we had a takehome test. i shall do some problems from this test.

1. a sub n = (-2/3) ^ n

you plug in values starting at one, then two, three, etc...
& find your answer.

2. determine the convergence/divergence of the sequence with given nth term. if the sequence converges, find its limit.
a sub n = 3^n/5^n
geometric test. rewrite as (3/5) ^ n.
abs. value of r < 1 then it converges. 3/5 = r. after that, take the limit of the original problem. & it converges to that #.

these are just some examples. can someone explain integral test to me?

Combined late and now bloggg..

Alright..so i'll start off by saying these holidays went by entirely too fast..and obviously, you can't post blogs on iphones..because i actually did try in the airport going to new york..and it doesn't work...your keyboard does not pull up when you press in the blog box..odd. i know.

but anyway, lets get this thing started.

So, lets go over some throw back stuff..

1. Substitution: when the derivative is in the equation
ex: Scosxsinx
u = sinx
du = cos x
S u du
=1/2u^2 = 1/2sin^2(x) + c

2. By-Parts: when the derivative is not in the equation but you can sort of manipulate to figure it out...
ex: Sarctan(x)
u = arctan(x) dv = dx
du = 1/ 1+x^2 v = x
= arctan(x) (x) - S x/1+x^2
= arctan(x) (x) - 1/2ln(2)

3. Partial Fractions: when you have a fraction where the bottom can be broken up or factored
ex: umm..i can't think of one bc im not so good at these
But basically you have to break up the bottom and separate it to different fractions with differetn letters "naming" the fractions then you solve different systems to find your letters and then plug in to find the answer.

So, i've been working on my packet and i'm having alot of trouble remembering this stuff...and i'm getting kinda confused on some things..so come prepared on monday-wednesday..cause i'll have PLENTY questions for the test thursday :)

11/17 (late blog)

Sorry I am late, I just hate computers and avoid them at all cost :).

Taylor Polynomials and Approximations:

The form of a convergent power series:

"In this section you will study a general procedure for deriving the power series for a function that has derivatives of all orders. The following theorem gives the form that every convergent power series must take."

If f is represent by a power series f(x) = E an(x-c)^n for all x in an open interval l containing c, then an = f^(n)(c)/n! and
f(x) = f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +... .

Definition of Taylor and Maclaurin Series:

If a function f has derivatives of all orders at x = c, then the series
E(from n=0 to infinity) f^n(c)/n! * (x-c)^n = f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +...
is called the Taylor series for f(x) at c. Moreover, if c = 0, then the series is the Maclaurin series for f.

"If you know the pattern for the coefficients of the Taylor polynomials for a function, you can extend the pattern easily to form the corresponding Taylor series."

The convergence of a Tyalor series will always equal f^n(c)/n! * (x-c)^n if lim(as n -> infinity) Rn = 0.

Guidelines for Finding A Taylor Series:

1.) Differentiate f(x) several times and evaluate each derivative at c.
f(c), f'(c), f''(c), f'''(c), ... , f^n(c), ...

2.) Use the sequence developed in the first step to form the Taylor coefficients an = f^n(c)/n!, and determine the interval of convergence for the resulting power series
f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)^2 +...+ f^n(c)/n! * (x-c)^n +... .

3.) Within the interval of convergence, determine whether the series converges to f(x).



Everyone should look at and put to memory the chart of page 684 about power series for elementary functions.

11/14 post

what we did this week in calc class, we basically went over some more stuff with power series and taylor polynomials and maclaurin serires and what not. same old chapter nine stuff that we been doing for the past 2 weeks or whatever.

ok so POWER SERIES:
what you do for this is
1. do ratio test
2. set lim of abs value less than 1
3. solve

i went over taylor polynomials and all that last week. so let's go over derivative rules, since we been using that lately
sin = cos
cox = -sin
tan = sec^2
sec = sectan
1/x^2 = -2/x^3
xsinx = product rule
x/cosx = quotient rule

that is just a few examples of some derivative formulas for ya.

lim rules as n approaches infinity
1. if top degree > bottom degree = +/- infinity
2. if top degree < bottom degree = 0
3. if top degree = bottom degree = divide leading coefficients

yayyyyyyyyyyyyy :)

Post # 11

Okayy, so I'm going to go over some derivative rules and identities because while I was doing the homework I quickly realized that I do NOT remember how to do derivatives because I'm so in integral mode..

So, let's get it.

**The formula for quotient rule is vu^1-uv^1/v^2 or the derivative of the top times the bottom – the derivative of the bottom times the top over the bottom squared.

An example is sin x / x-1. Take the derivative of the top which is (cos x) times the bottom (x-1) – the derivative of the bottom (1) times the top (sin x) over the bottom squared (x-1) ^2.

From there, it is just simple algebra.
The answer comes out to (cos x) (x-1)-sin x/(x-1) ^2.

*You do not use quotient rule when there is only an x on the bottom.
*You just bring the x to the top and make the exponent negative then use the formula U^n.

**Another thing is the product rule..if everyone remembers it.

(first)(derivative of second) + (second)(derivative of first)

**Third, you need to remember when to use chain rule..

*when you have something inside something or something raised to something with a variable..

Also, remember all the trig functions are THE OTHER way around..

Like

Sin = cos

Cos =-sin

Tan = sec^2

Sec = sectan

And so on…

The thing I need most help with is where to go after root test when doing power series…like I get an answer then take the limit then what? And also, what are the quiz orders and when are we taking the HUGE test on everything? THANKSSSS J