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.
Monday, December 13, 2010
Sunday, December 12, 2010
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*
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*
Thursday, December 2, 2010
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
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
Tuesday, November 30, 2010
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
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
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
Monday, November 29, 2010
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?
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?
Sunday, November 28, 2010
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 :)
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 :)
Wednesday, November 17, 2010
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.
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.
Monday, November 15, 2010
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 :)
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 :)
Sunday, November 14, 2010
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
Blog
Power Series is something relatively easy, and I found the homework for both this and thee Maclaurin/Taylor Polynomials to be extremely redundant. However, there's like only two basic rules for Power Series.
1.Do the ratio test.
2 set the limit of the absolut value less than 1
3. solve.
Basically this is just a review on the ratio test. HOWEVER. Say I have after the ratio test
limit as n-inf. of abs(x^2/2!)
Now this is the thing. the limit of the abs value is set to less than one right? well, if I plug in infinity, it'll give me inf over a number. which is just infinity. Therefore it diverges. HOWEVER,once again, if I plug in say 1 for x, I'll be left with 1/2 which is less than 1 but not greater than -1 (coming from the absolute value thing where you put -ve < inside of abs< +ve) value . if I plug in 0, I'll be left we something less than 1. therefore at both x=1 and 0, the polynomial converges (aka no infinity)
For what I do not get, and perhaps a question for Brob is exactly how this will be phrased on the AP. Also, I would like to know what to do with the graph ones?? Thanks oh so much.
1.Do the ratio test.
2 set the limit of the absolut value less than 1
3. solve.
Basically this is just a review on the ratio test. HOWEVER. Say I have after the ratio test
limit as n-inf. of abs(x^2/2!)
Now this is the thing. the limit of the abs value is set to less than one right? well, if I plug in infinity, it'll give me inf over a number. which is just infinity. Therefore it diverges. HOWEVER,once again, if I plug in say 1 for x, I'll be left with 1/2 which is less than 1 but not greater than -1 (coming from the absolute value thing where you put -ve < inside of abs< +ve) value . if I plug in 0, I'll be left we something less than 1. therefore at both x=1 and 0, the polynomial converges (aka no infinity)
For what I do not get, and perhaps a question for Brob is exactly how this will be phrased on the AP. Also, I would like to know what to do with the graph ones?? Thanks oh so much.
Monday, November 8, 2010
Post #11?
Well we worked on the Taylor and Maclaurin Polynomial/Series something things.
Taylor:
Pn(x) = f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2! + ... (f^n(c)(x-c)^n)/n!
*C is used b/c it is not centered a zero.
Maclaurin:
It is simply the same formula, but 0 replaces the c.
*This is one is centered at zero.
This will be a very short blog because....I don't understand this really.
I don't have my notebook..so I can't really think of problem. But I do know you follow the formula like you plug in c for the equation so that's the first term, then do + take the derivative and also put (x-c)
Can someone show me example pleassssse? I just really need an easy step by step for this.
Taylor:
Pn(x) = f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2! + ... (f^n(c)(x-c)^n)/n!
*C is used b/c it is not centered a zero.
Maclaurin:
It is simply the same formula, but 0 replaces the c.
*This is one is centered at zero.
This will be a very short blog because....I don't understand this really.
I don't have my notebook..so I can't really think of problem. But I do know you follow the formula like you plug in c for the equation so that's the first term, then do + take the derivative and also put (x-c)
Can someone show me example pleassssse? I just really need an easy step by step for this.
11/7 post.
so this week in calc bc we finally quit learning the hard stuff in chapter nine, (like sequences and series) and moved on to something that i'm starting to understand better! thank you! haha.
we learned taylor polynomial and some other stuff. so this is the formula thing for it...
Pn(x) = f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2! + ... (f^n(c)(x-c)^n)/n!
also we learned somehting called Macclaurin series. this uses the same formula, except everywhere you see c, you put 0. because Macclaurin's formula is centered at 0.
it was all-in-all a pretty simple week. you just follow that formula. i don't think i had any questions. let me just throw in a little something though to make this blog a little longer.
limit 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.
yayyyyyyy :D
we learned taylor polynomial and some other stuff. so this is the formula thing for it...
Pn(x) = f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2! + ... (f^n(c)(x-c)^n)/n!
also we learned somehting called Macclaurin series. this uses the same formula, except everywhere you see c, you put 0. because Macclaurin's formula is centered at 0.
it was all-in-all a pretty simple week. you just follow that formula. i don't think i had any questions. let me just throw in a little something though to make this blog a little longer.
limit 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.
yayyyyyyy :D
Sunday, November 7, 2010
Post for 11/7
So this week in the wonderful world of Calculus BC: Taylor Polynomial and Approximations.
I would just like to say that I have absolutley no clue at all what these are or how you do them. So that is pretty much my question for the week.
After teaching Mu A practice I realized that I am kind of rusty with my derivatives.
So today's post will be about derivatives.
d/dx [uv] = uv' + vu'
d/dx [u/v] = (vu' - uv') / v^2
d/dx [sinu] = (cosu)(u')
d/dx [cosu] = -(sinu)(u')
d/dx [tanu] = (secu)^2(u')
d/dx [cotu] = -(cscu)^2(u')
d/dx [secu] = (secu)(tanu)(u')
d/dx [cscu] = -(cscu)(cotu)(u')
d/dx [ln(u)] = u' / u
d/dx [u] = (u)(u') / (u)
d/dx [e^u] = e^u * u'
One rule for derivative that the Mu A's weren't really getting was chain rule. I told that the way I remember it was to work from the outside in.
Peace Out,
Ryan
I would just like to say that I have absolutley no clue at all what these are or how you do them. So that is pretty much my question for the week.
After teaching Mu A practice I realized that I am kind of rusty with my derivatives.
So today's post will be about derivatives.
d/dx [uv] = uv' + vu'
d/dx [u/v] = (vu' - uv') / v^2
d/dx [sinu] = (cosu)(u')
d/dx [cosu] = -(sinu)(u')
d/dx [tanu] = (secu)^2(u')
d/dx [cotu] = -(cscu)^2(u')
d/dx [secu] = (secu)(tanu)(u')
d/dx [cscu] = -(cscu)(cotu)(u')
d/dx [ln(u)] = u' / u
d/dx [u] = (u)(u') / (u)
d/dx [e^u] = e^u * u'
One rule for derivative that the Mu A's weren't really getting was chain rule. I told that the way I remember it was to work from the outside in.
Peace Out,
Ryan
Yet another blog.
Let me just say that today I sat down and did all my homework. Rather attempted all my homework. I found myself terribly confused once you get to remainders and such. I know for a fact that I need help on that. However I do know how to find whatever degree polynomial functions.
So Taylor is the series your generating. it's given as:
Pn(x)=f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2!...
C is going to be any number really, and you're just going to plug in the number to the original (they give you it) and then take the derivative and plug into the formula.
Now the difference between Maclaurin is that Maclaurin is centered at 0, meaning that your c is zero. so you're just going to start off by plugging in zero to the original the just taking the derivative and repeatedly plugging in the zero. And after approximating a value to the degree the problem it tells you to, you're get your answer (use this when you have like cos(1.1), or something you know you cant to without a calculator easily)
For stuff I DO NOT GET. I do not get how to find the remainder. Also, what do you do if you're approximating something and you find a pattern? like after the 4th term you get what you started with? Just a couple of questions. :DD
So Taylor is the series your generating. it's given as:
Pn(x)=f(c) + F'(c)(x-c) + (f''(c)(x-c)^2)/2!...
C is going to be any number really, and you're just going to plug in the number to the original (they give you it) and then take the derivative and plug into the formula.
Now the difference between Maclaurin is that Maclaurin is centered at 0, meaning that your c is zero. so you're just going to start off by plugging in zero to the original the just taking the derivative and repeatedly plugging in the zero. And after approximating a value to the degree the problem it tells you to, you're get your answer (use this when you have like cos(1.1), or something you know you cant to without a calculator easily)
For stuff I DO NOT GET. I do not get how to find the remainder. Also, what do you do if you're approximating something and you find a pattern? like after the 4th term you get what you started with? Just a couple of questions. :DD
Monday, November 1, 2010
Post # 10
So, since i hardly ever ask questions..and always in need of knowledge. I'll ask many questions to hopefully be answered.
1. When do you know if a test fails you and you need to do something else.
2. What tests are "inconclusive"?
3. What does inconclusive mean?
4. What came first: the chicken or the egg?
5. How do you do series with trig functions..like cos(pi/x)..?
6. How do you do the word problem things?
For the most part, i understood this chapter..these six questions always confused me though.. and i think on the test today i began to become more worried about that than what i needed to be worried about!
1. When do you know if a test fails you and you need to do something else.
2. What tests are "inconclusive"?
3. What does inconclusive mean?
4. What came first: the chicken or the egg?
5. How do you do series with trig functions..like cos(pi/x)..?
6. How do you do the word problem things?
For the most part, i understood this chapter..these six questions always confused me though.. and i think on the test today i began to become more worried about that than what i needed to be worried about!
Post #10
Sequence:
take the limit to see if it diverges or converges
+if it has a limit it converges
+if you have a type of infinity it diverges
Finding terms: plug into equation
Partial Sums: add the term before it
Series:
Arithmetic sum: n(t+tn)/2
Geometric sum: t1/1-r
nth term test:
take the limit, may have to use L"H rule
+if you get zero must use different test
+if you get a number it diverges
Integral Test:
if you can integrate it easily
+if x is greater than or equal to one it converges
+anything else diverges
p-series test:
1/n^p
+if p is greater than 1 it converges
+if it is less than or equal to 1 it diverges
Geometric:
+if absolute value of r is less than 1 it converges
+greater than or equal to one it diverges
direct comparison test:
compare it to something easier
you then might get a geometric or p-series
use the test that works
+if you get it converges, stop
+if it diverges you must know ?
Limit Comparison Test:
compare to something easier
use another test
+divide original by compared
+take limit, if +ve number it converges
+if infinity, it diverges
alternating series:
will have like (-1)^n+1
+take the alternating part out for new
+take limit, must get zero
*if you get a number it diverges
+now add +1 to each n, less than or equal to, new one
+if true converges
+if not true diverges
Ratio Test:
usually used with ! or variable exponents
+add +1 to each n divided by the original
+things should cancel
* remember 2^n+1 can be written has 2^n 2^1 and (n+1)! is like n!(n+1)
+take limit
+less than 1 it converges, greater than 1 or infinity it diverges
+if you get one it is inconclusive
root test:
used when something is raised to the n
+take the nth root of the absolute value of the original
+take limit
+if you get less than 1 it converges, if you et greater than one or infinity it diverges
+if you get one it is inconclusive
Absolute Convergence: if absolute value of an converges then an converges
Conditional Convergence: if an converges but absolute vale does not converge
take the limit to see if it diverges or converges
+if it has a limit it converges
+if you have a type of infinity it diverges
Finding terms: plug into equation
Partial Sums: add the term before it
Series:
Arithmetic sum: n(t+tn)/2
Geometric sum: t1/1-r
nth term test:
take the limit, may have to use L"H rule
+if you get zero must use different test
+if you get a number it diverges
Integral Test:
if you can integrate it easily
+if x is greater than or equal to one it converges
+anything else diverges
p-series test:
1/n^p
+if p is greater than 1 it converges
+if it is less than or equal to 1 it diverges
Geometric:
+if absolute value of r is less than 1 it converges
+greater than or equal to one it diverges
direct comparison test:
compare it to something easier
you then might get a geometric or p-series
use the test that works
+if you get it converges, stop
+if it diverges you must know ?
Limit Comparison Test:
compare to something easier
use another test
+divide original by compared
+take limit, if +ve number it converges
+if infinity, it diverges
alternating series:
will have like (-1)^n+1
+take the alternating part out for new
+take limit, must get zero
*if you get a number it diverges
+now add +1 to each n, less than or equal to, new one
+if true converges
+if not true diverges
Ratio Test:
usually used with ! or variable exponents
+add +1 to each n divided by the original
+things should cancel
* remember 2^n+1 can be written has 2^n 2^1 and (n+1)! is like n!(n+1)
+take limit
+less than 1 it converges, greater than 1 or infinity it diverges
+if you get one it is inconclusive
root test:
used when something is raised to the n
+take the nth root of the absolute value of the original
+take limit
+if you get less than 1 it converges, if you et greater than one or infinity it diverges
+if you get one it is inconclusive
Absolute Convergence: if absolute value of an converges then an converges
Conditional Convergence: if an converges but absolute vale does not converge
Maleries post
Okay. So I, for one, would like to know WHEN DOES THIS CHAPTER END??? I am BEYOND tired of sequences and series....
AHH.. okay. Math right? well perhaps this will help us in future endeavors.
Okay, so first, to see if a SERIES converges or diverges, you follow that chart.
First things first, nth term. you just take the limit as x goes to infinity. If you get anything other than zero, the series diverges right of the bat. If you get zero, the nth term test is inconclusive.
Next, you can determine whether or not it's a pseries, geometric, or something you can integrate.
PSERIES is when its n raised to some exponent
GEOMETRIC is when it's some fraction raised to the n
Integral is when its something easy..so say ln integration would be easy to do (i.e. 1/n)...or if you feel like messing with by parts, go for it.
Pseries-if p(exponent) is greater than one it converges, if its less than or equal to it diverges
Geometric-if abs(r) (your thing being raised) is less than one, it converges. Not it diverges.
Integral-if you get after integrating infinity anywhere, it diverges.# it converges
If the above do not apply you have a couple of options.
There's the ROOT TEST (where you just force a root...I'm a little shady on this one). RATIO TEST (where you add one to every n and put that over your original and take the limit) ALTERNATING SERIES TEST (which is a little tricky..check your book.) DIRECT COMPARISON TEST (compare it to something bigger and try to use pseries, geo, integral, etc.) and LIMIT COMPARISON TEST where you just compare it to something and put your original over what you're comparing it to and take the limit.)
AHH.. okay. Math right? well perhaps this will help us in future endeavors.
Okay, so first, to see if a SERIES converges or diverges, you follow that chart.
First things first, nth term. you just take the limit as x goes to infinity. If you get anything other than zero, the series diverges right of the bat. If you get zero, the nth term test is inconclusive.
Next, you can determine whether or not it's a pseries, geometric, or something you can integrate.
PSERIES is when its n raised to some exponent
GEOMETRIC is when it's some fraction raised to the n
Integral is when its something easy..so say ln integration would be easy to do (i.e. 1/n)...or if you feel like messing with by parts, go for it.
Pseries-if p(exponent) is greater than one it converges, if its less than or equal to it diverges
Geometric-if abs(r) (your thing being raised) is less than one, it converges. Not it diverges.
Integral-if you get after integrating infinity anywhere, it diverges.# it converges
If the above do not apply you have a couple of options.
There's the ROOT TEST (where you just force a root...I'm a little shady on this one). RATIO TEST (where you add one to every n and put that over your original and take the limit) ALTERNATING SERIES TEST (which is a little tricky..check your book.) DIRECT COMPARISON TEST (compare it to something bigger and try to use pseries, geo, integral, etc.) and LIMIT COMPARISON TEST where you just compare it to something and put your original over what you're comparing it to and take the limit.)
Sunday, October 31, 2010
post 10.
soooo, we got a big test tomorrow in calc. so let's go over what is gonna be on that!
p-series:
used whenever you have n^#.
if # is 1 it converges.
geometric series:
used whenever you have #^n
if abs.value of # <> 1, continue on to testing..
you would divide original by what you compared it to. then take the limit as n goes to infinty of that. & if you get # > 1, it diverges. if not, it is inconclusive.
nth term test:
you use this when you don't exactly know what else to do & you are just testing.
take the lim as x goes to infinity.
if you get anything besides 0, it diverges. if you get 0, it's inconclusive.
integral test:
this is used whenever you have absolutely nothing else to do. you take the integral from n to infinity
if you get infinity, it diverges. if you get a number, it converges.
alternating series test:
used whenever you have -1 or -2 raised to n.
you take out that portion of the problem, then continue on. but if you have -2, you just take out the negative.
after taking it out, you take the limit as n goes to infinity.
if you get anything besides 0, it diverges. if you get 0, continue on & add one to every n.
then compare to the original after you took out the -# ^ n by doing (n+1 equation) ^ if above is true, then converges. if false, then diverges.
sum of geometric series:
first term/1-r
sum of nongeometric series:
n(t1 + tn)/2
ratio test:
use if it tells you to.. pretty much.
add one to each n. then divide <-- that by the original. then take lim as n goes to inifinty.
if you get # <> 1 or infinity, it diverges.
(also you use this if you have a factorial!)
NOW for what I don't understand...
-direct comparison test
-limit comparison test
-absolute convergence
-conditional convergence
p-series:
used whenever you have n^#.
if # is 1 it converges.
geometric series:
used whenever you have #^n
if abs.value of # <> 1, continue on to testing..
you would divide original by what you compared it to. then take the limit as n goes to infinty of that. & if you get # > 1, it diverges. if not, it is inconclusive.
nth term test:
you use this when you don't exactly know what else to do & you are just testing.
take the lim as x goes to infinity.
if you get anything besides 0, it diverges. if you get 0, it's inconclusive.
integral test:
this is used whenever you have absolutely nothing else to do. you take the integral from n to infinity
if you get infinity, it diverges. if you get a number, it converges.
alternating series test:
used whenever you have -1 or -2 raised to n.
you take out that portion of the problem, then continue on. but if you have -2, you just take out the negative.
after taking it out, you take the limit as n goes to infinity.
if you get anything besides 0, it diverges. if you get 0, continue on & add one to every n.
then compare to the original after you took out the -# ^ n by doing (n+1 equation) ^ if above is true, then converges. if false, then diverges.
sum of geometric series:
first term/1-r
sum of nongeometric series:
n(t1 + tn)/2
ratio test:
use if it tells you to.. pretty much.
add one to each n. then divide <-- that by the original. then take lim as n goes to inifinty.
if you get # <> 1 or infinity, it diverges.
(also you use this if you have a factorial!)
NOW for what I don't understand...
-direct comparison test
-limit comparison test
-absolute convergence
-conditional convergence
Monday, October 25, 2010
Post #9
Well does it converge or diverge?
You should probably try the nth term test before trying another method.
P-Series: n^p
if p is greater than 1-->converge
if less than or eqaul to 1-->diverge
Geometric: (1/2)^n
if the absoultue value of r(which is the n) is less than 1-->converge
if the absoultue value of r greater than 1-->diverge
Intergral Test: Probably use when everything fails.
First, take the limit from 0 to infinity.
Second, integrate like normal.
Third, plug in 0 and infinity.
Finally, take the limit.
If you still get any form of infinity-->diverge
if you get a number-->converge.
Comparing: You can compare the problem to an eaiser one.
1. direct comparison
2. limit comparison
My Question: I need someone to explain the difference between direct and limit (the steps and everything). I'm afraid that I'm mixing them up or using a combination of both. Please help.
You should probably try the nth term test before trying another method.
P-Series: n^p
if p is greater than 1-->converge
if less than or eqaul to 1-->diverge
Geometric: (1/2)^n
if the absoultue value of r(which is the n) is less than 1-->converge
if the absoultue value of r greater than 1-->diverge
Intergral Test: Probably use when everything fails.
First, take the limit from 0 to infinity.
Second, integrate like normal.
Third, plug in 0 and infinity.
Finally, take the limit.
If you still get any form of infinity-->diverge
if you get a number-->converge.
Comparing: You can compare the problem to an eaiser one.
1. direct comparison
2. limit comparison
My Question: I need someone to explain the difference between direct and limit (the steps and everything). I'm afraid that I'm mixing them up or using a combination of both. Please help.
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