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.

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.

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

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

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

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!

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

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.)

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